940:. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003) and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down.
7343:
4138:
Usually the solution of an SDE requires a probabilistic setting, as the integral implicit in the solution is a stochastic integral. If it were possible to deal with the differential equation path by path, one would not need to define a stochastic integral and one could develop a theory independently
7543:
4490:
that coincides for example with the Ito integral with probability one for a particular choice of the iterated
Brownian integral. Other definitions of the iterated integral lead to deterministic pathwise equivalents of different stochastic integrals, like the Stratonovich integral. This has been used
4125:
through the concept of
Schwartz morphism, see also the related 2-jet interpretation of Ito SDEs on manifolds based on the jet-bundle. This interpretation is helpful when trying to optimally approximate the solution of an SDE given on a large space with the solutions of an SDE given on a submanifold
4120:
up to indistinguishable processes. Although
Stratonovich SDEs are the natural choice for SDEs on manifolds, given that they satisfy the chain rule and that their drift and diffusion coefficients behave as vector fields under changes of coordinates, there are cases where Ito calculus on manifolds is
2118:
Generalizing the geometric
Brownian motion, it is also possible to define SDEs admitting strong solutions and whose distribution is a convex combination of densities coming from different geometric Brownian motions or Black Scholes models, obtaining a single SDE whose solutions is distributed as a
1360:
For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition. Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when
974:
In physics, SDEs have wide applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be
869:
is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time. The
Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with
8116:
1134:
4289:
905:, this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure, leading to a N=2 supersymmetric model closely related to
7072:
3740:
1822:
5349:
8900:
Kuznetsov, D.F. (2023). Strong approximation of iterated Itô and
Stratonovich stochastic integrals: Method of generalized multiple Fourier series. Application to numerical integration of Itô SDEs and semilinear SPDEs. Differ. Uravn. Protsesy Upr., no. 1. DOI:
1321:
are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case in which
7684:
5831:
4499:
As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in
4978:
4115:
1618:
877:
An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the
Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the
4823:
4378:). However, a direct path-wise interpretation of the SDE is not possible, as the Brownian motion paths have unbounded variation and are nowhere differentiable with probability one, so that there is no naive way to give meaning to terms like
7066:
3515:
8501:
which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as
1365:. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a
6268:
2102:
2339:
1947:
The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution Both require the existence of a process
2493:
7354:
8242:
2199:
809:, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician
4679:
1999:). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. The
8312:
5922:
1366:
6776:
8711:
Kunita, H. (2004). Stochastic
Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser Boston.
5643:
5100:
4601:
6408:
7338:{\displaystyle X_{t}=\Phi _{t,t_{0}}\left(X_{t_{0}}+\int _{t_{0}}^{t}\Phi _{s,t_{0}}^{-1}(c(s)-b(s)\mathrm {d} (s))\mathrm {d} s+\int _{t_{0}}^{t}\Phi _{s,t_{0}}^{-1}\mathrm {d} (s)\mathrm {d} W_{s}\right)}
4145:
7953:
2552:
7940:
6602:
3651:
983:
There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:
979:
to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.
8477:
7777:
2897:
990:
1656:
5199:
2603:
5557:
5210:
7864:
4376:
3615:
1997:
841:. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral. Such a mathematical definition was first proposed by
4458:
4417:
2740:
6665:
5866:
3052:
910:
8825:
Armstrong, J., Brigo, D. and Rossi
Ferrucci, E. (2019), Optimal approximation of SDEs on submanifolds: the Itô-vector and Itô-jet projections. Proc. London Math. Soc., 119: 176-213.
3838:
3311:
3178:
874:, although it is possible and in some cases preferable to model random motion on manifolds through Itô SDEs, for example when trying to optimally approximate SDEs on submanifolds.
6814:
6504:
3392:
3244:
8913:
Rybakov, K.A. (2023). Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics. Mathematics, vol. 11, 4047. DOI:
6850:
5753:
4488:
4318:
2665:
1252:
5980:
5951:
5747:
5698:
4829:
3930:
3002:
766:
7556:
9103:
Armstrong, J., Bellani, C., Brigo, D. and Cass, T. (2021). Option pricing models without probability: a rough paths approach. Mathematical
Finance, vol. 31, pages 1494–1521.
8364:
6699:
6360:
3987:
3979:
1492:
1465:
1279:
1355:
909:. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e.,
6876:
6438:
3141:
6083:
1507:
891:
4702:
6924:
6301:
5386:
3400:
6328:
6057:
6031:
6011:
4338:
1216:
6899:
6628:
6118:
3264:
3078:
2951:
1919:
and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function
1163:
3360:
2775:
2634:
9382:
3791:
3340:
6126:
2015:
8384:
8139:
7707:
6458:
5718:
5669:
3950:
3901:
3880:
3858:
3762:
3639:
3538:
3198:
3101:
2921:
2818:
2798:
2687:
2406:
2386:
2255:
1641:
1319:
1299:
1187:
7538:{\displaystyle \Phi _{t,t_{0}}=\exp \left(\int _{t_{0}}^{t}\left(a(s)-{\frac {b^{2}(s)}{2}}\right)\mathrm {d} s+\int _{t_{0}}^{t}b(s)\mathrm {d} W_{s}\right)}
2411:
735:
4464:
theory, while adding a chosen definition of iterated integrals of Brownian motion, it is possible to define a deterministic rough integral for every single
8888:
Artemiev, S.S., Averina, T.A. (1997). Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, Utrecht, The Netherlands. DOI:
2352:. Under this hypothesis, the methodologies developed by Marcello Minenna determines prediction interval able to identify abnormal return that could hide
2232:, is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values of
8147:
9917:
2135:
932:
is almost surely nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the
797:
was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known as
9741:
9091:
Friz, P. and Hairer, M. (2020). A Course on Rough Paths with an Introduction to Regularity Structures, 2nd ed., Springer-Verlag, Heidelberg, DOI
9004:
Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183,
2228:, but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process,
697:
4607:
438:
10344:
9337:
8253:
9874:
9854:
5871:
8488:
6713:
10258:
5573:
5005:
2239:
A generalization of stochastic differential equations with the Fisk-Stratonovich integral to semimartingales with jumps are the SDEs of
10608:
4535:
181:
6365:
4284:{\displaystyle \mathrm {d} X_{t}(\omega )=\mu (X_{t}(\omega ),t)\,\mathrm {d} t+\sigma (X_{t}(\omega ),t)\,\mathrm {d} B_{t}(\omega )}
726:, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to
10175:
8564:
8539:
8111:{\displaystyle \mathrm {d} X_{t}=\left(\alpha f(X_{t})+{\frac {1}{2}}f(X_{t})f'(X_{t})\right)\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}}
3735:{\displaystyle \{\zeta <\infty \}\subset \left\{\lim \limits _{t\nearrow \zeta }X_{t}=\infty {\text{ in }}{\widehat {M}}\right\}}
9859:
9050:
1429:
312:
2498:
10185:
9869:
7875:
6516:
817:
and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist
9266:
Higham., Desmond J. (January 2001). "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations".
2127:
10227:
10124:
890:. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of
353:
10414:
10404:
10250:
9942:
9927:
9309:
9199:
9171:
9122:
9073:
8876:
Kloeden, P.E., Platen E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin, Heidelberg. DOI:
243:
9304:
Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM,
2119:
mixture dynamics of lognormal distributions of different Black Scholes models. This leads to models that can deal with the
1129:{\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=F(x(t))+\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t),\,}
10314:
10278:
8392:
7715:
1817:{\displaystyle X_{t+s}-X_{t}=\int _{t}^{t+s}\mu (X_{u},u)\mathrm {d} u+\int _{t}^{t+s}\sigma (X_{u},u)\,\mathrm {d} B_{u}.}
690:
261:
145:
2825:
10582:
10319:
564:
218:
10231:
9429:
9330:
8661:
Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.
8928:"Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters"
5344:{\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} B_{t}{\mbox{ for }}t\in ;}
5136:
2561:
10384:
9239:
8696:
8633:
1854:(but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length
239:
191:
5467:
4126:
of that space, in that a Stratonovich based projection does not result to be optimal. This has been applied to the
1440:
for solving stochastic differential equations. This notation makes the exotic nature of the random function of time
10235:
10219:
10134:
9962:
9932:
9354:
1647:(standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding
937:
906:
226:
201:
7788:
4343:
3543:
1964:
10613:
10334:
10299:
10268:
10263:
9699:
9616:
9021:
Steven Marcus (1981), "Modeling and approximation of stochastic differential equation driven by semimartigales",
8493:
In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the
1409:
683:
8497:
on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological
4422:
4381:
2696:
10273:
9902:
9897:
9704:
9601:
8609:
8574:
6637:
5836:
3007:
2247:
1362:
902:
838:
830:
786:
618:
171:
8792:, Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology 6 (2000), no. 1, 43-84.
4419:, precluding also a naive path-wise definition of the stochastic integral as an integral against every single
769:
are conjugate to stochastic differential equations. Stochastic differential equations can also be extended to
343:
10623:
10618:
10587:
10364:
10200:
10099:
10084:
9623:
9496:
9412:
9323:
5429:
3796:
3269:
1381:
358:
186:
176:
10359:
10239:
4460:. However, motivated by the Wong-Zakai result for limits of solutions of SDEs with regular noise and using
3146:
1957:
that solves the integral equation version of the SDE. The difference between the two lies in the underlying
10369:
8605:
6787:
6463:
5826:{\displaystyle \alpha :\mathbb {R} _{+}\times U\to \operatorname {Lin} (\mathbb {R} ^{n};\mathbb {R} ^{d})}
3365:
3203:
957:
887:
865:
are related, but different, objects and the choice between them depends on the application considered. The
633:
484:
387:
274:
196:
10374:
10010:
6362:
be some initial condition, meaning it is a measurable function with respect to the initial σ-algebra. Let
1498:. The mathematical formulation treats this complication with less ambiguity than the physics formulation.
1392:
gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by
1369:
that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.
9972:
9556:
9501:
9417:
6819:
4973:{\displaystyle {\big |}\mu (x,t)-\mu (y,t){\big |}+{\big |}\sigma (x,t)-\sigma (y,t){\big |}\leq D|x-y|;}
2000:
319:
234:
10304:
7679:{\displaystyle \mathrm {d} X_{t}={\frac {1}{2}}f(X_{t})f'(X_{t})\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}}
4467:
4297:
2639:
10309:
10294:
9937:
9907:
9474:
9372:
8617:
8511:
4110:{\displaystyle f(X_{t})=f(X_{0})+\int _{0}^{t}(\mathrm {d} f)_{X}A(X)\circ \mathrm {d} Z,\quad t\geq 0}
2606:
1377:
1221:
883:
524:
392:
8727:"The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors"
5956:
5927:
5723:
5674:
3906:
2956:
960:, Rosenbrock method, and methods based on different representations of iterated stochastic integrals.
10389:
10190:
10104:
10089:
10020:
9596:
9479:
9377:
6411:
2007:
1397:
949:
495:
473:
9288:
8320:
6670:
6333:
3955:
1470:
1443:
1257:
638:
10223:
10109:
9611:
9586:
9531:
8958:
Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI
1613:{\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} B_{t},}
1373:
1325:
976:
489:
397:
8839:
Parisi, G.; Sourlas, N. (1979). "Random Magnetic Fields, Supersymmetry, and Negative Dimensions".
4818:{\displaystyle {\big |}\mu (x,t){\big |}+{\big |}\sigma (x,t){\big |}\leq C{\big (}1+|x|{\big )};}
10524:
10514:
10329:
10205:
9987:
9912:
9726:
9591:
9447:
9402:
9036:
7061:{\displaystyle \mathrm {d} X_{t}=(a(t)X_{t}+c(t))\mathrm {d} t+(b(t)X_{t}+d(t))\mathrm {d} W_{t}}
6855:
6417:
3510:{\displaystyle f(X_{\tau })=f(x_{0})+\int _{0}^{\tau }(\mathrm {d} f)_{X}A(X)\circ \mathrm {d} Z}
3106:
2349:
2112:
569:
559:
551:
507:
348:
9009:
6062:
1405:
1385:
10466:
10394:
9819:
9809:
9653:
9283:
8808:(2018). Intrinsic stochastic differential equations as jets. Proc. R. Soc. A., 474: 20170559,
850:
818:
382:
6276:
5355:
790:
10489:
10471:
10451:
10446:
10165:
9997:
9977:
9824:
9767:
9606:
9516:
8559:
6313:
6303:
is the Euclidean norm. This condition guarantees the existence and uniqueness of a so-called
6036:
6016:
5996:
4323:
2365:
1192:
879:
862:
854:
770:
719:
628:
613:
502:
445:
427:
266:
22:
6884:
6613:
6088:
3249:
3057:
2930:
1428:(and in many applications of probability theory, for instance in signal processing with the
1142:
10564:
10519:
10509:
10195:
10170:
10139:
10119:
9957:
9879:
9864:
9731:
9275:
8848:
6263:{\displaystyle |\alpha (s,y)-\alpha (s,x)|\leq L(t,K)|y-x|,\quad x,y\in K,\;0\leq s\leq t,}
5127:
3345:
2748:
2612:
2246:
An innovative application in stochastic finance derives from the usage of the equation for
2097:{\displaystyle \mathrm {d} X_{t}=\mu X_{t}\,\mathrm {d} t+\sigma X_{t}\,\mathrm {d} B_{t}.}
1495:
1433:
1413:
1384:. It tells how the probability distribution function evolves in time similarly to how the
837:
variable. In most cases, SDEs are understood as continuous time limit of the corresponding
739:
514:
450:
423:
3767:
3316:
2334:{\displaystyle \mathrm {d} R_{t}=\mu R_{t}\,\mathrm {d} t+\sigma _{t}\,\mathrm {d} B_{t}.}
1254:
is a set of vector fields that define the coupling of the system to Gaussian white noise,
801:. Some of these early examples were linear stochastic differential equations, also called
8:
10559:
10399:
10324:
10129:
9889:
9799:
9689:
8776:
Michel Emery (1989). Stochastic calculus in manifolds. Springer Berlin, Heidelberg. Doi
5567:
The stochastic differential equation above is only a special case of a more general form
4685:
1868:
814:
546:
531:
432:
296:
135:
102:
93:
9279:
8852:
2204:
was used by Louis Bachelier as the first model for stock prices in 1900, known today as
10529:
10494:
10409:
10379:
10210:
10149:
10144:
9967:
9804:
9469:
9407:
9346:
9160:
8754:
8554:
8369:
8124:
7692:
6443:
5703:
5654:
3935:
3886:
3865:
3843:
3747:
3624:
3523:
3183:
3086:
2906:
2803:
2783:
2672:
2488:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in \mathbb {R} _{+}},P)}
2391:
2371:
1831:
1626:
1425:
1393:
1389:
1361:
noise is multiplicative and when the SDE is understood as a continuous time limit of a
1304:
1284:
1172:
727:
723:
541:
536:
419:
10549:
9762:
9679:
9648:
9541:
9521:
9511:
9367:
9362:
9305:
9254:
9235:
9195:
9185:. Mathematics and its Applications (46). Dordrecht: Kluwer Academic Publishers Group.
9167:
9118:
9069:
9005:
8789:
8746:
8692:
8639:
8629:
8569:
8544:
8530:
noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.
8515:
8494:
4127:
3618:
2211:
There are also more general stochastic differential equations where the coefficients
1958:
1937:
1843:
1648:
1437:
1401:
969:
898:
802:
598:
377:
112:
10354:
10005:
8684:
653:
10569:
10456:
10339:
10215:
9952:
9709:
9684:
9633:
9484:
9437:
9293:
9218:
8987:
8939:
8856:
8758:
8738:
8621:
8549:
8519:
4122:
2555:
2120:
663:
648:
9561:
9115:
Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale
9066:
Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale
8978:(2002). "Lognormal-mixture dynamics and calibration to market volatility smiles".
8237:{\displaystyle \mathrm {d} X_{t}=\alpha f(X_{t})\mathrm {d} t+f(X_{t})\circ W_{t}}
2368:
and for this purpose one uses the Fisk-Stratonovich integral. Consider a manifold
10534:
10434:
10419:
10180:
10114:
9792:
9736:
9719:
9464:
9249:
Seifedine Kadry (2007). "A Solution of Linear Stochastic Differential Equation".
8902:
8527:
5425:
4508:
2205:
1941:
1828:
953:
921:
798:
794:
782:
750:
603:
519:
46:
10349:
9581:
9153:. Mathematics in Science and Engineering (169). Orlando, FL: Academic Press Inc.
5562:
2194:{\displaystyle \mathrm {d} X_{t}=\mu \,\mathrm {d} t+\sigma \,\mathrm {d} B_{t}}
1847:
933:
866:
858:
846:
762:
658:
10539:
10504:
10424:
10030:
9777:
9694:
9663:
9658:
9638:
9628:
9571:
9546:
9526:
9491:
9459:
9442:
9223:
Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences
8975:
8944:
8927:
8860:
1872:
1644:
929:
925:
754:
623:
608:
414:
402:
121:
9566:
9297:
8991:
8742:
8713:
10602:
10441:
9982:
9814:
9772:
9714:
9536:
9452:
9392:
9258:
9051:"The detection of Market Abuse on financial markets: a quantitative approach"
8971:
8805:
8750:
8726:
8625:
8498:
5392:
2900:
806:
9092:
8877:
8777:
8643:
4674:{\displaystyle \sigma :\mathbb {R} ^{n}\times \to \mathbb {R} ^{n\times m};}
2236:, the defining equation is called a stochastic delay differential equation.
948:
Numerical methods for solving stochastic differential equations include the
842:
810:
10499:
10461:
10015:
9947:
9836:
9831:
9643:
9576:
9551:
9387:
8503:
4491:
for example in financial mathematics to price options without probability.
2353:
758:
643:
593:
479:
107:
10544:
10079:
10063:
10058:
10053:
10043:
9846:
9787:
9782:
9746:
9506:
9397:
8809:
5986:
More generally one can also look at stochastic differential equations on
4461:
4121:
preferable. A theory of Ito calculus on manifolds was first developed by
1467:
in the physics formulation more explicit. In strict mathematical terms,
1436:) is slightly different. It is also the notation used in publications on
1166:
834:
746:
731:
51:
8959:
8889:
8307:{\displaystyle \mathrm {d} Y_{t}=\alpha \mathrm {d} t+\mathrm {d} W_{t}}
6330:
is continuous and satisfies the above local Lipschitz condition and let
5993:
Whether the solution of this equation explodes depends on the choice of
2243:. The Marcus integral is an extension of McShane's stochastic calculus.
1218:
is a flow vector field representing deterministic law of evolution, and
10554:
10094:
10038:
9922:
9875:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
9315:
8914:
8523:
8507:
5917:{\displaystyle \operatorname {Lin} (\mathbb {R} ^{n};\mathbb {R} ^{d})}
2344:
which is the equation for the dynamics of the return of the price of a
668:
745:
SDEs have a random differential that is in the most basic case random
10048:
8826:
4505:
2364:
More generally one can extend the theory of stochastic calculus onto
1851:
1404:(for example, the Fokker-Planck equation can be transformed into the
871:
409:
130:
73:
63:
8689:
Stochastic Differential Equations: An Introduction with Applications
6771:{\displaystyle \mathrm {d} Y=\alpha (t,Y)\mathrm {d} X^{\zeta _{n}}}
897:
In physical science, there is an ambiguity in the usage of the term
9142:
5987:
5638:{\displaystyle \mathrm {d} Y_{t}=\alpha (t,Y_{t})\mathrm {d} X_{t}}
1896:
5095:{\displaystyle |\sigma |^{2}=\sum _{i,j=1}^{n}|\sigma _{ij}|^{2}.}
9192:
An Informal Introduction to Stochastic Calculus with Applications
8614:
Diffusions, Markov Processes and Martingales, Vol 2: Ito Calculus
4596:{\displaystyle \mu :\mathbb {R} ^{n}\times \to \mathbb {R} ^{n};}
757:. However, other types of random behaviour are possible, such as
84:
79:
68:
9117:(in German). Vieweg+Teubner Verlag Wiesbaden. pp. 297–299.
6403:{\displaystyle \zeta :\Omega \to {\overline {\mathbb {R} }}_{+}}
5204:
Then the stochastic differential equation/initial value problem
9183:
Nonlinear stochastic systems theory and applications to physics
9068:(in German). Vieweg+Teubner Verlag Wiesbaden. p. 364-365.
928:
was discovered to be exceptionally complex mathematically. The
849:. Another construction was later proposed by Russian physicist
1419:
781:
Stochastic differential equations originated in the theory of
8518:
explains the associated long-range dynamical behavior, i.e.,
5563:
General case: local Lipschitz condition and maximal solutions
2345:
2108:
1388:
gives the time evolution of the quantum wave function or the
9253:. USA: WSEAS TRANSACTIONS on MATHEMATICS, April 2007.: 618.
2547:{\displaystyle ({\mathcal {F}}_{t})_{t\in \mathbb {R} _{+}}}
1927:
is called the diffusion coefficient. The stochastic process
833:
with the right hand side perturbed by a term dependent on a
7935:{\displaystyle h(x)=\int ^{x}{\frac {\mathrm {d} s}{f(s)}}}
6597:{\displaystyle dY_{t}=\alpha (t,Y_{t})dX_{t},\quad Y_{0}=F}
4139:
of probability theory. This points to considering the SDE
3342:
is a real-valued semimartingale and for each stopping time
9855:
Autoregressive conditional heteroskedasticity (ARCH) model
9053:. Consob – The Italian Securities and Exchange Commission.
4294:
as a single deterministic differential equation for every
1400:
that draws on the analogy between statistical physics and
9097:
2107:
which is the equation for the dynamics of the price of a
9383:
Independent and identically distributed random variables
8980:
International Journal of Theoretical and Applied Finance
4494:
1494:
cannot be chosen as an ordinary function, but only as a
9234:. Singapore: World Scientific Publishing. p. 212.
9194:. Singapore: World Scientific Publishing. p. 315.
1372:
In physics, the main method of solution is to find the
821:, leading to a calculus similar to ordinary calculus.
9860:
Autoregressive integrated moving average (ARIMA) model
5311:
8821:
8819:
8817:
8472:{\displaystyle X_{t}=h^{-1}(\alpha t+W_{t}+h(X_{0}))}
8395:
8372:
8323:
8256:
8150:
8127:
7956:
7878:
7791:
7772:{\displaystyle \mathrm {d} X_{t}=f(X_{t})\circ W_{t}}
7718:
7695:
7559:
7357:
7075:
6927:
6887:
6858:
6822:
6790:
6716:
6673:
6640:
6616:
6519:
6466:
6446:
6420:
6368:
6336:
6316:
6279:
6129:
6091:
6065:
6039:
6019:
5999:
5959:
5930:
5874:
5839:
5756:
5726:
5706:
5677:
5657:
5576:
5470:
5358:
5213:
5139:
5008:
4832:
4705:
4610:
4538:
4470:
4425:
4384:
4346:
4326:
4300:
4148:
3990:
3958:
3938:
3909:
3889:
3883:. Given a maximal solution we can extend the time of
3868:
3846:
3799:
3770:
3750:
3654:
3627:
3546:
3526:
3403:
3368:
3348:
3319:
3272:
3252:
3206:
3186:
3149:
3109:
3089:
3060:
3010:
2959:
2933:
2909:
2828:
2806:
2786:
2751:
2699:
2675:
2642:
2615:
2564:
2501:
2414:
2394:
2374:
2258:
2138:
2018:
1967:
1827:
The equation above characterizes the behavior of the
1659:
1629:
1510:
1473:
1446:
1380:(FPE). The Fokker–Planck equation is a deterministic
1328:
1307:
1287:
1260:
1224:
1195:
1175:
1145:
993:
829:
The most common form of SDEs in the literature is an
16:
Differential equations involving stochastic processes
9232:
Elementary Stochastic Calculus: with Finance in View
9143:
An Introduction to Stochastic Differential Equations
9112:
9063:
8616:(2nd ed., Cambridge Mathematical Library ed.).
6908:
6033:
satisfies some local Lipschitz condition, i.e., for
4340:
is the sample space in the given probability space (
2892:{\displaystyle A:M\times E\to TM,(x,e)\mapsto A(x)e}
2219:
depend not only on the present value of the process
1376:
function as a function of time using the equivalent
845:
in the 1940s, leading to what is known today as the
4522:; the proof may be found in Øksendal (2003, §5.2).
9159:
8814:
8724:
8657:
8655:
8653:
8471:
8378:
8358:
8306:
8236:
8133:
8110:
7934:
7858:
7771:
7701:
7678:
7537:
7337:
7060:
6893:
6870:
6844:
6808:
6770:
6693:
6659:
6622:
6596:
6498:
6452:
6432:
6402:
6354:
6322:
6295:
6262:
6112:
6077:
6051:
6025:
6005:
5974:
5945:
5916:
5860:
5825:
5741:
5712:
5692:
5663:
5637:
5551:
5380:
5343:
5193:
5094:
4972:
4817:
4673:
4595:
4482:
4452:
4411:
4370:
4332:
4312:
4283:
4109:
3973:
3944:
3924:
3895:
3874:
3852:
3832:
3785:
3756:
3734:
3633:
3609:
3532:
3509:
3386:
3354:
3334:
3305:
3258:
3238:
3192:
3172:
3135:
3095:
3072:
3046:
2996:
2945:
2915:
2891:
2812:
2792:
2769:
2734:
2681:
2659:
2628:
2597:
2546:
2487:
2400:
2380:
2333:
2193:
2096:
1991:
1816:
1635:
1612:
1486:
1459:
1349:
1313:
1293:
1273:
1246:
1210:
1181:
1157:
1128:
8932:Journal of Mathematical Analysis and Applications
1408:by rescaling a few variables) or by writing down
10600:
9742:Stochastic chains with memory of variable length
9113:Hackenbroch, Wolfgang; Thalmaier, Anton (1994).
9064:Hackenbroch, Wolfgang; Thalmaier, Anton (1994).
8600:
8598:
8596:
8594:
8592:
8590:
5194:{\displaystyle \mathbb {E} {\big }<+\infty .}
5109:be a random variable that is independent of the
2598:{\displaystyle {\widehat {M}}=M\cup \{\infty \}}
2115:options pricing model of financial mathematics.
730:various behaviours of stochastic models such as
9248:
8650:
1923:is referred to as the drift coefficient, while
1189:, assumed to be a differentiable manifold, the
886:, an equation describing the time evolution of
9229:
9217:
8903:https://doi.org/10.21638/11701/spbu35.2023.110
8731:Journal of Dynamics and Differential Equations
8604:
8386:is defined as before. Its general solution is
5552:{\displaystyle \mathbb {E} \left<+\infty .}
3764:-almost surely. It follows from the fact that
9331:
9087:
9085:
9020:
8970:
8838:
8587:
5174:
5147:
4937:
4891:
4881:
4835:
4807:
4781:
4768:
4743:
4733:
4708:
691:
8489:Supersymmetric theory of stochastic dynamics
7859:{\displaystyle X_{t}=h^{-1}(W_{t}+h(X_{0}))}
6803:
6791:
4371:{\displaystyle \Omega ,\,{\mathcal {F}},\,P}
3667:
3655:
3610:{\displaystyle (df)_{X}:T_{x}M\to T_{f(x)}M}
3167:
3150:
2592:
2586:
2348:under the hypothesis that returns display a
1992:{\displaystyle \Omega ,\,{\mathcal {F}},\,P}
911:(overdamped) Langevin SDEs are never chaotic
870:geometric problems such as random motion on
9208:
8725:Imkeller, Peter; Schmalfuss, Björn (2001).
8714:https://doi.org/10.1007/978-1-4612-2054-1_6
6916:
1420:Use in probability and mathematical finance
1416:of the probability distribution function.
9870:Autoregressive–moving-average (ARMA) model
9338:
9324:
9082:
8872:
8870:
8800:
8798:
8679:
8677:
8675:
8673:
8671:
8669:
8667:
6241:
4453:{\displaystyle \mathrm {d} B_{t}(\omega )}
4412:{\displaystyle \mathrm {d} B_{t}(\omega )}
2735:{\displaystyle \mathrm {d} X=A(X)\circ dZ}
738:or physical systems that are subjected to
698:
684:
9287:
9093:https://doi.org/10.1007/978-3-030-41556-3
8943:
8894:
8878:https://doi.org/10.1007/978-3-662-12616-5
8778:https://doi.org/10.1007/978-3-642-75051-9
8565:Stochastic partial differential equations
8540:Backward stochastic differential equation
8482:
7945:
7548:
6660:{\displaystyle \zeta _{n}\nearrow \zeta }
6385:
5962:
5933:
5901:
5886:
5861:{\displaystyle U\subset \mathbb {R} ^{d}}
5848:
5810:
5795:
5765:
5729:
5680:
5523:
5472:
5294:
5257:
5141:
4652:
4619:
4580:
4547:
4364:
4353:
4256:
4210:
4130:, leading to optimal projection filters.
3912:
3047:{\displaystyle A(\cdot )e\in \Gamma (TM)}
2532:
2464:
2312:
2290:
2175:
2160:
2075:
2050:
2006:An important example is the equation for
1985:
1974:
1795:
1591:
1554:
1396:simulation. Other techniques include the
1125:
9345:
8683:
6703:stopped stochastic differential equation
9265:
9180:
9162:Nonlinear stochastic operator equations
9157:
9148:
8907:
8882:
8867:
8795:
8772:
8770:
8768:
8664:
3833:{\displaystyle f\in C_{c}^{\infty }(M)}
3306:{\displaystyle f\in C_{c}^{\infty }(M)}
2388:, some finite-dimensional vector space
1867:changes its value by an amount that is
722:in which one or more of the terms is a
10601:
10176:Doob's martingale convergence theorems
9213:. Chichester: Wiley. pp. 523–527.
8925:
8141:is equivalent to the Stratonovich SDE
7709:is equivalent to the Stratonovich SDE
3173:{\displaystyle \{{\mathcal {F}}_{t}\}}
943:
916:
9928:Constant elasticity of variance (CEV)
9918:Chan–Karolyi–Longstaff–Sanders (CKLS)
9319:
9189:
8960:https://doi.org/10.1007/3-540-30591-2
8890:https://doi.org/10.1515/9783110944662
8810:http://doi.org/10.1098/rspa.2017.0559
6809:{\displaystyle \{\zeta <\infty \}}
6499:{\displaystyle (Y_{t})_{t<\zeta }}
5924:is the space of all linear maps from
5833:is a map from some open nonempty set
4495:Existence and uniqueness of solutions
3387:{\displaystyle 0\leq \tau <\zeta }
3239:{\displaystyle (X_{t})_{t<\zeta }}
1165:is the position in the system in its
9209:Teugels, J.; Sund, B., eds. (2004).
8915:https://doi.org/10.3390/math11194047
8765:
8121:for a given differentiable function
7689:for a given differentiable function
2669:stochastic differential equation on
2359:
2003:makes a connection between the two.
1367:uniquely defined mathematical object
146:List of named differential equations
9166:. Orlando, FL: Academic Press Inc.
6845:{\displaystyle Y_{t}\to \partial U}
6634:for one (and hence all) announcing
3645:if the life time is maximal, i.e.,
853:, leading to what is known as the
219:Dependent and independent variables
13:
10415:Skorokhod's representation theorem
10196:Law of large numbers (weak/strong)
9135:
8827:https://doi.org/10.1112/plms.12226
8290:
8279:
8258:
8192:
8152:
8094:
8064:
7958:
7908:
7720:
7662:
7632:
7561:
7516:
7471:
7359:
7316:
7302:
7271:
7238:
7221:
7160:
7090:
7044:
6990:
6929:
6913:Explicitly solvable SDEs include:
6836:
6800:
6747:
6718:
6375:
6343:
5671:is a continuous semimartingale in
5621:
5578:
5543:
5525:
5296:
5259:
5215:
5185:
4483:{\displaystyle \omega \in \Omega }
4477:
4427:
4386:
4356:
4347:
4327:
4313:{\displaystyle \omega \in \Omega }
4307:
4258:
4212:
4150:
4087:
4054:
3816:
3707:
3664:
3500:
3467:
3289:
3156:
3029:
2701:
2660:{\displaystyle {\mathcal {F}}_{0}}
2646:
2589:
2508:
2440:
2426:
2418:
2314:
2292:
2260:
2177:
2162:
2140:
2077:
2052:
2020:
1977:
1968:
1797:
1739:
1593:
1556:
1512:
1501:A typical equation is of the form
1017:
998:
975:viewed as a generalization of the
901:. While Langevin SDEs can be of a
888:probability distribution functions
749:calculated as the derivative of a
14:
10635:
10609:Stochastic differential equations
10385:Martingale representation theorem
9251:Wseas Transactions on Mathematics
9211:Encyclopedia of Actuarial Science
9039:. Risk Magazine. 2 November 2004.
6909:Some explicitly solvable examples
5720:is a continuous semimartingal in
4133:
1247:{\displaystyle g_{\alpha }\in TX}
963:
10430:Stochastic differential equation
10320:Doob's optional stopping theorem
10315:Doob–Meyer decomposition theorem
5975:{\displaystyle \mathbb {R} ^{d}}
5946:{\displaystyle \mathbb {R} ^{n}}
5742:{\displaystyle \mathbb {R} ^{d}}
5693:{\displaystyle \mathbb {R} ^{n}}
5126: ≥ 0, and with finite
4688:for which there exist constants
3925:{\displaystyle \mathbb {R} _{+}}
2997:{\displaystyle A(x):E\to T_{x}M}
938:Stratonovich stochastic calculus
907:supersymmetric quantum mechanics
813:, who introduced the concept of
712:stochastic differential equation
354:Carathéodory's existence theorem
10300:Convergence of random variables
10186:Fisher–Tippett–Gnedenko theorem
9106:
9057:
9043:
9029:
9025:, vol. 4, pp. 223–245
9014:
8998:
8964:
8952:
8919:
8832:
6577:
6222:
4097:
2408:, a filtered probability space
1410:ordinary differential equations
839:stochastic difference equations
765:or semimartingales with jumps.
9898:Binomial options pricing model
9145:American Mathematical Society.
8782:
8718:
8705:
8575:Stochastic difference equation
8466:
8463:
8450:
8422:
8359:{\displaystyle Y_{t}=h(X_{t})}
8353:
8340:
8218:
8205:
8188:
8175:
8090:
8077:
8055:
8042:
8031:
8018:
7999:
7986:
7926:
7920:
7888:
7882:
7853:
7850:
7837:
7818:
7753:
7740:
7658:
7645:
7628:
7615:
7604:
7591:
7512:
7506:
7456:
7450:
7431:
7425:
7312:
7306:
7234:
7231:
7225:
7217:
7211:
7202:
7196:
7190:
7040:
7037:
7031:
7012:
7006:
7000:
6986:
6983:
6977:
6958:
6952:
6946:
6862:
6833:
6743:
6731:
6694:{\displaystyle Y^{\zeta _{n}}}
6651:
6558:
6539:
6481:
6467:
6378:
6355:{\displaystyle F:\Omega \to U}
6346:
6289:
6281:
6215:
6201:
6197:
6185:
6175:
6171:
6159:
6150:
6138:
6131:
6107:
6095:
5911:
5881:
5820:
5790:
5781:
5617:
5598:
5513:
5497:
5335:
5323:
5291:
5272:
5254:
5235:
5162:
5153:
5079:
5060:
5019:
5010:
4963:
4949:
4932:
4920:
4911:
4899:
4876:
4864:
4855:
4843:
4801:
4793:
4763:
4751:
4728:
4716:
4647:
4644:
4632:
4575:
4572:
4560:
4447:
4441:
4406:
4400:
4278:
4272:
4253:
4244:
4238:
4225:
4207:
4198:
4192:
4179:
4170:
4164:
4080:
4074:
4062:
4050:
4029:
4016:
4007:
3994:
3974:{\displaystyle {\widehat {M}}}
3827:
3821:
3780:
3774:
3686:
3599:
3593:
3582:
3557:
3547:
3493:
3487:
3475:
3463:
3442:
3429:
3420:
3407:
3329:
3323:
3300:
3294:
3266:, s.t. for each test function
3221:
3207:
3041:
3032:
3020:
3014:
2978:
2969:
2963:
2883:
2877:
2871:
2868:
2856:
2844:
2764:
2752:
2720:
2714:
2520:
2502:
2482:
2452:
2434:
2415:
1792:
1773:
1735:
1716:
1588:
1569:
1551:
1532:
1487:{\displaystyle \xi ^{\alpha }}
1460:{\displaystyle \xi ^{\alpha }}
1363:stochastic difference equation
1338:
1332:
1274:{\displaystyle \xi ^{\alpha }}
1119:
1113:
1100:
1097:
1091:
1085:
1048:
1045:
1039:
1033:
1011:
1005:
831:ordinary differential equation
824:
441: / Integral solutions
1:
10365:Kolmogorov continuity theorem
10201:Law of the iterated logarithm
8580:
7782:which has a general solution
4518:-dimensional Brownian motion
1382:partial differential equation
1350:{\displaystyle g(x)\propto x}
892:stochastic evolution operator
776:
767:Random differential equations
10370:Kolmogorov extension theorem
10049:Generalized queueing network
9557:Interacting particle systems
6389:
3932:and after a continuation of
485:Exponential response formula
231:Coupled / Decoupled
7:
9502:Continuous-time random walk
8533:
6871:{\displaystyle t\to \zeta }
6816:we have almost surely that
6433:{\displaystyle \zeta >0}
4529: > 0, and let
3679:
3136:{\displaystyle X_{0}=x_{0}}
988:
10:
10640:
10510:Extreme value theory (EVT)
10310:Doob decomposition theorem
9602:Ornstein–Uhlenbeck process
9373:Chinese restaurant process
9141:Evans, Lawrence C (2013).
8945:10.1016/j.jmaa.2013.01.027
8861:10.1103/PhysRevLett.43.744
8618:Cambridge University Press
8512:self-organized criticality
8486:
6078:{\displaystyle K\subset U}
3840:is a semimartingale, that
2607:one-point compactification
2248:Ornstein–Uhlenbeck process
2128:arithmetic Brownian motion
2123:in financial mathematics.
1842:as the sum of an ordinary
967:
10578:
10482:
10390:Optional stopping theorem
10287:
10249:
10191:Large deviation principle
10158:
10072:
10029:
9996:
9943:Heath–Jarrow–Morton (HJM)
9888:
9880:Moving-average (MA) model
9865:Autoregressive (AR) model
9845:
9755:
9690:Hidden Markov model (HMM)
9672:
9624:Schramm–Loewner evolution
9428:
9353:
9298:10.1137/S0036144500378302
8992:10.1142/S0219024902001511
6412:predictable stopping time
3083:A solution to the SDE on
2008:geometric Brownian motion
619:Józef Maria Hoene-Wroński
565:Undetermined coefficients
474:Method of characteristics
359:Cauchy–Kowalevski theorem
10305:Doléans-Dade exponential
10135:Progressively measurable
9933:Cox–Ingersoll–Ross (CIR)
9225:. Springer. p. 415.
9181:Adomian, George (1989).
9158:Adomian, George (1986).
9149:Adomian, George (1983).
9037:"Detecting Market Abuse"
8626:10.1017/CBO9780511805141
6917:Linear SDE: General case
6296:{\displaystyle |\cdot |}
5381:{\displaystyle X_{0}=Z;}
1374:probability distribution
977:dynamical systems theory
958:Runge–Kutta method (SDE)
344:Picard–Lindelöf theorem
338:Existence and uniqueness
10525:Mathematical statistics
10515:Large deviations theory
10345:Infinitesimal generator
10206:Maximal ergodic theorem
10125:Piecewise-deterministic
9727:Random dynamical system
9592:Markov additive process
9230:Thomas Mikosch (1998).
8841:Physical Review Letters
8788:Zdzisław Brzeźniak and
8743:10.1023/a:1016673307045
6460:-valued semimartingale
6323:{\displaystyle \alpha }
6052:{\displaystyle t\geq 0}
6026:{\displaystyle \alpha }
6006:{\displaystyle \alpha }
4333:{\displaystyle \Omega }
3793:for each test function
3103:with initial condition
2820:-valued semimartingale,
2350:Log-normal distribution
2126:The simpler SDE called
2001:Yamada–Watanabe theorem
1858:the stochastic process
1211:{\displaystyle F\in TX}
934:Itô stochastic calculus
805:after French physicist
570:Variation of parameters
560:Separation of variables
349:Peano existence theorem
10614:Differential equations
10360:Karhunen–Loève theorem
10295:Cameron–Martin formula
10259:Burkholder–Davis–Gundy
9654:Variance gamma process
9190:Calin, Ovidiu (2015).
8483:SDEs and supersymmetry
8473:
8380:
8360:
8308:
8247:which is reducible to
8238:
8135:
8112:
7946:Reducible SDEs: Case 2
7936:
7860:
7773:
7703:
7680:
7549:Reducible SDEs: Case 1
7539:
7339:
7062:
6895:
6894:{\displaystyle \zeta }
6872:
6846:
6810:
6772:
6695:
6661:
6624:
6623:{\displaystyle \zeta }
6598:
6500:
6454:
6434:
6404:
6356:
6324:
6297:
6264:
6114:
6113:{\displaystyle L(t,K)}
6079:
6053:
6027:
6007:
5976:
5947:
5918:
5862:
5827:
5743:
5714:
5694:
5665:
5639:
5553:
5399:-continuous solution (
5382:
5345:
5195:
5113:-algebra generated by
5096:
5058:
4987: ∈ and all
4974:
4819:
4675:
4597:
4484:
4454:
4413:
4372:
4334:
4314:
4285:
4111:
3975:
3946:
3926:
3897:
3876:
3854:
3834:
3787:
3758:
3736:
3635:
3611:
3540:-almost surely, where
3534:
3511:
3388:
3356:
3336:
3307:
3260:
3259:{\displaystyle \zeta }
3240:
3194:
3174:
3137:
3097:
3074:
3073:{\displaystyle e\in E}
3048:
2998:
2947:
2946:{\displaystyle x\in M}
2917:
2893:
2814:
2794:
2771:
2736:
2683:
2661:
2630:
2599:
2548:
2489:
2402:
2382:
2366:differential manifolds
2335:
2195:
2098:
1993:
1818:
1637:
1614:
1488:
1461:
1378:Fokker–Planck equation
1351:
1315:
1301:is a linear space and
1295:
1275:
1248:
1212:
1183:
1167:phase (or state) space
1159:
1158:{\displaystyle x\in X}
1130:
1074:
884:Fokker–Planck equation
771:differential manifolds
639:Carl David Tolmé Runge
182:Differential-algebraic
23:Differential equations
10490:Actuarial mathematics
10452:Uniform integrability
10447:Stratonovich integral
10375:Lévy–Prokhorov metric
10279:Marcinkiewicz–Zygmund
10166:Central limit theorem
9768:Gaussian random field
9597:McKean–Vlasov process
9517:Dyson Brownian motion
9378:Galton–Watson process
8560:Stochastic volatility
8474:
8381:
8361:
8309:
8239:
8136:
8113:
7937:
7861:
7774:
7704:
7681:
7540:
7340:
7063:
6896:
6873:
6847:
6811:
6773:
6701:is a solution to the
6696:
6662:
6625:
6599:
6501:
6455:
6435:
6405:
6357:
6325:
6298:
6265:
6115:
6080:
6059:and some compact set
6054:
6028:
6008:
5977:
5948:
5919:
5863:
5828:
5744:
5715:
5695:
5666:
5640:
5554:
5383:
5346:
5196:
5097:
5032:
4975:
4820:
4676:
4598:
4485:
4455:
4414:
4373:
4335:
4315:
4286:
4112:
3976:
3947:
3927:
3898:
3877:
3855:
3835:
3788:
3759:
3737:
3636:
3612:
3535:
3512:
3389:
3357:
3355:{\displaystyle \tau }
3337:
3308:
3261:
3241:
3195:
3175:
3138:
3098:
3075:
3049:
2999:
2948:
2918:
2899:is a homomorphism of
2894:
2815:
2795:
2772:
2770:{\displaystyle (A,Z)}
2737:
2684:
2662:
2631:
2629:{\displaystyle x_{0}}
2600:
2549:
2490:
2403:
2383:
2336:
2196:
2099:
1994:
1819:
1638:
1615:
1489:
1462:
1424:The notation used in
1352:
1316:
1296:
1276:
1249:
1213:
1184:
1160:
1131:
1054:
950:Euler–Maruyama method
880:Smoluchowski equation
863:Stratonovich integral
855:Stratonovich integral
720:differential equation
629:Augustin-Louis Cauchy
614:Joseph-Louis Lagrange
446:Numerical integration
428:Exponential stability
291:Relation to processes
10624:Mathematical finance
10619:Stochastic processes
10565:Time series analysis
10520:Mathematical finance
10405:Reflection principle
9732:Regenerative process
9532:Fleming–Viot process
9347:Stochastic processes
8691:. Berlin: Springer.
8520:the butterfly effect
8393:
8370:
8321:
8254:
8148:
8125:
7954:
7876:
7789:
7716:
7693:
7557:
7355:
7073:
6925:
6901:is also a so-called
6885:
6856:
6820:
6788:
6714:
6671:
6667:the stopped process
6638:
6614:
6517:
6464:
6444:
6418:
6366:
6334:
6314:
6277:
6127:
6089:
6063:
6037:
6017:
5997:
5957:
5928:
5872:
5837:
5754:
5724:
5704:
5675:
5655:
5574:
5468:
5356:
5211:
5137:
5006:
4830:
4703:
4686:measurable functions
4608:
4536:
4468:
4423:
4382:
4344:
4324:
4298:
4146:
3988:
3956:
3936:
3907:
3887:
3866:
3844:
3797:
3786:{\displaystyle f(X)}
3768:
3748:
3652:
3625:
3544:
3524:
3401:
3366:
3346:
3335:{\displaystyle f(X)}
3317:
3270:
3250:
3204:
3184:
3147:
3107:
3087:
3058:
3008:
2957:
2931:
2907:
2826:
2804:
2784:
2749:
2697:
2673:
2640:
2613:
2562:
2499:
2412:
2392:
2372:
2256:
2136:
2016:
1965:
1940:, and satisfies the
1869:normally distributed
1657:
1627:
1508:
1496:generalized function
1471:
1444:
1434:mathematical finance
1412:for the statistical
1406:Schrödinger equation
1386:Schrödinger equation
1326:
1305:
1285:
1258:
1222:
1193:
1173:
1143:
991:
753:or more generally a
740:thermal fluctuations
736:random growth models
451:Dirac delta function
187:Integro-differential
10560:Stochastic analysis
10400:Quadratic variation
10395:Prokhorov's theorem
10330:Feynman–Kac formula
9800:Markov random field
9448:Birth–death process
9280:2001SIAMR..43..525H
8926:Slavík, A. (2013).
8853:1979PhRvL..43..744P
7502:
7416:
7300:
7269:
7189:
7158:
5495:
4049:
3820:
3462:
3293:
1769:
1712:
944:Numerical solutions
917:Stochastic calculus
815:stochastic integral
793:in 1905, although
791:Marian Smoluchowski
547:Perturbation theory
542:Integral transforms
433:Rate of convergence
299:(discrete analogue)
136:Population dynamics
103:Continuum mechanics
94:Applied mathematics
10530:Probability theory
10410:Skorokhod integral
10380:Malliavin calculus
9963:Korn-Kreer-Lenssen
9847:Time series models
9810:Pitman–Yor process
9151:Stochastic systems
8685:Øksendal, Bernt K.
8555:Stochastic process
8495:differential forms
8469:
8376:
8356:
8304:
8234:
8131:
8108:
7932:
7856:
7769:
7699:
7676:
7535:
7481:
7395:
7335:
7270:
7248:
7159:
7137:
7058:
6891:
6868:
6842:
6806:
6768:
6691:
6657:
6620:
6594:
6496:
6450:
6430:
6400:
6352:
6320:
6293:
6260:
6110:
6085:and some constant
6075:
6049:
6023:
6003:
5972:
5943:
5914:
5858:
5823:
5739:
5710:
5690:
5661:
5635:
5549:
5481:
5378:
5341:
5315:
5191:
5092:
4970:
4815:
4671:
4593:
4480:
4450:
4409:
4368:
4330:
4310:
4281:
4107:
4035:
3971:
3942:
3922:
3893:
3872:
3862:semimartingale on
3850:
3830:
3806:
3783:
3754:
3732:
3693:
3631:
3607:
3530:
3507:
3448:
3384:
3352:
3332:
3303:
3279:
3256:
3236:
3190:
3170:
3133:
3093:
3070:
3044:
2994:
2943:
2913:
2889:
2810:
2790:
2767:
2732:
2679:
2657:
2626:
2595:
2544:
2485:
2398:
2378:
2331:
2191:
2094:
1989:
1832:stochastic process
1814:
1749:
1692:
1633:
1610:
1484:
1457:
1426:probability theory
1390:diffusion equation
1347:
1311:
1291:
1271:
1244:
1208:
1179:
1155:
1126:
803:Langevin equations
785:, in the work of
724:stochastic process
537:Integrating factor
378:Initial conditions
313:Stochastic partial
10596:
10595:
10550:Signal processing
10269:Doob's upcrossing
10264:Doob's martingale
10228:Engelbert–Schmidt
10171:Donsker's theorem
10105:Feller-continuous
9973:Rendleman–Bartter
9763:Dirichlet process
9680:Branching process
9649:Telegraph process
9542:Geometric process
9522:Empirical process
9512:Diffusion process
9368:Branching process
9363:Bernoulli process
9310:978-1-611976-42-7
9201:978-981-4678-93-3
9173:978-0-12-044375-8
9124:978-3-519-02229-9
9075:978-3-519-02229-9
8804:Armstrong J. and
8570:Diffusion process
8545:Langevin dynamics
8516:Goldstone theorem
8379:{\displaystyle h}
8134:{\displaystyle f}
8013:
7930:
7702:{\displaystyle f}
7586:
7463:
6453:{\displaystyle U}
6440:almost surely. A
6392:
5713:{\displaystyle Y}
5664:{\displaystyle X}
5314:
4514:and driven by an
4128:filtering problem
3968:
3945:{\displaystyle f}
3896:{\displaystyle X}
3875:{\displaystyle M}
3853:{\displaystyle X}
3757:{\displaystyle P}
3724:
3713:
3678:
3634:{\displaystyle X}
3533:{\displaystyle P}
3193:{\displaystyle M}
3096:{\displaystyle M}
2916:{\displaystyle M}
2813:{\displaystyle E}
2793:{\displaystyle Z}
2682:{\displaystyle M}
2574:
2401:{\displaystyle E}
2381:{\displaystyle M}
2360:SDEs on manifolds
1959:probability space
1938:diffusion process
1844:Lebesgue integral
1649:integral equation
1636:{\displaystyle B}
1438:numerical methods
1430:filtering problem
1402:quantum mechanics
1314:{\displaystyle g}
1294:{\displaystyle X}
1182:{\displaystyle X}
1025:
970:Langevin equation
903:more general form
708:
707:
599:Gottfried Leibniz
490:Finite difference
282:
281:
143:
142:
113:Dynamical systems
10631:
10570:Machine learning
10457:Usual hypotheses
10340:Girsanov theorem
10325:Dynkin's formula
10090:Continuous paths
9998:Actuarial models
9938:Garman–Kohlhagen
9908:Black–Karasinski
9903:Black–Derman–Toy
9890:Financial models
9756:Fields and other
9685:Gaussian process
9634:Sigma-martingale
9438:Additive process
9340:
9333:
9326:
9317:
9316:
9301:
9291:
9262:
9245:
9226:
9214:
9205:
9186:
9177:
9165:
9154:
9129:
9128:
9110:
9104:
9101:
9095:
9089:
9080:
9079:
9061:
9055:
9054:
9047:
9041:
9040:
9033:
9027:
9026:
9018:
9012:
9002:
8996:
8995:
8968:
8962:
8956:
8950:
8949:
8947:
8923:
8917:
8911:
8905:
8898:
8892:
8886:
8880:
8874:
8865:
8864:
8836:
8830:
8823:
8812:
8802:
8793:
8786:
8780:
8774:
8763:
8762:
8722:
8716:
8709:
8703:
8702:
8681:
8662:
8659:
8648:
8647:
8602:
8550:Local volatility
8478:
8476:
8475:
8470:
8462:
8461:
8443:
8442:
8421:
8420:
8405:
8404:
8385:
8383:
8382:
8377:
8365:
8363:
8362:
8357:
8352:
8351:
8333:
8332:
8313:
8311:
8310:
8305:
8303:
8302:
8293:
8282:
8271:
8270:
8261:
8243:
8241:
8240:
8235:
8233:
8232:
8217:
8216:
8195:
8187:
8186:
8165:
8164:
8155:
8140:
8138:
8137:
8132:
8117:
8115:
8114:
8109:
8107:
8106:
8097:
8089:
8088:
8067:
8062:
8058:
8054:
8053:
8041:
8030:
8029:
8014:
8006:
7998:
7997:
7971:
7970:
7961:
7941:
7939:
7938:
7933:
7931:
7929:
7915:
7911:
7905:
7903:
7902:
7865:
7863:
7862:
7857:
7849:
7848:
7830:
7829:
7817:
7816:
7801:
7800:
7778:
7776:
7775:
7770:
7768:
7767:
7752:
7751:
7733:
7732:
7723:
7708:
7706:
7705:
7700:
7685:
7683:
7682:
7677:
7675:
7674:
7665:
7657:
7656:
7635:
7627:
7626:
7614:
7603:
7602:
7587:
7579:
7574:
7573:
7564:
7544:
7542:
7541:
7536:
7534:
7530:
7529:
7528:
7519:
7501:
7496:
7495:
7494:
7474:
7469:
7465:
7464:
7459:
7449:
7448:
7438:
7415:
7410:
7409:
7408:
7380:
7379:
7378:
7377:
7344:
7342:
7341:
7336:
7334:
7330:
7329:
7328:
7319:
7305:
7299:
7291:
7290:
7289:
7268:
7263:
7262:
7261:
7241:
7224:
7188:
7180:
7179:
7178:
7157:
7152:
7151:
7150:
7133:
7132:
7131:
7130:
7111:
7110:
7109:
7108:
7085:
7084:
7067:
7065:
7064:
7059:
7057:
7056:
7047:
7024:
7023:
6993:
6970:
6969:
6942:
6941:
6932:
6900:
6898:
6897:
6892:
6877:
6875:
6874:
6869:
6851:
6849:
6848:
6843:
6832:
6831:
6815:
6813:
6812:
6807:
6777:
6775:
6774:
6769:
6767:
6766:
6765:
6764:
6750:
6721:
6700:
6698:
6697:
6692:
6690:
6689:
6688:
6687:
6666:
6664:
6663:
6658:
6650:
6649:
6629:
6627:
6626:
6621:
6603:
6601:
6600:
6595:
6587:
6586:
6573:
6572:
6557:
6556:
6532:
6531:
6508:maximal solution
6505:
6503:
6502:
6497:
6495:
6494:
6479:
6478:
6459:
6457:
6456:
6451:
6439:
6437:
6436:
6431:
6409:
6407:
6406:
6401:
6399:
6398:
6393:
6388:
6383:
6361:
6359:
6358:
6353:
6329:
6327:
6326:
6321:
6305:maximal solution
6302:
6300:
6299:
6294:
6292:
6284:
6269:
6267:
6266:
6261:
6218:
6204:
6178:
6134:
6119:
6117:
6116:
6111:
6084:
6082:
6081:
6076:
6058:
6056:
6055:
6050:
6032:
6030:
6029:
6024:
6012:
6010:
6009:
6004:
5981:
5979:
5978:
5973:
5971:
5970:
5965:
5952:
5950:
5949:
5944:
5942:
5941:
5936:
5923:
5921:
5920:
5915:
5910:
5909:
5904:
5895:
5894:
5889:
5867:
5865:
5864:
5859:
5857:
5856:
5851:
5832:
5830:
5829:
5824:
5819:
5818:
5813:
5804:
5803:
5798:
5774:
5773:
5768:
5748:
5746:
5745:
5740:
5738:
5737:
5732:
5719:
5717:
5716:
5711:
5699:
5697:
5696:
5691:
5689:
5688:
5683:
5670:
5668:
5667:
5662:
5644:
5642:
5641:
5636:
5634:
5633:
5624:
5616:
5615:
5591:
5590:
5581:
5558:
5556:
5555:
5550:
5536:
5532:
5528:
5522:
5521:
5516:
5510:
5509:
5500:
5494:
5489:
5475:
5387:
5385:
5384:
5379:
5368:
5367:
5350:
5348:
5347:
5342:
5316:
5312:
5309:
5308:
5299:
5284:
5283:
5262:
5247:
5246:
5228:
5227:
5218:
5200:
5198:
5197:
5192:
5178:
5177:
5171:
5170:
5165:
5156:
5151:
5150:
5144:
5101:
5099:
5098:
5093:
5088:
5087:
5082:
5076:
5075:
5063:
5057:
5052:
5028:
5027:
5022:
5013:
4979:
4977:
4976:
4971:
4966:
4952:
4941:
4940:
4895:
4894:
4885:
4884:
4839:
4838:
4824:
4822:
4821:
4816:
4811:
4810:
4804:
4796:
4785:
4784:
4772:
4771:
4747:
4746:
4737:
4736:
4712:
4711:
4680:
4678:
4677:
4672:
4667:
4666:
4655:
4628:
4627:
4622:
4602:
4600:
4599:
4594:
4589:
4588:
4583:
4556:
4555:
4550:
4489:
4487:
4486:
4481:
4459:
4457:
4456:
4451:
4440:
4439:
4430:
4418:
4416:
4415:
4410:
4399:
4398:
4389:
4377:
4375:
4374:
4369:
4360:
4359:
4339:
4337:
4336:
4331:
4319:
4317:
4316:
4311:
4290:
4288:
4287:
4282:
4271:
4270:
4261:
4237:
4236:
4215:
4191:
4190:
4163:
4162:
4153:
4123:Laurent Schwartz
4116:
4114:
4113:
4108:
4090:
4070:
4069:
4057:
4048:
4043:
4028:
4027:
4006:
4005:
3980:
3978:
3977:
3972:
3970:
3969:
3961:
3951:
3949:
3948:
3943:
3931:
3929:
3928:
3923:
3921:
3920:
3915:
3902:
3900:
3899:
3894:
3881:
3879:
3878:
3873:
3859:
3857:
3856:
3851:
3839:
3837:
3836:
3831:
3819:
3814:
3792:
3790:
3789:
3784:
3763:
3761:
3760:
3755:
3741:
3739:
3738:
3733:
3731:
3727:
3726:
3725:
3717:
3714:
3711:
3703:
3702:
3692:
3643:maximal solution
3640:
3638:
3637:
3632:
3616:
3614:
3613:
3608:
3603:
3602:
3578:
3577:
3565:
3564:
3539:
3537:
3536:
3531:
3516:
3514:
3513:
3508:
3503:
3483:
3482:
3470:
3461:
3456:
3441:
3440:
3419:
3418:
3393:
3391:
3390:
3385:
3361:
3359:
3358:
3353:
3341:
3339:
3338:
3333:
3312:
3310:
3309:
3304:
3292:
3287:
3265:
3263:
3262:
3257:
3246:up to life time
3245:
3243:
3242:
3237:
3235:
3234:
3219:
3218:
3200:-valued process
3199:
3197:
3196:
3191:
3179:
3177:
3176:
3171:
3166:
3165:
3160:
3159:
3143:is a continuous
3142:
3140:
3139:
3134:
3132:
3131:
3119:
3118:
3102:
3100:
3099:
3094:
3079:
3077:
3076:
3071:
3053:
3051:
3050:
3045:
3003:
3001:
3000:
2995:
2990:
2989:
2952:
2950:
2949:
2944:
2922:
2920:
2919:
2914:
2898:
2896:
2895:
2890:
2819:
2817:
2816:
2811:
2800:is a continuous
2799:
2797:
2796:
2791:
2776:
2774:
2773:
2768:
2741:
2739:
2738:
2733:
2704:
2688:
2686:
2685:
2680:
2666:
2664:
2663:
2658:
2656:
2655:
2650:
2649:
2635:
2633:
2632:
2627:
2625:
2624:
2604:
2602:
2601:
2596:
2576:
2575:
2567:
2556:usual conditions
2553:
2551:
2550:
2545:
2543:
2542:
2541:
2540:
2535:
2518:
2517:
2512:
2511:
2494:
2492:
2491:
2486:
2475:
2474:
2473:
2472:
2467:
2450:
2449:
2444:
2443:
2430:
2429:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2340:
2338:
2337:
2332:
2327:
2326:
2317:
2311:
2310:
2295:
2289:
2288:
2273:
2272:
2263:
2200:
2198:
2197:
2192:
2190:
2189:
2180:
2165:
2153:
2152:
2143:
2121:volatility smile
2103:
2101:
2100:
2095:
2090:
2089:
2080:
2074:
2073:
2055:
2049:
2048:
2033:
2032:
2023:
1998:
1996:
1995:
1990:
1981:
1980:
1823:
1821:
1820:
1815:
1810:
1809:
1800:
1785:
1784:
1768:
1757:
1742:
1728:
1727:
1711:
1700:
1688:
1687:
1675:
1674:
1642:
1640:
1639:
1634:
1619:
1617:
1616:
1611:
1606:
1605:
1596:
1581:
1580:
1559:
1544:
1543:
1525:
1524:
1515:
1493:
1491:
1490:
1485:
1483:
1482:
1466:
1464:
1463:
1458:
1456:
1455:
1398:path integration
1356:
1354:
1353:
1348:
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1270:
1269:
1253:
1251:
1250:
1245:
1234:
1233:
1217:
1215:
1214:
1209:
1188:
1186:
1185:
1180:
1164:
1162:
1161:
1156:
1135:
1133:
1132:
1127:
1112:
1111:
1084:
1083:
1073:
1068:
1026:
1024:
1020:
1014:
1001:
995:
700:
693:
686:
664:Phyllis Nicolson
649:Rudolf Lipschitz
532:Green's function
508:Infinite element
499:
464:Solution methods
442:
300:
211:By variable type
165:
164:
47:Natural sciences
40:
39:
19:
18:
10639:
10638:
10634:
10633:
10632:
10630:
10629:
10628:
10599:
10598:
10597:
10592:
10574:
10535:Queueing theory
10478:
10420:Skorokhod space
10283:
10274:Kunita–Watanabe
10245:
10211:Sanov's theorem
10181:Ergodic theorem
10154:
10150:Time-reversible
10068:
10031:Queueing models
10025:
10021:Sparre–Anderson
10011:Cramér–Lundberg
9992:
9978:SABR volatility
9884:
9841:
9793:Boolean network
9751:
9737:Renewal process
9668:
9617:Non-homogeneous
9607:Poisson process
9497:Contact process
9460:Brownian motion
9430:Continuous time
9424:
9418:Maximal entropy
9349:
9344:
9289:10.1.1.137.6375
9242:
9202:
9174:
9138:
9136:Further reading
9133:
9132:
9125:
9111:
9107:
9102:
9098:
9090:
9083:
9076:
9062:
9058:
9049:
9048:
9044:
9035:
9034:
9030:
9019:
9015:
9003:
8999:
8976:Mercurio, Fabio
8969:
8965:
8957:
8953:
8924:
8920:
8912:
8908:
8899:
8895:
8887:
8883:
8875:
8868:
8847:(11): 744–745.
8837:
8833:
8824:
8815:
8803:
8796:
8787:
8783:
8775:
8766:
8723:
8719:
8710:
8706:
8699:
8682:
8665:
8660:
8651:
8636:
8610:Williams, David
8603:
8588:
8583:
8536:
8491:
8485:
8457:
8453:
8438:
8434:
8413:
8409:
8400:
8396:
8394:
8391:
8390:
8371:
8368:
8367:
8347:
8343:
8328:
8324:
8322:
8319:
8318:
8298:
8294:
8289:
8278:
8266:
8262:
8257:
8255:
8252:
8251:
8228:
8224:
8212:
8208:
8191:
8182:
8178:
8160:
8156:
8151:
8149:
8146:
8145:
8126:
8123:
8122:
8102:
8098:
8093:
8084:
8080:
8063:
8049:
8045:
8034:
8025:
8021:
8005:
7993:
7989:
7979:
7975:
7966:
7962:
7957:
7955:
7952:
7951:
7948:
7916:
7907:
7906:
7904:
7898:
7894:
7877:
7874:
7873:
7844:
7840:
7825:
7821:
7809:
7805:
7796:
7792:
7790:
7787:
7786:
7763:
7759:
7747:
7743:
7728:
7724:
7719:
7717:
7714:
7713:
7694:
7691:
7690:
7670:
7666:
7661:
7652:
7648:
7631:
7622:
7618:
7607:
7598:
7594:
7578:
7569:
7565:
7560:
7558:
7555:
7554:
7551:
7524:
7520:
7515:
7497:
7490:
7486:
7485:
7470:
7444:
7440:
7439:
7437:
7421:
7417:
7411:
7404:
7400:
7399:
7394:
7390:
7373:
7369:
7362:
7358:
7356:
7353:
7352:
7324:
7320:
7315:
7301:
7292:
7285:
7281:
7274:
7264:
7257:
7253:
7252:
7237:
7220:
7181:
7174:
7170:
7163:
7153:
7146:
7142:
7141:
7126:
7122:
7121:
7117:
7116:
7112:
7104:
7100:
7093:
7089:
7080:
7076:
7074:
7071:
7070:
7052:
7048:
7043:
7019:
7015:
6989:
6965:
6961:
6937:
6933:
6928:
6926:
6923:
6922:
6919:
6911:
6886:
6883:
6882:
6857:
6854:
6853:
6827:
6823:
6821:
6818:
6817:
6789:
6786:
6785:
6760:
6756:
6755:
6751:
6746:
6717:
6715:
6712:
6711:
6683:
6679:
6678:
6674:
6672:
6669:
6668:
6645:
6641:
6639:
6636:
6635:
6615:
6612:
6611:
6582:
6578:
6568:
6564:
6552:
6548:
6527:
6523:
6518:
6515:
6514:
6484:
6480:
6474:
6470:
6465:
6462:
6461:
6445:
6442:
6441:
6419:
6416:
6415:
6394:
6384:
6382:
6381:
6367:
6364:
6363:
6335:
6332:
6331:
6315:
6312:
6311:
6288:
6280:
6278:
6275:
6274:
6214:
6200:
6174:
6130:
6128:
6125:
6124:
6090:
6087:
6086:
6064:
6061:
6060:
6038:
6035:
6034:
6018:
6015:
6014:
5998:
5995:
5994:
5966:
5961:
5960:
5958:
5955:
5954:
5937:
5932:
5931:
5929:
5926:
5925:
5905:
5900:
5899:
5890:
5885:
5884:
5873:
5870:
5869:
5852:
5847:
5846:
5838:
5835:
5834:
5814:
5809:
5808:
5799:
5794:
5793:
5769:
5764:
5763:
5755:
5752:
5751:
5733:
5728:
5727:
5725:
5722:
5721:
5705:
5702:
5701:
5684:
5679:
5678:
5676:
5673:
5672:
5656:
5653:
5652:
5629:
5625:
5620:
5611:
5607:
5586:
5582:
5577:
5575:
5572:
5571:
5565:
5524:
5517:
5512:
5511:
5505:
5501:
5496:
5490:
5485:
5480:
5476:
5471:
5469:
5466:
5465:
5452:
5439:
5415:
5363:
5359:
5357:
5354:
5353:
5313: for
5310:
5304:
5300:
5295:
5279:
5275:
5258:
5242:
5238:
5223:
5219:
5214:
5212:
5209:
5208:
5173:
5172:
5166:
5161:
5160:
5152:
5146:
5145:
5140:
5138:
5135:
5134:
5121:
5083:
5078:
5077:
5068:
5064:
5059:
5053:
5036:
5023:
5018:
5017:
5009:
5007:
5004:
5003:
4962:
4948:
4936:
4935:
4890:
4889:
4880:
4879:
4834:
4833:
4831:
4828:
4827:
4806:
4805:
4800:
4792:
4780:
4779:
4767:
4766:
4742:
4741:
4732:
4731:
4707:
4706:
4704:
4701:
4700:
4656:
4651:
4650:
4623:
4618:
4617:
4609:
4606:
4605:
4584:
4579:
4578:
4551:
4546:
4545:
4537:
4534:
4533:
4509:Euclidean space
4497:
4469:
4466:
4465:
4435:
4431:
4426:
4424:
4421:
4420:
4394:
4390:
4385:
4383:
4380:
4379:
4355:
4354:
4345:
4342:
4341:
4325:
4322:
4321:
4299:
4296:
4295:
4266:
4262:
4257:
4232:
4228:
4211:
4186:
4182:
4158:
4154:
4149:
4147:
4144:
4143:
4136:
4086:
4065:
4061:
4053:
4044:
4039:
4023:
4019:
4001:
3997:
3989:
3986:
3985:
3960:
3959:
3957:
3954:
3953:
3937:
3934:
3933:
3916:
3911:
3910:
3908:
3905:
3904:
3888:
3885:
3884:
3867:
3864:
3863:
3845:
3842:
3841:
3815:
3810:
3798:
3795:
3794:
3769:
3766:
3765:
3749:
3746:
3745:
3716:
3715:
3710:
3698:
3694:
3682:
3677:
3673:
3653:
3650:
3649:
3626:
3623:
3622:
3589:
3585:
3573:
3569:
3560:
3556:
3545:
3542:
3541:
3525:
3522:
3521:
3499:
3478:
3474:
3466:
3457:
3452:
3436:
3432:
3414:
3410:
3402:
3399:
3398:
3367:
3364:
3363:
3347:
3344:
3343:
3318:
3315:
3314:
3288:
3283:
3271:
3268:
3267:
3251:
3248:
3247:
3224:
3220:
3214:
3210:
3205:
3202:
3201:
3185:
3182:
3181:
3161:
3155:
3154:
3153:
3148:
3145:
3144:
3127:
3123:
3114:
3110:
3108:
3105:
3104:
3088:
3085:
3084:
3059:
3056:
3055:
3009:
3006:
3005:
2985:
2981:
2958:
2955:
2954:
2932:
2929:
2928:
2908:
2905:
2904:
2827:
2824:
2823:
2805:
2802:
2801:
2785:
2782:
2781:
2750:
2747:
2746:
2700:
2698:
2695:
2694:
2674:
2671:
2670:
2667:-measurable. A
2651:
2645:
2644:
2643:
2641:
2638:
2637:
2620:
2616:
2614:
2611:
2610:
2566:
2565:
2563:
2560:
2559:
2554:satisfying the
2536:
2531:
2530:
2523:
2519:
2513:
2507:
2506:
2505:
2500:
2497:
2496:
2468:
2463:
2462:
2455:
2451:
2445:
2439:
2438:
2437:
2425:
2424:
2413:
2410:
2409:
2393:
2390:
2389:
2373:
2370:
2369:
2362:
2322:
2318:
2313:
2306:
2302:
2291:
2284:
2280:
2268:
2264:
2259:
2257:
2254:
2253:
2227:
2206:Bachelier model
2185:
2181:
2176:
2161:
2148:
2144:
2139:
2137:
2134:
2133:
2085:
2081:
2076:
2069:
2065:
2051:
2044:
2040:
2028:
2024:
2019:
2017:
2014:
2013:
1976:
1975:
1966:
1963:
1962:
1956:
1942:Markov property
1935:
1910:
1886:
1866:
1841:
1829:continuous time
1805:
1801:
1796:
1780:
1776:
1758:
1753:
1738:
1723:
1719:
1701:
1696:
1683:
1679:
1664:
1660:
1658:
1655:
1654:
1628:
1625:
1624:
1601:
1597:
1592:
1576:
1572:
1555:
1539:
1535:
1520:
1516:
1511:
1509:
1506:
1505:
1478:
1474:
1472:
1469:
1468:
1451:
1447:
1445:
1442:
1441:
1422:
1327:
1324:
1323:
1306:
1303:
1302:
1286:
1283:
1282:
1265:
1261:
1259:
1256:
1255:
1229:
1225:
1223:
1220:
1219:
1194:
1191:
1190:
1174:
1171:
1170:
1144:
1141:
1140:
1107:
1103:
1079:
1075:
1069:
1058:
1016:
1015:
997:
996:
994:
992:
989:
972:
966:
954:Milstein method
946:
922:Brownian motion
919:
899:"Langevin SDEs"
827:
799:Bachelier model
795:Louis Bachelier
787:Albert Einstein
783:Brownian motion
779:
751:Brownian motion
704:
675:
674:
673:
604:Jacob Bernoulli
588:
575:
574:
556:
525:Petrov–Galerkin
493:
478:
465:
457:
456:
455:
437:
383:Boundary values
372:
364:
363:
339:
326:
325:
324:
298:
292:
284:
283:
271:
248:
206:
162:
149:
148:
144:
122:Social sciences
78:
56:
37:
17:
12:
11:
5:
10637:
10627:
10626:
10621:
10616:
10611:
10594:
10593:
10591:
10590:
10585:
10583:List of topics
10579:
10576:
10575:
10573:
10572:
10567:
10562:
10557:
10552:
10547:
10542:
10540:Renewal theory
10537:
10532:
10527:
10522:
10517:
10512:
10507:
10505:Ergodic theory
10502:
10497:
10495:Control theory
10492:
10486:
10484:
10480:
10479:
10477:
10476:
10475:
10474:
10469:
10459:
10454:
10449:
10444:
10439:
10438:
10437:
10427:
10425:Snell envelope
10422:
10417:
10412:
10407:
10402:
10397:
10392:
10387:
10382:
10377:
10372:
10367:
10362:
10357:
10352:
10347:
10342:
10337:
10332:
10327:
10322:
10317:
10312:
10307:
10302:
10297:
10291:
10289:
10285:
10284:
10282:
10281:
10276:
10271:
10266:
10261:
10255:
10253:
10247:
10246:
10244:
10243:
10224:Borel–Cantelli
10213:
10208:
10203:
10198:
10193:
10188:
10183:
10178:
10173:
10168:
10162:
10160:
10159:Limit theorems
10156:
10155:
10153:
10152:
10147:
10142:
10137:
10132:
10127:
10122:
10117:
10112:
10107:
10102:
10097:
10092:
10087:
10082:
10076:
10074:
10070:
10069:
10067:
10066:
10061:
10056:
10051:
10046:
10041:
10035:
10033:
10027:
10026:
10024:
10023:
10018:
10013:
10008:
10002:
10000:
9994:
9993:
9991:
9990:
9985:
9980:
9975:
9970:
9965:
9960:
9955:
9950:
9945:
9940:
9935:
9930:
9925:
9920:
9915:
9910:
9905:
9900:
9894:
9892:
9886:
9885:
9883:
9882:
9877:
9872:
9867:
9862:
9857:
9851:
9849:
9843:
9842:
9840:
9839:
9834:
9829:
9828:
9827:
9822:
9812:
9807:
9802:
9797:
9796:
9795:
9790:
9780:
9778:Hopfield model
9775:
9770:
9765:
9759:
9757:
9753:
9752:
9750:
9749:
9744:
9739:
9734:
9729:
9724:
9723:
9722:
9717:
9712:
9707:
9697:
9695:Markov process
9692:
9687:
9682:
9676:
9674:
9670:
9669:
9667:
9666:
9664:Wiener sausage
9661:
9659:Wiener process
9656:
9651:
9646:
9641:
9639:Stable process
9636:
9631:
9629:Semimartingale
9626:
9621:
9620:
9619:
9614:
9604:
9599:
9594:
9589:
9584:
9579:
9574:
9572:Jump diffusion
9569:
9564:
9559:
9554:
9549:
9547:Hawkes process
9544:
9539:
9534:
9529:
9527:Feller process
9524:
9519:
9514:
9509:
9504:
9499:
9494:
9492:Cauchy process
9489:
9488:
9487:
9482:
9477:
9472:
9467:
9457:
9456:
9455:
9445:
9443:Bessel process
9440:
9434:
9432:
9426:
9425:
9423:
9422:
9421:
9420:
9415:
9410:
9405:
9395:
9390:
9385:
9380:
9375:
9370:
9365:
9359:
9357:
9351:
9350:
9343:
9342:
9335:
9328:
9320:
9314:
9313:
9302:
9274:(3): 525–546.
9263:
9246:
9240:
9227:
9219:C. W. Gardiner
9215:
9206:
9200:
9187:
9178:
9172:
9155:
9146:
9137:
9134:
9131:
9130:
9123:
9105:
9096:
9081:
9074:
9056:
9042:
9028:
9013:
8997:
8986:(4): 427–446.
8972:Brigo, Damiano
8963:
8951:
8938:(1): 261–274.
8918:
8906:
8893:
8881:
8866:
8831:
8813:
8794:
8790:K. D. Elworthy
8781:
8764:
8737:(2): 215–249.
8717:
8704:
8697:
8663:
8649:
8634:
8606:Rogers, L.C.G.
8585:
8584:
8582:
8579:
8578:
8577:
8572:
8567:
8562:
8557:
8552:
8547:
8542:
8535:
8532:
8487:Main article:
8484:
8481:
8480:
8479:
8468:
8465:
8460:
8456:
8452:
8449:
8446:
8441:
8437:
8433:
8430:
8427:
8424:
8419:
8416:
8412:
8408:
8403:
8399:
8375:
8355:
8350:
8346:
8342:
8339:
8336:
8331:
8327:
8315:
8314:
8301:
8297:
8292:
8288:
8285:
8281:
8277:
8274:
8269:
8265:
8260:
8245:
8244:
8231:
8227:
8223:
8220:
8215:
8211:
8207:
8204:
8201:
8198:
8194:
8190:
8185:
8181:
8177:
8174:
8171:
8168:
8163:
8159:
8154:
8130:
8119:
8118:
8105:
8101:
8096:
8092:
8087:
8083:
8079:
8076:
8073:
8070:
8066:
8061:
8057:
8052:
8048:
8044:
8040:
8037:
8033:
8028:
8024:
8020:
8017:
8012:
8009:
8004:
8001:
7996:
7992:
7988:
7985:
7982:
7978:
7974:
7969:
7965:
7960:
7947:
7944:
7943:
7942:
7928:
7925:
7922:
7919:
7914:
7910:
7901:
7897:
7893:
7890:
7887:
7884:
7881:
7867:
7866:
7855:
7852:
7847:
7843:
7839:
7836:
7833:
7828:
7824:
7820:
7815:
7812:
7808:
7804:
7799:
7795:
7780:
7779:
7766:
7762:
7758:
7755:
7750:
7746:
7742:
7739:
7736:
7731:
7727:
7722:
7698:
7687:
7686:
7673:
7669:
7664:
7660:
7655:
7651:
7647:
7644:
7641:
7638:
7634:
7630:
7625:
7621:
7617:
7613:
7610:
7606:
7601:
7597:
7593:
7590:
7585:
7582:
7577:
7572:
7568:
7563:
7550:
7547:
7546:
7545:
7533:
7527:
7523:
7518:
7514:
7511:
7508:
7505:
7500:
7493:
7489:
7484:
7480:
7477:
7473:
7468:
7462:
7458:
7455:
7452:
7447:
7443:
7436:
7433:
7430:
7427:
7424:
7420:
7414:
7407:
7403:
7398:
7393:
7389:
7386:
7383:
7376:
7372:
7368:
7365:
7361:
7346:
7345:
7333:
7327:
7323:
7318:
7314:
7311:
7308:
7304:
7298:
7295:
7288:
7284:
7280:
7277:
7273:
7267:
7260:
7256:
7251:
7247:
7244:
7240:
7236:
7233:
7230:
7227:
7223:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7192:
7187:
7184:
7177:
7173:
7169:
7166:
7162:
7156:
7149:
7145:
7140:
7136:
7129:
7125:
7120:
7115:
7107:
7103:
7099:
7096:
7092:
7088:
7083:
7079:
7068:
7055:
7051:
7046:
7042:
7039:
7036:
7033:
7030:
7027:
7022:
7018:
7014:
7011:
7008:
7005:
7002:
6999:
6996:
6992:
6988:
6985:
6982:
6979:
6976:
6973:
6968:
6964:
6960:
6957:
6954:
6951:
6948:
6945:
6940:
6936:
6931:
6918:
6915:
6910:
6907:
6903:explosion time
6890:
6880:
6879:
6867:
6864:
6861:
6841:
6838:
6835:
6830:
6826:
6805:
6802:
6799:
6796:
6793:
6781:
6780:
6779:
6778:
6763:
6759:
6754:
6749:
6745:
6742:
6739:
6736:
6733:
6730:
6727:
6724:
6720:
6706:
6705:
6686:
6682:
6677:
6656:
6653:
6648:
6644:
6619:
6605:
6604:
6593:
6590:
6585:
6581:
6576:
6571:
6567:
6563:
6560:
6555:
6551:
6547:
6544:
6541:
6538:
6535:
6530:
6526:
6522:
6493:
6490:
6487:
6483:
6477:
6473:
6469:
6449:
6429:
6426:
6423:
6397:
6391:
6387:
6380:
6377:
6374:
6371:
6351:
6348:
6345:
6342:
6339:
6319:
6291:
6287:
6283:
6271:
6270:
6259:
6256:
6253:
6250:
6247:
6244:
6240:
6237:
6234:
6231:
6228:
6225:
6221:
6217:
6213:
6210:
6207:
6203:
6199:
6196:
6193:
6190:
6187:
6184:
6181:
6177:
6173:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6133:
6120:the condition
6109:
6106:
6103:
6100:
6097:
6094:
6074:
6071:
6068:
6048:
6045:
6042:
6022:
6002:
5984:
5983:
5969:
5964:
5940:
5935:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5880:
5877:
5855:
5850:
5845:
5842:
5822:
5817:
5812:
5807:
5802:
5797:
5792:
5789:
5786:
5783:
5780:
5777:
5772:
5767:
5762:
5759:
5749:
5736:
5731:
5709:
5687:
5682:
5660:
5646:
5645:
5632:
5628:
5623:
5619:
5614:
5610:
5606:
5603:
5600:
5597:
5594:
5589:
5585:
5580:
5564:
5561:
5560:
5559:
5548:
5545:
5542:
5539:
5535:
5531:
5527:
5520:
5515:
5508:
5504:
5499:
5493:
5488:
5484:
5479:
5474:
5448:
5435:
5411:
5407:) ↦
5389:
5388:
5377:
5374:
5371:
5366:
5362:
5351:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5307:
5303:
5298:
5293:
5290:
5287:
5282:
5278:
5274:
5271:
5268:
5265:
5261:
5256:
5253:
5250:
5245:
5241:
5237:
5234:
5231:
5226:
5222:
5217:
5202:
5201:
5190:
5187:
5184:
5181:
5176:
5169:
5164:
5159:
5155:
5149:
5143:
5117:
5103:
5102:
5091:
5086:
5081:
5074:
5071:
5067:
5062:
5056:
5051:
5048:
5045:
5042:
5039:
5035:
5031:
5026:
5021:
5016:
5012:
4981:
4980:
4969:
4965:
4961:
4958:
4955:
4951:
4947:
4944:
4939:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4893:
4888:
4883:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4837:
4825:
4814:
4809:
4803:
4799:
4795:
4791:
4788:
4783:
4778:
4775:
4770:
4765:
4762:
4759:
4756:
4753:
4750:
4745:
4740:
4735:
4730:
4727:
4724:
4721:
4718:
4715:
4710:
4682:
4681:
4670:
4665:
4662:
4659:
4654:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4626:
4621:
4616:
4613:
4603:
4592:
4587:
4582:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4554:
4549:
4544:
4541:
4496:
4493:
4479:
4476:
4473:
4449:
4446:
4443:
4438:
4434:
4429:
4408:
4405:
4402:
4397:
4393:
4388:
4367:
4363:
4358:
4352:
4349:
4329:
4309:
4306:
4303:
4292:
4291:
4280:
4277:
4274:
4269:
4265:
4260:
4255:
4252:
4249:
4246:
4243:
4240:
4235:
4231:
4227:
4224:
4221:
4218:
4214:
4209:
4206:
4203:
4200:
4197:
4194:
4189:
4185:
4181:
4178:
4175:
4172:
4169:
4166:
4161:
4157:
4152:
4135:
4134:As rough paths
4132:
4118:
4117:
4106:
4103:
4100:
4096:
4093:
4089:
4085:
4082:
4079:
4076:
4073:
4068:
4064:
4060:
4056:
4052:
4047:
4042:
4038:
4034:
4031:
4026:
4022:
4018:
4015:
4012:
4009:
4004:
4000:
3996:
3993:
3967:
3964:
3941:
3919:
3914:
3892:
3871:
3849:
3829:
3826:
3823:
3818:
3813:
3809:
3805:
3802:
3782:
3779:
3776:
3773:
3753:
3743:
3742:
3730:
3723:
3720:
3712: in
3709:
3706:
3701:
3697:
3691:
3688:
3685:
3681:
3676:
3672:
3669:
3666:
3663:
3660:
3657:
3630:
3606:
3601:
3598:
3595:
3592:
3588:
3584:
3581:
3576:
3572:
3568:
3563:
3559:
3555:
3552:
3549:
3529:
3518:
3517:
3506:
3502:
3498:
3495:
3492:
3489:
3486:
3481:
3477:
3473:
3469:
3465:
3460:
3455:
3451:
3447:
3444:
3439:
3435:
3431:
3428:
3425:
3422:
3417:
3413:
3409:
3406:
3383:
3380:
3377:
3374:
3371:
3351:
3331:
3328:
3325:
3322:
3302:
3299:
3296:
3291:
3286:
3282:
3278:
3275:
3255:
3233:
3230:
3227:
3223:
3217:
3213:
3209:
3189:
3169:
3164:
3158:
3152:
3130:
3126:
3122:
3117:
3113:
3092:
3069:
3066:
3063:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3004:is linear and
2993:
2988:
2984:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2942:
2939:
2936:
2925:
2924:
2912:
2901:vector bundles
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2821:
2809:
2789:
2766:
2763:
2760:
2757:
2754:
2743:
2742:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2703:
2678:
2654:
2648:
2623:
2619:
2594:
2591:
2588:
2585:
2582:
2579:
2573:
2570:
2539:
2534:
2529:
2526:
2522:
2516:
2510:
2504:
2484:
2481:
2478:
2471:
2466:
2461:
2458:
2454:
2448:
2442:
2436:
2433:
2428:
2423:
2420:
2417:
2397:
2377:
2361:
2358:
2342:
2341:
2330:
2325:
2321:
2316:
2309:
2305:
2301:
2298:
2294:
2287:
2283:
2279:
2276:
2271:
2267:
2262:
2223:
2202:
2201:
2188:
2184:
2179:
2174:
2171:
2168:
2164:
2159:
2156:
2151:
2147:
2142:
2105:
2104:
2093:
2088:
2084:
2079:
2072:
2068:
2064:
2061:
2058:
2054:
2047:
2043:
2039:
2036:
2031:
2027:
2022:
1988:
1984:
1979:
1973:
1970:
1952:
1931:
1906:
1882:
1862:
1837:
1825:
1824:
1813:
1808:
1804:
1799:
1794:
1791:
1788:
1783:
1779:
1775:
1772:
1767:
1764:
1761:
1756:
1752:
1748:
1745:
1741:
1737:
1734:
1731:
1726:
1722:
1718:
1715:
1710:
1707:
1704:
1699:
1695:
1691:
1686:
1682:
1678:
1673:
1670:
1667:
1663:
1645:Wiener process
1632:
1621:
1620:
1609:
1604:
1600:
1595:
1590:
1587:
1584:
1579:
1575:
1571:
1568:
1565:
1562:
1558:
1553:
1550:
1547:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1514:
1481:
1477:
1454:
1450:
1421:
1418:
1346:
1343:
1340:
1337:
1334:
1331:
1310:
1290:
1268:
1264:
1243:
1240:
1237:
1232:
1228:
1207:
1204:
1201:
1198:
1178:
1154:
1151:
1148:
1137:
1136:
1124:
1121:
1118:
1115:
1110:
1106:
1102:
1099:
1096:
1093:
1090:
1087:
1082:
1078:
1072:
1067:
1064:
1061:
1057:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1023:
1019:
1013:
1010:
1007:
1004:
1000:
965:
964:Use in physics
962:
945:
942:
930:Wiener process
926:Wiener process
918:
915:
826:
823:
778:
775:
763:Lévy processes
759:jump processes
755:semimartingale
706:
705:
703:
702:
695:
688:
680:
677:
676:
672:
671:
666:
661:
656:
654:Ernst Lindelöf
651:
646:
641:
636:
631:
626:
624:Joseph Fourier
621:
616:
611:
609:Leonhard Euler
606:
601:
596:
590:
589:
586:
585:
582:
581:
577:
576:
573:
572:
567:
562:
555:
554:
549:
544:
539:
534:
529:
528:
527:
517:
512:
511:
510:
503:Finite element
500:
496:Crank–Nicolson
487:
482:
476:
471:
467:
466:
463:
462:
459:
458:
454:
453:
448:
443:
435:
430:
417:
415:Phase portrait
412:
407:
406:
405:
403:Cauchy problem
400:
395:
390:
380:
374:
373:
371:General topics
370:
369:
366:
365:
362:
361:
356:
351:
346:
340:
337:
336:
333:
332:
328:
327:
323:
322:
317:
316:
315:
304:
303:
302:
293:
290:
289:
286:
285:
280:
279:
278:
277:
270:
269:
264:
258:
255:
254:
250:
249:
247:
246:
244:Nonhomogeneous
237:
232:
229:
223:
222:
221:
213:
212:
208:
207:
205:
204:
199:
194:
189:
184:
179:
174:
168:
163:
160:
159:
156:
155:
154:Classification
151:
150:
141:
140:
139:
138:
133:
125:
124:
118:
117:
116:
115:
110:
105:
97:
96:
90:
89:
88:
87:
82:
76:
71:
66:
58:
57:
55:
54:
49:
43:
38:
35:
34:
31:
30:
26:
25:
15:
9:
6:
4:
3:
2:
10636:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10606:
10604:
10589:
10586:
10584:
10581:
10580:
10577:
10571:
10568:
10566:
10563:
10561:
10558:
10556:
10553:
10551:
10548:
10546:
10543:
10541:
10538:
10536:
10533:
10531:
10528:
10526:
10523:
10521:
10518:
10516:
10513:
10511:
10508:
10506:
10503:
10501:
10498:
10496:
10493:
10491:
10488:
10487:
10485:
10481:
10473:
10470:
10468:
10465:
10464:
10463:
10460:
10458:
10455:
10453:
10450:
10448:
10445:
10443:
10442:Stopping time
10440:
10436:
10433:
10432:
10431:
10428:
10426:
10423:
10421:
10418:
10416:
10413:
10411:
10408:
10406:
10403:
10401:
10398:
10396:
10393:
10391:
10388:
10386:
10383:
10381:
10378:
10376:
10373:
10371:
10368:
10366:
10363:
10361:
10358:
10356:
10353:
10351:
10348:
10346:
10343:
10341:
10338:
10336:
10333:
10331:
10328:
10326:
10323:
10321:
10318:
10316:
10313:
10311:
10308:
10306:
10303:
10301:
10298:
10296:
10293:
10292:
10290:
10286:
10280:
10277:
10275:
10272:
10270:
10267:
10265:
10262:
10260:
10257:
10256:
10254:
10252:
10248:
10241:
10237:
10233:
10232:Hewitt–Savage
10229:
10225:
10221:
10217:
10216:Zero–one laws
10214:
10212:
10209:
10207:
10204:
10202:
10199:
10197:
10194:
10192:
10189:
10187:
10184:
10182:
10179:
10177:
10174:
10172:
10169:
10167:
10164:
10163:
10161:
10157:
10151:
10148:
10146:
10143:
10141:
10138:
10136:
10133:
10131:
10128:
10126:
10123:
10121:
10118:
10116:
10113:
10111:
10108:
10106:
10103:
10101:
10098:
10096:
10093:
10091:
10088:
10086:
10083:
10081:
10078:
10077:
10075:
10071:
10065:
10062:
10060:
10057:
10055:
10052:
10050:
10047:
10045:
10042:
10040:
10037:
10036:
10034:
10032:
10028:
10022:
10019:
10017:
10014:
10012:
10009:
10007:
10004:
10003:
10001:
9999:
9995:
9989:
9986:
9984:
9981:
9979:
9976:
9974:
9971:
9969:
9966:
9964:
9961:
9959:
9956:
9954:
9951:
9949:
9946:
9944:
9941:
9939:
9936:
9934:
9931:
9929:
9926:
9924:
9921:
9919:
9916:
9914:
9913:Black–Scholes
9911:
9909:
9906:
9904:
9901:
9899:
9896:
9895:
9893:
9891:
9887:
9881:
9878:
9876:
9873:
9871:
9868:
9866:
9863:
9861:
9858:
9856:
9853:
9852:
9850:
9848:
9844:
9838:
9835:
9833:
9830:
9826:
9823:
9821:
9818:
9817:
9816:
9815:Point process
9813:
9811:
9808:
9806:
9803:
9801:
9798:
9794:
9791:
9789:
9786:
9785:
9784:
9781:
9779:
9776:
9774:
9773:Gibbs measure
9771:
9769:
9766:
9764:
9761:
9760:
9758:
9754:
9748:
9745:
9743:
9740:
9738:
9735:
9733:
9730:
9728:
9725:
9721:
9718:
9716:
9713:
9711:
9708:
9706:
9703:
9702:
9701:
9698:
9696:
9693:
9691:
9688:
9686:
9683:
9681:
9678:
9677:
9675:
9671:
9665:
9662:
9660:
9657:
9655:
9652:
9650:
9647:
9645:
9642:
9640:
9637:
9635:
9632:
9630:
9627:
9625:
9622:
9618:
9615:
9613:
9610:
9609:
9608:
9605:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9573:
9570:
9568:
9565:
9563:
9562:Itô diffusion
9560:
9558:
9555:
9553:
9550:
9548:
9545:
9543:
9540:
9538:
9537:Gamma process
9535:
9533:
9530:
9528:
9525:
9523:
9520:
9518:
9515:
9513:
9510:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9486:
9483:
9481:
9478:
9476:
9473:
9471:
9468:
9466:
9463:
9462:
9461:
9458:
9454:
9451:
9450:
9449:
9446:
9444:
9441:
9439:
9436:
9435:
9433:
9431:
9427:
9419:
9416:
9414:
9411:
9409:
9408:Self-avoiding
9406:
9404:
9401:
9400:
9399:
9396:
9394:
9393:Moran process
9391:
9389:
9386:
9384:
9381:
9379:
9376:
9374:
9371:
9369:
9366:
9364:
9361:
9360:
9358:
9356:
9355:Discrete time
9352:
9348:
9341:
9336:
9334:
9329:
9327:
9322:
9321:
9318:
9311:
9307:
9303:
9299:
9295:
9290:
9285:
9281:
9277:
9273:
9269:
9264:
9260:
9256:
9252:
9247:
9243:
9241:981-02-3543-7
9237:
9233:
9228:
9224:
9220:
9216:
9212:
9207:
9203:
9197:
9193:
9188:
9184:
9179:
9175:
9169:
9164:
9163:
9156:
9152:
9147:
9144:
9140:
9139:
9126:
9120:
9116:
9109:
9100:
9094:
9088:
9086:
9077:
9071:
9067:
9060:
9052:
9046:
9038:
9032:
9024:
9017:
9011:
9007:
9001:
8993:
8989:
8985:
8981:
8977:
8973:
8967:
8961:
8955:
8946:
8941:
8937:
8933:
8929:
8922:
8916:
8910:
8904:
8897:
8891:
8885:
8879:
8873:
8871:
8862:
8858:
8854:
8850:
8846:
8842:
8835:
8828:
8822:
8820:
8818:
8811:
8807:
8801:
8799:
8791:
8785:
8779:
8773:
8771:
8769:
8760:
8756:
8752:
8748:
8744:
8740:
8736:
8732:
8728:
8721:
8715:
8708:
8700:
8698:3-540-04758-1
8694:
8690:
8686:
8680:
8678:
8676:
8674:
8672:
8670:
8668:
8658:
8656:
8654:
8645:
8641:
8637:
8635:0-521-77594-9
8631:
8627:
8623:
8619:
8615:
8611:
8607:
8601:
8599:
8597:
8595:
8593:
8591:
8586:
8576:
8573:
8571:
8568:
8566:
8563:
8561:
8558:
8556:
8553:
8551:
8548:
8546:
8543:
8541:
8538:
8537:
8531:
8529:
8525:
8521:
8517:
8514:etc. and the
8513:
8509:
8505:
8500:
8499:supersymmetry
8496:
8490:
8458:
8454:
8447:
8444:
8439:
8435:
8431:
8428:
8425:
8417:
8414:
8410:
8406:
8401:
8397:
8389:
8388:
8387:
8373:
8348:
8344:
8337:
8334:
8329:
8325:
8299:
8295:
8286:
8283:
8275:
8272:
8267:
8263:
8250:
8249:
8248:
8229:
8225:
8221:
8213:
8209:
8202:
8199:
8196:
8183:
8179:
8172:
8169:
8166:
8161:
8157:
8144:
8143:
8142:
8128:
8103:
8099:
8085:
8081:
8074:
8071:
8068:
8059:
8050:
8046:
8038:
8035:
8026:
8022:
8015:
8010:
8007:
8002:
7994:
7990:
7983:
7980:
7976:
7972:
7967:
7963:
7950:
7949:
7923:
7917:
7912:
7899:
7895:
7891:
7885:
7879:
7872:
7871:
7870:
7845:
7841:
7834:
7831:
7826:
7822:
7813:
7810:
7806:
7802:
7797:
7793:
7785:
7784:
7783:
7764:
7760:
7756:
7748:
7744:
7737:
7734:
7729:
7725:
7712:
7711:
7710:
7696:
7671:
7667:
7653:
7649:
7642:
7639:
7636:
7623:
7619:
7611:
7608:
7599:
7595:
7588:
7583:
7580:
7575:
7570:
7566:
7553:
7552:
7531:
7525:
7521:
7509:
7503:
7498:
7491:
7487:
7482:
7478:
7475:
7466:
7460:
7453:
7445:
7441:
7434:
7428:
7422:
7418:
7412:
7405:
7401:
7396:
7391:
7387:
7384:
7381:
7374:
7370:
7366:
7363:
7351:
7350:
7349:
7331:
7325:
7321:
7309:
7296:
7293:
7286:
7282:
7278:
7275:
7265:
7258:
7254:
7249:
7245:
7242:
7228:
7214:
7208:
7205:
7199:
7193:
7185:
7182:
7175:
7171:
7167:
7164:
7154:
7147:
7143:
7138:
7134:
7127:
7123:
7118:
7113:
7105:
7101:
7097:
7094:
7086:
7081:
7077:
7069:
7053:
7049:
7034:
7028:
7025:
7020:
7016:
7009:
7003:
6997:
6994:
6980:
6974:
6971:
6966:
6962:
6955:
6949:
6943:
6938:
6934:
6921:
6920:
6914:
6906:
6904:
6888:
6865:
6859:
6839:
6828:
6824:
6797:
6794:
6783:
6782:
6761:
6757:
6752:
6740:
6737:
6734:
6728:
6725:
6722:
6710:
6709:
6708:
6707:
6704:
6684:
6680:
6675:
6654:
6646:
6642:
6633:
6632:
6631:
6617:
6610:
6591:
6588:
6583:
6579:
6574:
6569:
6565:
6561:
6553:
6549:
6545:
6542:
6536:
6533:
6528:
6524:
6520:
6513:
6512:
6511:
6509:
6491:
6488:
6485:
6475:
6471:
6447:
6427:
6424:
6421:
6413:
6395:
6372:
6369:
6349:
6340:
6337:
6317:
6308:
6306:
6285:
6257:
6254:
6251:
6248:
6245:
6242:
6238:
6235:
6232:
6229:
6226:
6223:
6219:
6211:
6208:
6205:
6194:
6191:
6188:
6182:
6179:
6168:
6165:
6162:
6156:
6153:
6147:
6144:
6141:
6135:
6123:
6122:
6121:
6104:
6101:
6098:
6092:
6072:
6069:
6066:
6046:
6043:
6040:
6020:
6000:
5991:
5989:
5967:
5938:
5906:
5896:
5891:
5878:
5875:
5853:
5843:
5840:
5815:
5805:
5800:
5787:
5784:
5778:
5775:
5770:
5760:
5757:
5750:
5734:
5707:
5685:
5658:
5651:
5650:
5649:
5630:
5626:
5612:
5608:
5604:
5601:
5595:
5592:
5587:
5583:
5570:
5569:
5568:
5546:
5540:
5537:
5533:
5529:
5518:
5506:
5502:
5491:
5486:
5482:
5477:
5464:
5463:
5462:
5460:
5457: ≤
5456:
5451:
5447:
5443:
5440:generated by
5438:
5434:
5431:
5427:
5423:
5419:
5414:
5410:
5406:
5402:
5398:
5394:
5393:almost surely
5375:
5372:
5369:
5364:
5360:
5352:
5338:
5332:
5329:
5326:
5320:
5317:
5305:
5301:
5288:
5285:
5280:
5276:
5269:
5266:
5263:
5251:
5248:
5243:
5239:
5232:
5229:
5224:
5220:
5207:
5206:
5205:
5188:
5182:
5179:
5167:
5157:
5133:
5132:
5131:
5129:
5128:second moment
5125:
5120:
5116:
5112:
5108:
5089:
5084:
5072:
5069:
5065:
5054:
5049:
5046:
5043:
5040:
5037:
5033:
5029:
5024:
5014:
5002:
5001:
5000:
4998:
4995: ∈
4994:
4990:
4986:
4967:
4959:
4956:
4953:
4945:
4942:
4929:
4926:
4923:
4917:
4914:
4908:
4905:
4902:
4896:
4886:
4873:
4870:
4867:
4861:
4858:
4852:
4849:
4846:
4840:
4826:
4812:
4797:
4789:
4786:
4776:
4773:
4760:
4757:
4754:
4748:
4738:
4725:
4722:
4719:
4713:
4699:
4698:
4697:
4695:
4691:
4687:
4668:
4663:
4660:
4657:
4641:
4638:
4635:
4629:
4624:
4614:
4611:
4604:
4590:
4585:
4569:
4566:
4563:
4557:
4552:
4542:
4539:
4532:
4531:
4530:
4528:
4523:
4521:
4517:
4513:
4510:
4507:
4503:
4492:
4474:
4471:
4463:
4444:
4436:
4432:
4403:
4395:
4391:
4365:
4361:
4350:
4304:
4301:
4275:
4267:
4263:
4250:
4247:
4241:
4233:
4229:
4222:
4219:
4216:
4204:
4201:
4195:
4187:
4183:
4176:
4173:
4167:
4159:
4155:
4142:
4141:
4140:
4131:
4129:
4124:
4104:
4101:
4098:
4094:
4091:
4083:
4077:
4071:
4066:
4058:
4045:
4040:
4036:
4032:
4024:
4020:
4013:
4010:
4002:
3998:
3991:
3984:
3983:
3982:
3965:
3962:
3939:
3917:
3890:
3882:
3869:
3847:
3824:
3811:
3807:
3803:
3800:
3777:
3771:
3751:
3728:
3721:
3718:
3704:
3699:
3695:
3689:
3683:
3674:
3670:
3661:
3658:
3648:
3647:
3646:
3644:
3628:
3620:
3604:
3596:
3590:
3586:
3579:
3574:
3570:
3566:
3561:
3553:
3550:
3527:
3504:
3496:
3490:
3484:
3479:
3471:
3458:
3453:
3449:
3445:
3437:
3433:
3426:
3423:
3415:
3411:
3404:
3397:
3396:
3395:
3394:the equation
3381:
3378:
3375:
3372:
3369:
3349:
3326:
3320:
3297:
3284:
3280:
3276:
3273:
3253:
3231:
3228:
3225:
3215:
3211:
3187:
3162:
3128:
3124:
3120:
3115:
3111:
3090:
3081:
3067:
3064:
3061:
3038:
3035:
3026:
3023:
3017:
3011:
2991:
2986:
2982:
2975:
2972:
2966:
2960:
2940:
2937:
2934:
2910:
2902:
2886:
2880:
2874:
2865:
2862:
2859:
2853:
2850:
2847:
2841:
2838:
2835:
2832:
2829:
2822:
2807:
2787:
2780:
2779:
2778:
2761:
2758:
2755:
2729:
2726:
2723:
2717:
2711:
2708:
2705:
2693:
2692:
2691:
2689:
2676:
2652:
2621:
2617:
2608:
2583:
2580:
2577:
2571:
2568:
2557:
2537:
2527:
2524:
2514:
2479:
2476:
2469:
2459:
2456:
2446:
2431:
2421:
2395:
2375:
2367:
2357:
2356:phenomena.
2355:
2351:
2347:
2328:
2323:
2319:
2307:
2303:
2299:
2296:
2285:
2281:
2277:
2274:
2269:
2265:
2252:
2251:
2250:
2249:
2244:
2242:
2237:
2235:
2231:
2226:
2222:
2218:
2214:
2209:
2207:
2186:
2182:
2172:
2169:
2166:
2157:
2154:
2149:
2145:
2132:
2131:
2130:
2129:
2124:
2122:
2116:
2114:
2113:Black–Scholes
2110:
2091:
2086:
2082:
2070:
2066:
2062:
2059:
2056:
2045:
2041:
2037:
2034:
2029:
2025:
2012:
2011:
2010:
2009:
2004:
2002:
1986:
1982:
1971:
1960:
1955:
1951:
1945:
1943:
1939:
1934:
1930:
1926:
1922:
1918:
1914:
1909:
1905:
1901:
1898:
1894:
1890:
1885:
1881:
1877:
1874:
1870:
1865:
1861:
1857:
1853:
1849:
1845:
1840:
1836:
1833:
1830:
1811:
1806:
1802:
1789:
1786:
1781:
1777:
1770:
1765:
1762:
1759:
1754:
1750:
1746:
1743:
1732:
1729:
1724:
1720:
1713:
1708:
1705:
1702:
1697:
1693:
1689:
1684:
1680:
1676:
1671:
1668:
1665:
1661:
1653:
1652:
1651:
1650:
1646:
1630:
1607:
1602:
1598:
1585:
1582:
1577:
1573:
1566:
1563:
1560:
1548:
1545:
1540:
1536:
1529:
1526:
1521:
1517:
1504:
1503:
1502:
1499:
1497:
1479:
1475:
1452:
1448:
1439:
1435:
1431:
1427:
1417:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1370:
1368:
1364:
1358:
1344:
1341:
1335:
1329:
1308:
1288:
1266:
1262:
1241:
1238:
1235:
1230:
1226:
1205:
1202:
1199:
1196:
1176:
1168:
1152:
1149:
1146:
1122:
1116:
1108:
1104:
1094:
1088:
1080:
1076:
1070:
1065:
1062:
1059:
1055:
1051:
1042:
1036:
1030:
1027:
1021:
1008:
1002:
987:
986:
985:
981:
978:
971:
961:
959:
955:
951:
941:
939:
935:
931:
927:
923:
914:
912:
908:
904:
900:
895:
893:
889:
885:
881:
875:
873:
868:
864:
860:
856:
852:
848:
844:
840:
836:
832:
822:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
774:
772:
768:
764:
760:
756:
752:
748:
743:
741:
737:
733:
729:
725:
721:
717:
713:
701:
696:
694:
689:
687:
682:
681:
679:
678:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
591:
584:
583:
579:
578:
571:
568:
566:
563:
561:
558:
557:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
526:
523:
522:
521:
518:
516:
515:Finite volume
513:
509:
506:
505:
504:
501:
497:
491:
488:
486:
483:
481:
477:
475:
472:
469:
468:
461:
460:
452:
449:
447:
444:
440:
436:
434:
431:
429:
425:
421:
418:
416:
413:
411:
408:
404:
401:
399:
396:
394:
391:
389:
386:
385:
384:
381:
379:
376:
375:
368:
367:
360:
357:
355:
352:
350:
347:
345:
342:
341:
335:
334:
330:
329:
321:
318:
314:
311:
310:
309:
306:
305:
301:
295:
294:
288:
287:
276:
273:
272:
268:
265:
263:
260:
259:
257:
256:
252:
251:
245:
241:
238:
236:
233:
230:
228:
225:
224:
220:
217:
216:
215:
214:
210:
209:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
169:
167:
166:
158:
157:
153:
152:
147:
137:
134:
132:
129:
128:
127:
126:
123:
120:
119:
114:
111:
109:
106:
104:
101:
100:
99:
98:
95:
92:
91:
86:
83:
81:
77:
75:
72:
70:
67:
65:
62:
61:
60:
59:
53:
50:
48:
45:
44:
42:
41:
33:
32:
28:
27:
24:
21:
20:
10500:Econometrics
10462:Wiener space
10429:
10350:Itô integral
10251:Inequalities
10140:Self-similar
10110:Gauss–Markov
10100:Exchangeable
10080:Càdlàg paths
10016:Risk process
9968:LIBOR market
9837:Random graph
9832:Random field
9644:Superprocess
9582:Lévy process
9577:Jump process
9552:Hunt process
9388:Markov chain
9271:
9267:
9250:
9231:
9222:
9210:
9191:
9182:
9161:
9150:
9114:
9108:
9099:
9065:
9059:
9045:
9031:
9022:
9016:
9000:
8983:
8979:
8966:
8954:
8935:
8931:
8921:
8909:
8896:
8884:
8844:
8840:
8834:
8784:
8734:
8730:
8720:
8707:
8688:
8613:
8492:
8316:
8246:
8120:
7868:
7781:
7688:
7347:
6912:
6902:
6881:
6702:
6608:
6606:
6507:
6506:is called a
6309:
6304:
6272:
5992:
5985:
5647:
5566:
5458:
5454:
5449:
5445:
5441:
5436:
5432:
5421:
5420:) such that
5417:
5412:
5408:
5404:
5400:
5396:
5390:
5203:
5123:
5118:
5114:
5110:
5106:
5104:
4996:
4992:
4988:
4984:
4982:
4693:
4689:
4683:
4526:
4524:
4519:
4515:
4511:
4501:
4498:
4293:
4137:
4119:
3861:
3744:
3642:
3619:differential
3519:
3313:the process
3082:
2926:
2777:, such that
2744:
2668:
2363:
2354:market abuse
2343:
2245:
2240:
2238:
2233:
2229:
2224:
2220:
2216:
2212:
2210:
2203:
2125:
2117:
2106:
2005:
1953:
1949:
1946:
1936:is called a
1932:
1928:
1924:
1920:
1916:
1912:
1907:
1903:
1899:
1892:
1888:
1883:
1879:
1875:
1863:
1859:
1855:
1848:Itô integral
1838:
1834:
1826:
1622:
1500:
1423:
1371:
1359:
1138:
982:
973:
947:
920:
896:
876:
867:Itô calculus
859:Itô integral
851:Stratonovich
847:Itô calculus
828:
819:Stratonovich
780:
744:
732:stock prices
715:
711:
709:
659:Émile Picard
644:Martin Kutta
634:George Green
594:Isaac Newton
426: /
422: /
307:
242: /
108:Chaos theory
10545:Ruin theory
10483:Disciplines
10355:Itô's lemma
10130:Predictable
9805:Percolation
9788:Potts model
9783:Ising model
9747:White noise
9705:Differences
9567:Itô process
9507:Cox process
9403:Loop-erased
9398:Random walk
9268:SIAM Review
9023:Stochastics
6784:on the set
4506:dimensional
4462:rough paths
2241:Marcus type
1873:expectation
1394:Monte Carlo
835:white noise
825:Terminology
747:white noise
552:Runge–Kutta
297:Difference
240:Homogeneous
52:Engineering
10603:Categories
10555:Statistics
10335:Filtration
10236:Kolmogorov
10220:Blumenthal
10145:Stationary
10085:Continuous
10073:Properties
9958:Hull–White
9700:Martingale
9587:Local time
9475:Fractional
9453:pure birth
8581:References
8508:turbulence
6013:. Suppose
5430:filtration
4696:such that
3903:onto full
3641:. It is a
2745:is a pair
1643:denotes a
968:See also:
843:Kiyosi Itô
811:Kiyosi Itô
777:Background
669:John Crank
470:Inspection
424:Asymptotic
308:Stochastic
227:Autonomous
202:Non-linear
192:Fractional
10467:Classical
9480:Geometric
9470:Excursion
9284:CiteSeerX
9259:1109-2769
9010:1469-7688
8751:1040-7294
8528:crackling
8426:α
8415:−
8276:α
8222:∘
8170:α
7981:α
7896:∫
7811:−
7757:∘
7483:∫
7435:−
7397:∫
7388:
7360:Φ
7294:−
7272:Φ
7250:∫
7206:−
7183:−
7161:Φ
7139:∫
7091:Φ
6889:ζ
6866:ζ
6863:→
6837:∂
6834:→
6801:∞
6795:ζ
6758:ζ
6729:α
6681:ζ
6655:ζ
6652:↗
6643:ζ
6618:ζ
6609:life time
6537:α
6492:ζ
6422:ζ
6390:¯
6379:→
6376:Ω
6370:ζ
6347:→
6344:Ω
6318:α
6286:⋅
6252:≤
6246:≤
6233:∈
6209:−
6180:≤
6157:α
6154:−
6136:α
6070:⊂
6044:≥
6021:α
6001:α
5988:manifolds
5879:
5844:⊂
5788:
5782:→
5776:×
5758:α
5596:α
5544:∞
5483:∫
5321:∈
5270:σ
5233:μ
5186:∞
5066:σ
5034:∑
5015:σ
4957:−
4943:≤
4918:σ
4915:−
4897:σ
4862:μ
4859:−
4841:μ
4774:≤
4749:σ
4714:μ
4661:×
4648:→
4630:×
4612:σ
4576:→
4558:×
4540:μ
4478:Ω
4475:∈
4472:ω
4445:ω
4404:ω
4348:Ω
4328:Ω
4308:Ω
4305:∈
4302:ω
4276:ω
4242:ω
4223:σ
4196:ω
4177:μ
4168:ω
4102:≥
4084:∘
4037:∫
3966:^
3817:∞
3804:∈
3722:^
3708:∞
3690:ζ
3687:↗
3671:⊂
3665:∞
3659:ζ
3583:→
3497:∘
3459:τ
3450:∫
3416:τ
3382:ζ
3376:τ
3373:≤
3350:τ
3290:∞
3277:∈
3254:ζ
3232:ζ
3180:-adapted
3065:∈
3054:for each
3030:Γ
3027:∈
3018:⋅
2979:→
2938:∈
2927:For each
2872:↦
2845:→
2839:×
2724:∘
2590:∞
2584:∪
2572:^
2528:∈
2460:∈
2419:Ω
2304:σ
2278:μ
2173:σ
2158:μ
2063:σ
2038:μ
1969:Ω
1852:heuristic
1771:σ
1751:∫
1714:μ
1694:∫
1677:−
1567:σ
1530:μ
1480:α
1476:ξ
1453:α
1449:ξ
1342:∝
1267:α
1263:ξ
1236:∈
1231:α
1200:∈
1150:∈
1109:α
1105:ξ
1081:α
1060:α
1056:∑
872:manifolds
410:Wronskian
388:Dirichlet
131:Economics
74:Chemistry
64:Astronomy
10588:Category
10472:Abstract
10006:Bühlmann
9612:Compound
9221:(2004).
8806:Brigo D.
8687:(2003).
8644:42874839
8612:(2000).
8534:See also
8039:′
7612:′
6310:Suppose
5868:, where
5391:has a P-
4999:, where
4983:for all
4320:, where
2953:the map
2690:written
2558:and let
1897:variance
936:and the
807:Langevin
520:Galerkin
420:Lyapunov
331:Solution
275:Notation
267:Operator
253:Features
172:Ordinary
10095:Ergodic
9983:Vašíček
9825:Poisson
9485:Meander
9312:(2021).
9276:Bibcode
8849:Bibcode
8759:3120200
5428:to the
5426:adapted
5403:,
5395:unique
3981:we get
3617:is the
2605:be the
2111:in the
1915:)
1911:,
1891:)
1887:,
1846:and an
1432:and in
1414:moments
924:or the
882:or the
718:) is a
393:Neumann
177:Partial
85:Geology
80:Biology
69:Physics
10435:Tanaka
10120:Mixing
10115:Markov
9988:Wilkie
9953:Ho–Lee
9948:Heston
9720:Super-
9465:Bridge
9413:Biased
9308:
9286:
9257:
9238:
9198:
9170:
9121:
9072:
9008:
8757:
8749:
8695:
8642:
8632:
8366:where
8317:where
7869:where
7348:where
6273:where
5648:where
5461:, and
3520:holds
1623:where
1139:where
857:. The
580:People
492:
439:Series
197:Linear
36:Fields
10288:Tools
10064:M/M/c
10059:M/M/1
10054:M/G/1
10044:Fluid
9710:Local
8755:S2CID
8504:chaos
6852:with
6607:with
6414:with
6410:be a
5700:and
3860:is a
3362:with
2903:over
2495:with
2346:stock
2109:stock
1871:with
1281:. If
789:and
761:like
728:model
480:Euler
398:Robin
320:Delay
262:Order
235:Exact
161:Types
29:Scope
10240:Lévy
10039:Bulk
9923:Chen
9715:Sub-
9673:Both
9306:ISBN
9255:ISSN
9236:ISBN
9196:ISBN
9168:ISBN
9119:ISBN
9070:ISBN
9006:ISSN
8747:ISSN
8693:ISBN
8640:OCLC
8630:ISBN
8526:and
6798:<
6489:<
6425:>
5538:<
5444:and
5180:<
5105:Let
4991:and
4692:and
4525:Let
3662:<
3379:<
3229:<
2609:and
2215:and
1895:and
1850:. A
861:and
587:List
9820:Cox
9294:doi
8988:doi
8940:doi
8936:402
8857:doi
8739:doi
8622:doi
8524:1/f
7385:exp
6630:if
6510:of
5953:to
5876:Lin
5785:Lin
5424:is
4684:be
3952:on
3680:lim
3621:at
3080:.
2636:be
716:SDE
10605::
10238:,
10234:,
10230:,
10226:,
10222:,
9292:.
9282:.
9272:43
9270:.
9084:^
8982:.
8974:;
8934:.
8930:.
8869:^
8855:.
8845:43
8843:.
8816:^
8797:^
8767:^
8753:.
8745:.
8735:13
8733:.
8729:.
8666:^
8652:^
8638:.
8628:.
8620:.
8608:;
8589:^
8522:,
8510:,
8506:,
6905:.
6307:.
5990:.
5453:,
5130::
5122:,
2208:.
1944:.
1357:.
1169:,
956:,
952:,
913:.
894:.
773:.
742:.
734:,
710:A
10242:)
10218:(
9339:e
9332:t
9325:v
9300:.
9296::
9278::
9261:.
9244:.
9204:.
9176:.
9127:.
9078:.
8994:.
8990::
8984:5
8948:.
8942::
8863:.
8859::
8851::
8829:.
8761:.
8741::
8701:.
8646:.
8624::
8467:)
8464:)
8459:0
8455:X
8451:(
8448:h
8445:+
8440:t
8436:W
8432:+
8429:t
8423:(
8418:1
8411:h
8407:=
8402:t
8398:X
8374:h
8354:)
8349:t
8345:X
8341:(
8338:h
8335:=
8330:t
8326:Y
8300:t
8296:W
8291:d
8287:+
8284:t
8280:d
8273:=
8268:t
8264:Y
8259:d
8230:t
8226:W
8219:)
8214:t
8210:X
8206:(
8203:f
8200:+
8197:t
8193:d
8189:)
8184:t
8180:X
8176:(
8173:f
8167:=
8162:t
8158:X
8153:d
8129:f
8104:t
8100:W
8095:d
8091:)
8086:t
8082:X
8078:(
8075:f
8072:+
8069:t
8065:d
8060:)
8056:)
8051:t
8047:X
8043:(
8036:f
8032:)
8027:t
8023:X
8019:(
8016:f
8011:2
8008:1
8003:+
8000:)
7995:t
7991:X
7987:(
7984:f
7977:(
7973:=
7968:t
7964:X
7959:d
7927:)
7924:s
7921:(
7918:f
7913:s
7909:d
7900:x
7892:=
7889:)
7886:x
7883:(
7880:h
7854:)
7851:)
7846:0
7842:X
7838:(
7835:h
7832:+
7827:t
7823:W
7819:(
7814:1
7807:h
7803:=
7798:t
7794:X
7765:t
7761:W
7754:)
7749:t
7745:X
7741:(
7738:f
7735:=
7730:t
7726:X
7721:d
7697:f
7672:t
7668:W
7663:d
7659:)
7654:t
7650:X
7646:(
7643:f
7640:+
7637:t
7633:d
7629:)
7624:t
7620:X
7616:(
7609:f
7605:)
7600:t
7596:X
7592:(
7589:f
7584:2
7581:1
7576:=
7571:t
7567:X
7562:d
7532:)
7526:s
7522:W
7517:d
7513:)
7510:s
7507:(
7504:b
7499:t
7492:0
7488:t
7479:+
7476:s
7472:d
7467:)
7461:2
7457:)
7454:s
7451:(
7446:2
7442:b
7432:)
7429:s
7426:(
7423:a
7419:(
7413:t
7406:0
7402:t
7392:(
7382:=
7375:0
7371:t
7367:,
7364:t
7332:)
7326:s
7322:W
7317:d
7313:)
7310:s
7307:(
7303:d
7297:1
7287:0
7283:t
7279:,
7276:s
7266:t
7259:0
7255:t
7246:+
7243:s
7239:d
7235:)
7232:)
7229:s
7226:(
7222:d
7218:)
7215:s
7212:(
7209:b
7203:)
7200:s
7197:(
7194:c
7191:(
7186:1
7176:0
7172:t
7168:,
7165:s
7155:t
7148:0
7144:t
7135:+
7128:0
7124:t
7119:X
7114:(
7106:0
7102:t
7098:,
7095:t
7087:=
7082:t
7078:X
7054:t
7050:W
7045:d
7041:)
7038:)
7035:t
7032:(
7029:d
7026:+
7021:t
7017:X
7013:)
7010:t
7007:(
7004:b
7001:(
6998:+
6995:t
6991:d
6987:)
6984:)
6981:t
6978:(
6975:c
6972:+
6967:t
6963:X
6959:)
6956:t
6953:(
6950:a
6947:(
6944:=
6939:t
6935:X
6930:d
6878:.
6860:t
6840:U
6829:t
6825:Y
6804:}
6792:{
6762:n
6753:X
6748:d
6744:)
6741:Y
6738:,
6735:t
6732:(
6726:=
6723:Y
6719:d
6685:n
6676:Y
6647:n
6592:F
6589:=
6584:0
6580:Y
6575:,
6570:t
6566:X
6562:d
6559:)
6554:t
6550:Y
6546:,
6543:t
6540:(
6534:=
6529:t
6525:Y
6521:d
6486:t
6482:)
6476:t
6472:Y
6468:(
6448:U
6428:0
6396:+
6386:R
6373::
6350:U
6341::
6338:F
6290:|
6282:|
6258:,
6255:t
6249:s
6243:0
6239:,
6236:K
6230:y
6227:,
6224:x
6220:,
6216:|
6212:x
6206:y
6202:|
6198:)
6195:K
6192:,
6189:t
6186:(
6183:L
6176:|
6172:)
6169:x
6166:,
6163:s
6160:(
6151:)
6148:y
6145:,
6142:s
6139:(
6132:|
6108:)
6105:K
6102:,
6099:t
6096:(
6093:L
6073:U
6067:K
6047:0
6041:t
5982:.
5968:d
5963:R
5939:n
5934:R
5912:)
5907:d
5902:R
5897:;
5892:n
5887:R
5882:(
5854:d
5849:R
5841:U
5821:)
5816:d
5811:R
5806:;
5801:n
5796:R
5791:(
5779:U
5771:+
5766:R
5761::
5735:d
5730:R
5708:Y
5686:n
5681:R
5659:X
5631:t
5627:X
5622:d
5618:)
5613:t
5609:Y
5605:,
5602:t
5599:(
5593:=
5588:t
5584:Y
5579:d
5547:.
5541:+
5534:]
5530:t
5526:d
5519:2
5514:|
5507:t
5503:X
5498:|
5492:T
5487:0
5478:[
5473:E
5459:t
5455:s
5450:s
5446:B
5442:Z
5437:t
5433:F
5422:X
5418:ω
5416:(
5413:t
5409:X
5405:ω
5401:t
5397:t
5376:;
5373:Z
5370:=
5365:0
5361:X
5339:;
5336:]
5333:T
5330:,
5327:0
5324:[
5318:t
5306:t
5302:B
5297:d
5292:)
5289:t
5286:,
5281:t
5277:X
5273:(
5267:+
5264:t
5260:d
5255:)
5252:t
5249:,
5244:t
5240:X
5236:(
5230:=
5225:t
5221:X
5216:d
5189:.
5183:+
5175:]
5168:2
5163:|
5158:Z
5154:|
5148:[
5142:E
5124:s
5119:s
5115:B
5111:σ
5107:Z
5090:.
5085:2
5080:|
5073:j
5070:i
5061:|
5055:n
5050:1
5047:=
5044:j
5041:,
5038:i
5030:=
5025:2
5020:|
5011:|
4997:R
4993:y
4989:x
4985:t
4968:;
4964:|
4960:y
4954:x
4950:|
4946:D
4938:|
4933:)
4930:t
4927:,
4924:y
4921:(
4912:)
4909:t
4906:,
4903:x
4900:(
4892:|
4887:+
4882:|
4877:)
4874:t
4871:,
4868:y
4865:(
4856:)
4853:t
4850:,
4847:x
4844:(
4836:|
4813:;
4808:)
4802:|
4798:x
4794:|
4790:+
4787:1
4782:(
4777:C
4769:|
4764:)
4761:t
4758:,
4755:x
4752:(
4744:|
4739:+
4734:|
4729:)
4726:t
4723:,
4720:x
4717:(
4709:|
4694:D
4690:C
4669:;
4664:m
4658:n
4653:R
4645:]
4642:T
4639:,
4636:0
4633:[
4625:n
4620:R
4615::
4591:;
4586:n
4581:R
4573:]
4570:T
4567:,
4564:0
4561:[
4553:n
4548:R
4543::
4527:T
4520:B
4516:m
4512:R
4504:-
4502:n
4448:)
4442:(
4437:t
4433:B
4428:d
4407:)
4401:(
4396:t
4392:B
4387:d
4366:P
4362:,
4357:F
4351:,
4279:)
4273:(
4268:t
4264:B
4259:d
4254:)
4251:t
4248:,
4245:)
4239:(
4234:t
4230:X
4226:(
4220:+
4217:t
4213:d
4208:)
4205:t
4202:,
4199:)
4193:(
4188:t
4184:X
4180:(
4174:=
4171:)
4165:(
4160:t
4156:X
4151:d
4105:0
4099:t
4095:,
4092:Z
4088:d
4081:)
4078:X
4075:(
4072:A
4067:X
4063:)
4059:f
4055:d
4051:(
4046:t
4041:0
4033:+
4030:)
4025:0
4021:X
4017:(
4014:f
4011:=
4008:)
4003:t
3999:X
3995:(
3992:f
3963:M
3940:f
3918:+
3913:R
3891:X
3870:M
3848:X
3828:)
3825:M
3822:(
3812:c
3808:C
3801:f
3781:)
3778:X
3775:(
3772:f
3752:P
3729:}
3719:M
3705:=
3700:t
3696:X
3684:t
3675:{
3668:}
3656:{
3629:X
3605:M
3600:)
3597:x
3594:(
3591:f
3587:T
3580:M
3575:x
3571:T
3567::
3562:X
3558:)
3554:f
3551:d
3548:(
3528:P
3505:Z
3501:d
3494:)
3491:X
3488:(
3485:A
3480:X
3476:)
3472:f
3468:d
3464:(
3454:0
3446:+
3443:)
3438:0
3434:x
3430:(
3427:f
3424:=
3421:)
3412:X
3408:(
3405:f
3370:0
3330:)
3327:X
3324:(
3321:f
3301:)
3298:M
3295:(
3285:c
3281:C
3274:f
3226:t
3222:)
3216:t
3212:X
3208:(
3188:M
3168:}
3163:t
3157:F
3151:{
3129:0
3125:x
3121:=
3116:0
3112:X
3091:M
3068:E
3062:e
3042:)
3039:M
3036:T
3033:(
3024:e
3021:)
3015:(
3012:A
2992:M
2987:x
2983:T
2976:E
2973::
2970:)
2967:x
2964:(
2961:A
2941:M
2935:x
2923:.
2911:M
2887:e
2884:)
2881:x
2878:(
2875:A
2869:)
2866:e
2863:,
2860:x
2857:(
2854:,
2851:M
2848:T
2842:E
2836:M
2833::
2830:A
2808:E
2788:Z
2765:)
2762:Z
2759:,
2756:A
2753:(
2730:Z
2727:d
2721:)
2718:X
2715:(
2712:A
2709:=
2706:X
2702:d
2677:M
2653:0
2647:F
2622:0
2618:x
2593:}
2587:{
2581:M
2578:=
2569:M
2538:+
2533:R
2525:t
2521:)
2515:t
2509:F
2503:(
2483:)
2480:P
2477:,
2470:+
2465:R
2457:t
2453:)
2447:t
2441:F
2435:(
2432:,
2427:F
2422:,
2416:(
2396:E
2376:M
2329:.
2324:t
2320:B
2315:d
2308:t
2300:+
2297:t
2293:d
2286:t
2282:R
2275:=
2270:t
2266:R
2261:d
2234:X
2230:X
2225:t
2221:X
2217:σ
2213:μ
2187:t
2183:B
2178:d
2170:+
2167:t
2163:d
2155:=
2150:t
2146:X
2141:d
2092:.
2087:t
2083:B
2078:d
2071:t
2067:X
2060:+
2057:t
2053:d
2046:t
2042:X
2035:=
2030:t
2026:X
2021:d
1987:P
1983:,
1978:F
1972:,
1961:(
1954:t
1950:X
1933:t
1929:X
1925:σ
1921:μ
1917:δ
1913:t
1908:t
1904:X
1902:(
1900:σ
1893:δ
1889:t
1884:t
1880:X
1878:(
1876:μ
1864:t
1860:X
1856:δ
1839:t
1835:X
1812:.
1807:u
1803:B
1798:d
1793:)
1790:u
1787:,
1782:u
1778:X
1774:(
1766:s
1763:+
1760:t
1755:t
1747:+
1744:u
1740:d
1736:)
1733:u
1730:,
1725:u
1721:X
1717:(
1709:s
1706:+
1703:t
1698:t
1690:=
1685:t
1681:X
1672:s
1669:+
1666:t
1662:X
1631:B
1608:,
1603:t
1599:B
1594:d
1589:)
1586:t
1583:,
1578:t
1574:X
1570:(
1564:+
1561:t
1557:d
1552:)
1549:t
1546:,
1541:t
1537:X
1533:(
1527:=
1522:t
1518:X
1513:d
1345:x
1339:)
1336:x
1333:(
1330:g
1309:g
1289:X
1242:X
1239:T
1227:g
1206:X
1203:T
1197:F
1177:X
1153:X
1147:x
1123:,
1120:)
1117:t
1114:(
1101:)
1098:)
1095:t
1092:(
1089:x
1086:(
1077:g
1071:n
1066:1
1063:=
1052:+
1049:)
1046:)
1043:t
1040:(
1037:x
1034:(
1031:F
1028:=
1022:t
1018:d
1012:)
1009:t
1006:(
1003:x
999:d
714:(
699:e
692:t
685:v
498:)
494:(
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