3313: †0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball is bounded below by the harmonic function for all points inside the ball. Similarly, if a supermartingale and a martingale have equivalent expectations for a given time, the history of the supermartingale tends to be bounded below by the history of the martingale. Roughly speaking, the prefix "super-" is consistent because the current observation
65:
2397:
3062: â„ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function for all points inside the ball. Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the
3488:, you can look at the sequence so far and tell if it is time to stop. An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings (for example, he might leave only when he goes broke), but he can't choose to go or stay based on the outcome of games that haven't been played yet.
2187:
2409:, i.e. a group of species that are in a particular trophic level, competing for similar resources in a local area, the number of individuals of any particular species of fixed size is a function of (discrete) time, and may be viewed as a sequence of random variables. This sequence is a martingale under the
98:
wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly
2202:
a marble is randomly selected from the urn and replaced with several more of that same colour. For any given colour, the fraction of marbles in the urn with that colour is a martingale. For example, if currently 95% of the marbles are red then, though the next iteration is more likely to add red
1547:, such that the gambler wins $ 1 if the coin toss outcome is heads and loses $ 1 if the coin toss outcome is tails. The gambler's conditional expected fortune after the next game, given the history, is equal to his present fortune. This sequence is thus a martingale.
1774:
723:
1205:
1433:
1332:
536:
324:
3296:
3045:
2354:
1779:
622:
441:
229:
2203:
marbles than another color, this bias is exactly balanced out by the fact that adding more red marbles alters the fraction much less significantly than adding the same number of non-red marbles would.
3349:
Consider again the gambler who wins $ 1 when a coin comes up heads and loses $ 1 when the coin comes up tails. Suppose now that the coin may be biased, so that it comes up heads with probability
3709:
2182:{\displaystyle {\begin{aligned}E&=p(q/p)^{X_{n}+1}+q(q/p)^{X_{n}-1}\\&=p(q/p)(q/p)^{X_{n}}+q(p/q)(q/p)^{X_{n}}\\&=q(q/p)^{X_{n}}+p(q/p)^{X_{n}}=(q/p)^{X_{n}}=Y_{n}.\end{aligned}}}
1510:
3191:
2940:
2550:
2837:
3649:
934:
878:
1739:
1668:
3578:
2659:
829:
1123:
1029:
990:
964:
3598:
2604:
2577:
789:
4217:
4087:
3531:. That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used.
630:
2410:
2679:
2480:
901:
750:
4752:
1138:
4576:
2701:
but instead an upper or lower bound on the conditional expectation. These generalizations reflect the relationship between martingale theory and
5179:
4172:
1362:
4709:
4689:
1586:= 1, 2, 3, ... } is a martingale. This can be used to show that the gambler's total gain or loss varies roughly between plus or minus the
1528:
A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game of
1248:
449:
237:
5093:
3202:
2951:
2262:
99:
approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a
126:
among others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.
5010:
3743:
3387:. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that
4694:
3718:
which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.
5020:
4704:
577:
396:
184:
5062:
4959:
5249:
5239:
5085:
4777:
4762:
4015:
3993:
3904:
3836:
5458:
5149:
5113:
3753:
3360:
is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale.
5417:
5154:
122:, who also extended the definition to continuous martingales. Much of the original development of the theory was done by
4094:
3654:
5066:
4264:
4165:
3929:
3871:
5219:
4039:
3788:
2451:
1469:
5264:
5070:
5054:
4969:
4797:
4767:
4189:
3778:
3103:
2852:
135:
5169:
5134:
5103:
5098:
4451:
3748:
1002:
2485:
5108:
4737:
4732:
4539:
4436:
4003:
3783:
3374:
is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale.
3367:
is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale.
2783:
3603:
906:
71:
is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy.
5443:
5422:
5199:
5035:
4934:
4919:
4458:
4331:
4247:
4158:
3959:
2731:
17:
5194:
5074:
3342:
Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is
845:
5204:
4051:"Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis"
5209:
4845:
115:
4807:
4391:
4336:
4252:
3954:
3773:
3714:
The concept of a stopped martingale leads to a series of important theorems, including, for example, the
1683:
82:
31:
5139:
1621:
5448:
5144:
5129:
4772:
4742:
4309:
4207:
330:
3537:
2609:
794:
5224:
5025:
4939:
4924:
4855:
4431:
4314:
4212:
3715:
3980:
Entire issue dedicated to
Martingale probability theory (Laurent Mazliak and Glenn Shafer, Editors).
5058:
4944:
4446:
4421:
4366:
3504:
2690:
There are two generalizations of a martingale that also include cases when the current observation
333:
of the next observation, given all the past observations, is equal to the most recent observation.
3527:, ... but not that it is completely determined by the history of the process up to time
1106:
1012:
973:
5359:
5349:
5164:
5040:
4822:
4747:
4561:
4426:
4282:
4237:
4088:"Martingales and Stopping Times: Use of martingales in obtaining bounds and analyzing algorithms"
1451:
the probability measure (with respect to which the expectations are taken). It is possible that
1440:
942:
108:
68:
61:
of the next value in the sequence is equal to the present value, regardless of all prior values.
58:
3949:
1466:
In the Banach space setting the conditional expectation is also denoted in operator notation as
5301:
5229:
4654:
4644:
4488:
3384:
3303:
3828:
5324:
5306:
5286:
5281:
5000:
4832:
4812:
4659:
4602:
4441:
4351:
3728:
2206:
1447:
It is important to note that the property of being a martingale involves both the filtration
718:{\displaystyle \mathbf {E} (Y_{t}\mid \{X_{\tau },\tau \leq s\})=Y_{s}\quad \forall s\leq t.}
5453:
5399:
5354:
5344:
5030:
5005:
4974:
4954:
4792:
4714:
4699:
4566:
3583:
3052:
2685:
2582:
2555:
2457:
2406:
767:
4139:
8:
5394:
5234:
5159:
4964:
4724:
4634:
4524:
2840:
1070:
968:
5364:
5329:
5244:
5214:
5045:
4984:
4979:
4802:
4639:
4304:
4242:
4181:
4031:
Path
Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
2664:
2465:
1346:
886:
840:
735:
728:
This expresses the property that the conditional expectation of an observation at time
559:
549:
142:
104:
54:
38:
5384:
4597:
4514:
4483:
4376:
4356:
4346:
4202:
4197:
4143:
4035:
4011:
3989:
3925:
3900:
3832:
3821:
3332:
the future conditional expectation, and the process tends to decrease in future time.
3085:
the future conditional expectation, and the process tends to increase in future time.
2706:
1593:
1006:
5189:
4840:
4130:
2195:
111:, which is a martingale process, can be used to model the trajectory of such games.
5404:
5291:
5174:
5050:
4787:
4544:
4519:
4468:
4319:
4272:
4135:
4125:
4072:
4062:
3758:
3399:
2739:
2702:
2199:
1456:
87:
4396:
118:
in 1934, though he did not name it. The term "martingale" was introduced later by
5369:
5269:
5254:
5015:
4949:
4627:
4571:
4554:
4299:
4113:
3768:
3738:
3733:
3380:
2743:
2428:
1200:{\displaystyle \mathbf {E} _{\mathbb {P} }(\lVert Y_{t}\rVert _{S})<+\infty ;}
1038:
543:
150:
123:
50:
5184:
4416:
3495:
is defined by requiring only that the occurrence or non-occurrence of the event
5374:
5339:
5259:
4865:
4612:
4529:
4498:
4493:
4473:
4463:
4406:
4381:
4361:
4326:
4294:
4277:
4025:
3798:
4401:
4029:
1460:
5437:
5276:
4817:
4649:
4607:
4549:
4371:
4287:
4227:
3793:
3418:
3412:
2226:
1529:
100:
5334:
5296:
4850:
4782:
4671:
4666:
4478:
4411:
4386:
4222:
3856:
3763:
1455:
could be a martingale with respect to one measure but not another one; the
881:
5379:
4914:
4898:
4893:
4888:
4878:
4681:
4622:
4617:
4581:
4341:
4232:
4067:
4050:
1587:
1522:
1428:{\displaystyle Y_{s}=\mathbf {E} _{\mathbb {P} }(Y_{t}\mid \Sigma _{s}),}
94:. The simplest of these strategies was designed for a game in which the
4077:
2686:
Submartingales, supermartingales, and relationship to harmonic functions
336:
5389:
4929:
4873:
4757:
4710:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4150:
2210:
4883:
3967:
3484:. The intuition behind the definition is that at any particular time
2217:
is thought to be distributed according either to probability density
1597:
1575:
is the gambler's fortune from the prior example. Then the sequence {
1544:
1050:
1327:{\displaystyle \mathbf {E} _{\mathbb {P} }\left(\chi _{F}\right)=0,}
531:{\displaystyle \mathbf {E} (Y_{n+1}\mid X_{1},\ldots ,X_{n})=Y_{n}.}
319:{\displaystyle \mathbf {E} (X_{n+1}\mid X_{1},\ldots ,X_{n})=X_{n}.}
107:
of the bets eventually bankrupts its users due to finite bankrolls.
1088:
146:
46:
3442:, ... is a random variable Ï with the property that for each
3336:
3291:{\displaystyle \operatorname {E} \leq X_{s}\quad \forall s\leq t.}
3040:{\displaystyle \operatorname {E} \geq X_{s}\quad \forall s\leq t.}
2349:{\displaystyle Y_{n}=\prod _{i=1}^{n}{\frac {g(X_{i})}{f(X_{i})}}}
114:
The concept of martingale in probability theory was introduced by
95:
64:
3063:
2709:. Just as a continuous-time martingale satisfies E â
2697:
is not necessarily equal to the future conditional expectation
91:
2396:
1525:, in any number of dimensions, is an example of a martingale.
3972:
Electronic
Journal for History of Probability and Statistics
3864:
Electronic
Journal for History of Probability and Statistics
1674:
with "+" in case of "heads" and "â" in case of "tails". Let
3600:
is a stopping time, then the corresponding stopped process
2450: â„ 0 } is a continuous-time martingale with
4690:
Autoregressive conditional heteroskedasticity (ARCH) model
617:{\displaystyle \mathbf {E} (\vert Y_{t}\vert )<\infty }
436:{\displaystyle \mathbf {E} (\vert Y_{n}\vert )<\infty }
224:{\displaystyle \mathbf {E} (\vert X_{n}\vert )<\infty }
3984:
Baldi, Paolo; Mazliak, Laurent; Priouret, Pierre (1991).
3328:. Consequently, the current observation provides support
3081:. Consequently, the current observation provides support
2198:
contains a number of different-coloured marbles; at each
4218:
Independent and identically distributed random variables
1459:
offers a way to find a measure with respect to which an
3534:
One of the basic properties of martingales is that, if
2411:
unified neutral theory of biodiversity and biogeography
4695:
Autoregressive integrated moving average (ARIMA) model
3346:
a submartingale and a supermartingale is a martingale.
3197:
Likewise, a continuous-time supermartingale satisfies
2362:
If X is actually distributed according to the density
3657:
3606:
3586:
3540:
3205:
3106:
3066:"sub-" is consistent because the current observation
2954:
2855:
2786:
2667:
2612:
2585:
2558:
2488:
2468:
2265:
1777:
1686:
1624:
1472:
1365:
1251:
1141:
1109:
1015:
976:
945:
909:
889:
848:
797:
770:
738:
633:
580:
452:
399:
337:
Martingale sequences with respect to another sequence
240:
187:
3983:
2946:
Likewise, a continuous-time submartingale satisfies
3924:. American Mathematical Society. pp. 372â373.
3820:
3704:{\displaystyle X_{t}^{\tau }:=X_{\min\{\tau ,t\}}}
3703:
3643:
3592:
3572:
3290:
3185:
3039:
2934:
2831:
2673:
2653:
2598:
2571:
2544:
2474:
2348:
2181:
1758:= 1, 2, 3, ... } is a martingale with respect to {
1733:
1662:
1504:
1427:
1326:
1199:
1117:
1023:
984:
958:
928:
895:
872:
823:
783:
744:
717:
616:
530:
435:
318:
223:
3894:
3406:
5435:
4577:Stochastic chains with memory of variable length
3681:
3446:, the occurrence or non-occurrence of the event
3823:Money Management Strategies for Futures Traders
3421:with respect to a sequence of random variables
3337:Examples of submartingales and supermartingales
2381:=1, 2, 3,...} is a martingale with respect to {
1590:of the number of games of coin flipping played.
1505:{\displaystyle \mathbf {E} ^{\Sigma _{s}}Y_{t}}
2435:, then the compensated Poisson process {
4166:
4120:. Monographies des Probabilités (in French).
4118:Bulletin of the American Mathematical Society
4034:(4th ed.). Singapore: World Scientific.
3186:{\displaystyle \operatorname {E} \leq X_{n}.}
2935:{\displaystyle \operatorname {E} \geq X_{n}.}
4048:
3696:
3684:
3253:
3228:
3002:
2977:
1173:
1159:
917:
910:
680:
655:
602:
589:
421:
408:
209:
196:
4085:
3968:"The Splendors and Miseries of Martingales"
4705:Autoregressiveâmoving-average (ARMA) model
4173:
4159:
4114:"Ătude critique de la notion de collectif"
3324:(or equal to) the conditional expectation
3077:(or equal to) the conditional expectation
2545:{\displaystyle M=(M^{(1)},\dots ,M^{(d)})}
1515:
4129:
4076:
4066:
3899:(3rd ed.). Oxford University Press.
2832:{\displaystyle X_{1},X_{2},X_{3},\ldots }
1387:
1260:
1150:
1111:
1017:
978:
30:For the martingale betting strategy, see
4180:
4024:
4002:
3919:
3644:{\displaystyle (X_{t}^{\tau })_{t>0}}
2395:
929:{\displaystyle \lVert \cdot \rVert _{S}}
732:, given all the observations up to time
63:
3818:
3383:of a martingale is a submartingale, by
938:martingale with respect to a filtration
174:, ... that satisfies for any time
57:) for which, at a particular time, the
14:
5436:
5011:Doob's martingale convergence theorems
4093:. University of Athens. Archived from
3857:"The origins of the Word "Martingale""
3854:
3744:Doob's martingale convergence theorems
2221:or to a different probability density
873:{\displaystyle Y:T\times \Omega \to S}
752:, is equal to the observation at time
4763:Constant elasticity of variance (CEV)
4753:ChanâKarolyiâLongstaffâSanders (CKLS)
4154:
4111:
3402:of a martingale is a supermartingale.
834:
119:
3895:Grimmett, G.; Stirzaker, D. (2001).
3711:is also a (sub-/super-) martingale.
1604:of coming up heads and probability
1357:, this last condition is denoted as
764:). The second property implies that
2661:is a one-dimensional martingale in
1734:{\displaystyle Y_{n}=(q/p)^{X_{n}}}
24:
5250:Skorokhod's representation theorem
5031:Law of large numbers (weak/strong)
4049:Richard, Mark; Vecer, Jan (2021).
3580:is a (sub-/super-) martingale and
3273:
3206:
3107:
3022:
2955:
2856:
2400:Software-created martingale series
1663:{\displaystyle X_{n+1}=X_{n}\pm 1}
1481:
1410:
1191:
947:
861:
700:
611:
430:
218:
25:
5470:
5220:Martingale representation theorem
3789:Martingale representation theorem
1600:, i.e., biased, with probability
90:that was popular in 18th-century
5265:Stochastic differential equation
5155:Doob's optional stopping theorem
5150:DoobâMeyer decomposition theorem
3897:Probability and Random Processes
3779:Martingale central limit theorem
3754:DoobâMeyer decomposition theorem
3573:{\displaystyle (X_{t})_{t>0}}
3491:In some contexts the concept of
3398:is a martingale). Similarly, a
2654:{\displaystyle T_{i}(M)=M^{(i)}}
1475:
1381:
1355:Probability and Random Processes
1254:
1144:
824:{\displaystyle X_{1}\dots X_{n}}
635:
582:
454:
401:
242:
189:
5135:Convergence of random variables
5021:FisherâTippettâGnedenko theorem
4131:10.1090/S0002-9904-1939-07089-4
3877:from the original on 2012-01-31
3272:
3021:
1769:= 1, 2, 3, ... }. To show this
1539:is the gambler's fortune after
699:
4733:Binomial options pricing model
4010:. Cambridge University Press.
3913:
3888:
3848:
3812:
3784:Martingale difference sequence
3626:
3607:
3555:
3541:
3454:depends only on the values of
3407:Martingales and stopping times
3256:
3212:
3164:
3113:
3005:
2961:
2913:
2862:
2738: = 0 where Î is the
2646:
2640:
2629:
2623:
2539:
2534:
2528:
2509:
2503:
2495:
2340:
2327:
2319:
2306:
2143:
2128:
2109:
2094:
2072:
2057:
2028:
2013:
2010:
1996:
1974:
1959:
1956:
1942:
1907:
1892:
1864:
1849:
1836:
1785:
1715:
1700:
1419:
1393:
1353:. In Grimmett and Stirzaker's
1297:
1271:
1182:
1156:
864:
791:is measurable with respect to
683:
639:
605:
586:
509:
458:
424:
405:
297:
246:
212:
193:
129:
13:
1:
5200:Kolmogorov continuity theorem
5036:Law of the iterated logarithm
3986:Martingales and Markov Chains
3942:
3505:probabilistically independent
3090:Analogously, a discrete-time
2732:partial differential equation
1598:coin toss outcomes are unfair
5205:Kolmogorov extension theorem
4884:Generalized queueing network
4392:Interacting particle systems
4008:Probability with Martingales
3749:Doob's martingale inequality
1118:{\displaystyle \mathbb {P} }
1024:{\displaystyle \mathbb {P} }
985:{\displaystyle \mathbb {P} }
7:
4337:Continuous-time random walk
4086:Siminelakis, Paris (2010).
3955:Encyclopedia of Mathematics
3920:Bogachev, Vladimir (1998).
3855:Mansuy, Roger (June 2009).
3774:Martingale (betting system)
3721:
2843:random variables satisfying
2452:right-continuous/left-limit
1608: = 1 â
1596:'s martingale: Suppose the
1439:which is a general form of
959:{\displaystyle \Sigma _{*}}
341:More generally, a sequence
32:martingale (betting system)
27:Model in probability theory
10:
5475:
5345:Extreme value theory (EVT)
5145:Doob decomposition theorem
4437:OrnsteinâUhlenbeck process
4208:Chinese restaurant process
3410:
756:(of course, provided that
546:martingale with respect to
364:martingale with respect to
362: ... is said to be a
331:conditional expected value
75:
29:
5459:Paul LĂ©vy (mathematician)
5413:
5317:
5225:Optional stopping theorem
5122:
5084:
5026:Large deviation principle
4993:
4907:
4864:
4831:
4778:HeathâJarrowâMorton (HJM)
4723:
4715:Moving-average (MA) model
4700:Autoregressive (AR) model
4680:
4590:
4525:Hidden Markov model (HMM)
4507:
4459:SchrammâLoewner evolution
4263:
4188:
3827:. Wiley Finance. p.
3716:optional stopping theorem
2366:rather than according to
2253:be the "likelihood ratio"
5140:Doléans-Dade exponential
4970:Progressively measurable
4768:CoxâIngersollâRoss (CIR)
3805:
2772:) is also a martingale.
2759:, the resulting process
2755:and a harmonic function
2705:, that is, the study of
2207:Likelihood-ratio testing
134:A basic definition of a
5360:Mathematical statistics
5350:Large deviations theory
5180:Infinitesimal generator
5041:Maximal ergodic theorem
4960:Piecewise-deterministic
4562:Random dynamical system
4427:Markov additive process
3819:Balsara, N. J. (1992).
3302:In potential theory, a
3051:In potential theory, a
1516:Examples of martingales
1441:conditional expectation
109:Stopped Brownian motion
86:referred to a class of
69:Stopped Brownian motion
59:conditional expectation
5195:KarhunenâLoĂšve theorem
5130:CameronâMartin formula
5094:BurkholderâDavisâGundy
4489:Variance gamma process
4124:(11). Paris: 824â825.
3705:
3645:
3594:
3574:
3304:superharmonic function
3292:
3187:
3041:
2936:
2833:
2726:, a harmonic function
2675:
2655:
2600:
2573:
2546:
2476:
2427: â„ 0 } is a
2401:
2350:
2299:
2183:
1735:
1664:
1506:
1429:
1328:
1201:
1119:
1056:, the random variable
1025:
986:
960:
930:
897:
874:
839:In full generality, a
825:
785:
746:
719:
618:
532:
437:
320:
225:
72:
5325:Actuarial mathematics
5287:Uniform integrability
5282:Stratonovich integral
5210:LĂ©vyâProkhorov metric
5114:MarcinkiewiczâZygmund
5001:Central limit theorem
4603:Gaussian random field
4432:McKeanâVlasov process
4352:Dyson Brownian motion
4213:GaltonâWatson process
3706:
3646:
3595:
3593:{\displaystyle \tau }
3575:
3293:
3188:
3042:
2937:
2834:
2718: = 0 â
2676:
2656:
2601:
2599:{\displaystyle S^{d}}
2574:
2572:{\displaystyle S^{d}}
2547:
2482:-dimensional process
2477:
2399:
2351:
2279:
2184:
1736:
1665:
1507:
1430:
1329:
1202:
1120:
1026:
987:
961:
931:
898:
875:
826:
786:
784:{\displaystyle Y_{n}}
747:
720:
619:
533:
438:
387: ... if for all
321:
226:
67:
5444:Stochastic processes
5400:Time series analysis
5355:Mathematical finance
5240:Reflection principle
4567:Regenerative process
4367:FlemingâViot process
4182:Stochastic processes
4112:Ville, Jean (1939).
4068:10.3390/risks9020031
3988:. Chapman and Hall.
3655:
3604:
3584:
3538:
3203:
3104:
3053:subharmonic function
2952:
2853:
2784:
2665:
2610:
2583:
2556:
2486:
2466:
2407:ecological community
2263:
2213:: A random variable
1775:
1684:
1622:
1470:
1363:
1249:
1139:
1107:
1013:
974:
943:
907:
887:
846:
795:
768:
736:
631:
578:
450:
397:
238:
185:
5395:Stochastic analysis
5235:Quadratic variation
5230:Prokhorov's theorem
5165:FeynmanâKac formula
4635:Markov random field
4283:Birthâdeath process
3672:
3624:
3385:Jensen's inequality
2579:is a martingale in
1071:measurable function
1041:to the filtration ÎŁ
969:probability measure
880:taking values in a
141:is a discrete-time
5365:Probability theory
5245:Skorokhod integral
5215:Malliavin calculus
4798:Korn-Kreer-Lenssen
4682:Time series models
4645:PitmanâYor process
3729:Azuma's inequality
3701:
3658:
3641:
3610:
3590:
3570:
3288:
3183:
3037:
2932:
2829:
2740:Laplacian operator
2707:harmonic functions
2671:
2651:
2606:if each component
2596:
2569:
2542:
2472:
2402:
2346:
2179:
2177:
1731:
1660:
1502:
1425:
1347:indicator function
1324:
1197:
1115:
1021:
1005:of the underlying
982:
956:
926:
893:
870:
841:stochastic process
835:General definition
821:
781:
742:
715:
614:
568:such that for all
560:stochastic process
550:stochastic process
528:
433:
316:
221:
143:stochastic process
105:exponential growth
88:betting strategies
73:
55:stochastic process
39:probability theory
5449:Martingale theory
5431:
5430:
5385:Signal processing
5104:Doob's upcrossing
5099:Doob's martingale
5063:EngelbertâSchmidt
5006:Donsker's theorem
4940:Feller-continuous
4808:RendlemanâBartter
4598:Dirichlet process
4515:Branching process
4484:Telegraph process
4377:Geometric process
4357:Empirical process
4347:Diffusion process
4203:Branching process
4198:Bernoulli process
4017:978-0-521-40605-5
3995:978-1-584-88329-6
3922:Gaussian Measures
3906:978-0-19-857223-7
3838:978-0-471-52215-7
3475:, ...,
2674:{\displaystyle S}
2475:{\displaystyle d}
2458:Wald's martingale
2344:
1463:is a martingale.
1045:, i.e., for each
1009:(Ω, Σ,
1007:probability space
896:{\displaystyle S}
745:{\displaystyle s}
366:another sequence
16:(Redirected from
5466:
5405:Machine learning
5292:Usual hypotheses
5175:Girsanov theorem
5160:Dynkin's formula
4925:Continuous paths
4833:Actuarial models
4773:GarmanâKohlhagen
4743:BlackâKarasinski
4738:BlackâDermanâToy
4725:Financial models
4591:Fields and other
4520:Gaussian process
4469:Sigma-martingale
4273:Additive process
4175:
4168:
4161:
4152:
4151:
4147:
4133:
4108:
4106:
4105:
4099:
4092:
4082:
4080:
4070:
4045:
4021:
3999:
3979:
3963:
3936:
3935:
3917:
3911:
3910:
3892:
3886:
3885:
3883:
3882:
3876:
3861:
3852:
3846:
3845:
3826:
3816:
3759:Local martingale
3710:
3708:
3707:
3702:
3700:
3699:
3671:
3666:
3650:
3648:
3647:
3642:
3640:
3639:
3623:
3618:
3599:
3597:
3596:
3591:
3579:
3577:
3576:
3571:
3569:
3568:
3553:
3552:
3400:concave function
3297:
3295:
3294:
3289:
3271:
3270:
3240:
3239:
3224:
3223:
3192:
3190:
3189:
3184:
3179:
3178:
3163:
3162:
3144:
3143:
3131:
3130:
3046:
3044:
3043:
3038:
3020:
3019:
2989:
2988:
2973:
2972:
2941:
2939:
2938:
2933:
2928:
2927:
2912:
2911:
2893:
2892:
2880:
2879:
2838:
2836:
2835:
2830:
2822:
2821:
2809:
2808:
2796:
2795:
2776:A discrete-time
2703:potential theory
2680:
2678:
2677:
2672:
2660:
2658:
2657:
2652:
2650:
2649:
2622:
2621:
2605:
2603:
2602:
2597:
2595:
2594:
2578:
2576:
2575:
2570:
2568:
2567:
2551:
2549:
2548:
2543:
2538:
2537:
2513:
2512:
2481:
2479:
2478:
2473:
2355:
2353:
2352:
2347:
2345:
2343:
2339:
2338:
2322:
2318:
2317:
2301:
2298:
2293:
2275:
2274:
2188:
2186:
2185:
2180:
2178:
2171:
2170:
2158:
2157:
2156:
2155:
2138:
2124:
2123:
2122:
2121:
2104:
2087:
2086:
2085:
2084:
2067:
2047:
2043:
2042:
2041:
2040:
2023:
2006:
1989:
1988:
1987:
1986:
1969:
1952:
1932:
1928:
1927:
1920:
1919:
1902:
1885:
1884:
1877:
1876:
1859:
1835:
1834:
1816:
1815:
1803:
1802:
1740:
1738:
1737:
1732:
1730:
1729:
1728:
1727:
1710:
1696:
1695:
1669:
1667:
1666:
1661:
1653:
1652:
1640:
1639:
1511:
1509:
1508:
1503:
1501:
1500:
1491:
1490:
1489:
1488:
1478:
1457:Girsanov theorem
1434:
1432:
1431:
1426:
1418:
1417:
1405:
1404:
1392:
1391:
1390:
1384:
1375:
1374:
1333:
1331:
1330:
1325:
1314:
1310:
1309:
1308:
1296:
1295:
1283:
1282:
1265:
1264:
1263:
1257:
1225: <
1206:
1204:
1203:
1198:
1181:
1180:
1171:
1170:
1155:
1154:
1153:
1147:
1124:
1122:
1121:
1116:
1114:
1030:
1028:
1027:
1022:
1020:
991:
989:
988:
983:
981:
965:
963:
962:
957:
955:
954:
935:
933:
932:
927:
925:
924:
902:
900:
899:
894:
879:
877:
876:
871:
830:
828:
827:
822:
820:
819:
807:
806:
790:
788:
787:
782:
780:
779:
751:
749:
748:
743:
724:
722:
721:
716:
698:
697:
667:
666:
651:
650:
638:
623:
621:
620:
615:
601:
600:
585:
537:
535:
534:
529:
524:
523:
508:
507:
489:
488:
476:
475:
457:
442:
440:
439:
434:
420:
419:
404:
325:
323:
322:
317:
312:
311:
296:
295:
277:
276:
264:
263:
245:
230:
228:
227:
222:
208:
207:
192:
151:random variables
103:. However, the
51:random variables
21:
5474:
5473:
5469:
5468:
5467:
5465:
5464:
5463:
5434:
5433:
5432:
5427:
5409:
5370:Queueing theory
5313:
5255:Skorokhod space
5118:
5109:KunitaâWatanabe
5080:
5046:Sanov's theorem
5016:Ergodic theorem
4989:
4985:Time-reversible
4903:
4866:Queueing models
4860:
4856:SparreâAnderson
4846:CramĂ©râLundberg
4827:
4813:SABR volatility
4719:
4676:
4628:Boolean network
4586:
4572:Renewal process
4503:
4452:Non-homogeneous
4442:Poisson process
4332:Contact process
4295:Brownian motion
4265:Continuous time
4259:
4253:Maximal entropy
4184:
4179:
4103:
4101:
4097:
4090:
4042:
4026:Kleinert, Hagen
4018:
4004:Williams, David
3996:
3978:(1). June 2009.
3966:
3948:
3945:
3940:
3939:
3932:
3918:
3914:
3907:
3893:
3889:
3880:
3878:
3874:
3859:
3853:
3849:
3839:
3817:
3813:
3808:
3803:
3769:Markov property
3739:Doob martingale
3734:Brownian motion
3724:
3680:
3676:
3667:
3662:
3656:
3653:
3652:
3629:
3625:
3619:
3614:
3605:
3602:
3601:
3585:
3582:
3581:
3558:
3554:
3548:
3544:
3539:
3536:
3535:
3526:
3516:
3483:
3474:
3467:
3460:
3441:
3434:
3427:
3415:
3409:
3392:
3381:convex function
3339:
3318:
3266:
3262:
3235:
3231:
3219:
3215:
3204:
3201:
3200:
3174:
3170:
3158:
3154:
3139:
3135:
3120:
3116:
3105:
3102:
3101:
3092:supermartingale
3071:
3015:
3011:
2984:
2980:
2968:
2964:
2953:
2950:
2949:
2923:
2919:
2907:
2903:
2888:
2884:
2869:
2865:
2854:
2851:
2850:
2817:
2813:
2804:
2800:
2791:
2787:
2785:
2782:
2781:
2771:
2754:
2744:Brownian motion
2717:
2695:
2688:
2666:
2663:
2662:
2639:
2635:
2617:
2613:
2611:
2608:
2607:
2590:
2586:
2584:
2581:
2580:
2563:
2559:
2557:
2554:
2553:
2527:
2523:
2502:
2498:
2487:
2484:
2483:
2467:
2464:
2463:
2440:
2431:with intensity
2429:Poisson process
2421:
2386:
2375:
2334:
2330:
2323:
2313:
2309:
2302:
2300:
2294:
2283:
2270:
2266:
2264:
2261:
2260:
2252:
2244:is taken. Let
2243:
2234:
2176:
2175:
2166:
2162:
2151:
2147:
2146:
2142:
2134:
2117:
2113:
2112:
2108:
2100:
2080:
2076:
2075:
2071:
2063:
2045:
2044:
2036:
2032:
2031:
2027:
2019:
2002:
1982:
1978:
1977:
1973:
1965:
1948:
1930:
1929:
1915:
1911:
1910:
1906:
1898:
1872:
1868:
1867:
1863:
1855:
1839:
1830:
1826:
1811:
1807:
1792:
1788:
1778:
1776:
1773:
1772:
1763:
1752:
1723:
1719:
1718:
1714:
1706:
1691:
1687:
1685:
1682:
1681:
1648:
1644:
1629:
1625:
1623:
1620:
1619:
1580:
1573:
1562:
1555:
1537:
1518:
1496:
1492:
1484:
1480:
1479:
1474:
1473:
1471:
1468:
1467:
1413:
1409:
1400:
1396:
1386:
1385:
1380:
1379:
1370:
1366:
1364:
1361:
1360:
1343:
1304:
1300:
1291:
1287:
1278:
1274:
1270:
1266:
1259:
1258:
1253:
1252:
1250:
1247:
1246:
1238:
1176:
1172:
1166:
1162:
1149:
1148:
1143:
1142:
1140:
1137:
1136:
1110:
1108:
1105:
1104:
1102:
1085:
1068:
1061:
1044:
1016:
1014:
1011:
1010:
1000:
977:
975:
972:
971:
950:
946:
944:
941:
940:
920:
916:
908:
905:
904:
888:
885:
884:
847:
844:
843:
837:
815:
811:
802:
798:
796:
793:
792:
775:
771:
769:
766:
765:
737:
734:
733:
693:
689:
662:
658:
646:
642:
634:
632:
629:
628:
596:
592:
581:
579:
576:
575:
566:
556:
544:continuous-time
519:
515:
503:
499:
484:
480:
465:
461:
453:
451:
448:
447:
415:
411:
400:
398:
395:
394:
386:
379:
372:
361:
354:
347:
339:
307:
303:
291:
287:
272:
268:
253:
249:
241:
239:
236:
235:
203:
199:
188:
186:
183:
182:
173:
166:
159:
132:
124:Joseph Leo Doob
78:
35:
28:
23:
22:
15:
12:
11:
5:
5472:
5462:
5461:
5456:
5451:
5446:
5429:
5428:
5426:
5425:
5420:
5418:List of topics
5414:
5411:
5410:
5408:
5407:
5402:
5397:
5392:
5387:
5382:
5377:
5375:Renewal theory
5372:
5367:
5362:
5357:
5352:
5347:
5342:
5340:Ergodic theory
5337:
5332:
5330:Control theory
5327:
5321:
5319:
5315:
5314:
5312:
5311:
5310:
5309:
5304:
5294:
5289:
5284:
5279:
5274:
5273:
5272:
5262:
5260:Snell envelope
5257:
5252:
5247:
5242:
5237:
5232:
5227:
5222:
5217:
5212:
5207:
5202:
5197:
5192:
5187:
5182:
5177:
5172:
5167:
5162:
5157:
5152:
5147:
5142:
5137:
5132:
5126:
5124:
5120:
5119:
5117:
5116:
5111:
5106:
5101:
5096:
5090:
5088:
5082:
5081:
5079:
5078:
5059:BorelâCantelli
5048:
5043:
5038:
5033:
5028:
5023:
5018:
5013:
5008:
5003:
4997:
4995:
4994:Limit theorems
4991:
4990:
4988:
4987:
4982:
4977:
4972:
4967:
4962:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4922:
4917:
4911:
4909:
4905:
4904:
4902:
4901:
4896:
4891:
4886:
4881:
4876:
4870:
4868:
4862:
4861:
4859:
4858:
4853:
4848:
4843:
4837:
4835:
4829:
4828:
4826:
4825:
4820:
4815:
4810:
4805:
4800:
4795:
4790:
4785:
4780:
4775:
4770:
4765:
4760:
4755:
4750:
4745:
4740:
4735:
4729:
4727:
4721:
4720:
4718:
4717:
4712:
4707:
4702:
4697:
4692:
4686:
4684:
4678:
4677:
4675:
4674:
4669:
4664:
4663:
4662:
4657:
4647:
4642:
4637:
4632:
4631:
4630:
4625:
4615:
4613:Hopfield model
4610:
4605:
4600:
4594:
4592:
4588:
4587:
4585:
4584:
4579:
4574:
4569:
4564:
4559:
4558:
4557:
4552:
4547:
4542:
4532:
4530:Markov process
4527:
4522:
4517:
4511:
4509:
4505:
4504:
4502:
4501:
4499:Wiener sausage
4496:
4494:Wiener process
4491:
4486:
4481:
4476:
4474:Stable process
4471:
4466:
4464:Semimartingale
4461:
4456:
4455:
4454:
4449:
4439:
4434:
4429:
4424:
4419:
4414:
4409:
4407:Jump diffusion
4404:
4399:
4394:
4389:
4384:
4382:Hawkes process
4379:
4374:
4369:
4364:
4362:Feller process
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4327:Cauchy process
4324:
4323:
4322:
4317:
4312:
4307:
4302:
4292:
4291:
4290:
4280:
4278:Bessel process
4275:
4269:
4267:
4261:
4260:
4258:
4257:
4256:
4255:
4250:
4245:
4240:
4230:
4225:
4220:
4215:
4210:
4205:
4200:
4194:
4192:
4186:
4185:
4178:
4177:
4170:
4163:
4155:
4149:
4148:
4144:Review by Doob
4109:
4083:
4046:
4040:
4022:
4016:
4000:
3994:
3981:
3964:
3944:
3941:
3938:
3937:
3931:978-1470418694
3930:
3912:
3905:
3887:
3847:
3837:
3810:
3809:
3807:
3804:
3802:
3801:
3799:Semimartingale
3796:
3791:
3786:
3781:
3776:
3771:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3725:
3723:
3720:
3698:
3695:
3692:
3689:
3686:
3683:
3679:
3675:
3670:
3665:
3661:
3638:
3635:
3632:
3628:
3622:
3617:
3613:
3609:
3589:
3567:
3564:
3561:
3557:
3551:
3547:
3543:
3525: + 2
3521:
3515: + 1
3511:
3479:
3472:
3465:
3458:
3439:
3432:
3425:
3411:Main article:
3408:
3405:
3404:
3403:
3390:
3377:
3376:
3375:
3368:
3361:
3347:
3338:
3335:
3334:
3333:
3316:
3300:
3299:
3298:
3287:
3284:
3281:
3278:
3275:
3269:
3265:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3238:
3234:
3230:
3227:
3222:
3218:
3214:
3211:
3208:
3195:
3194:
3193:
3182:
3177:
3173:
3169:
3166:
3161:
3157:
3153:
3150:
3147:
3142:
3138:
3134:
3129:
3126:
3123:
3119:
3115:
3112:
3109:
3096:
3095:
3087:
3086:
3069:
3049:
3048:
3047:
3036:
3033:
3030:
3027:
3024:
3018:
3014:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2987:
2983:
2979:
2976:
2971:
2967:
2963:
2960:
2957:
2944:
2943:
2942:
2931:
2926:
2922:
2918:
2915:
2910:
2906:
2902:
2899:
2896:
2891:
2887:
2883:
2878:
2875:
2872:
2868:
2864:
2861:
2858:
2845:
2844:
2828:
2825:
2820:
2816:
2812:
2807:
2803:
2799:
2794:
2790:
2780:is a sequence
2767:
2750:
2730:satisfies the
2713:
2693:
2687:
2684:
2683:
2682:
2670:
2648:
2645:
2642:
2638:
2634:
2631:
2628:
2625:
2620:
2616:
2593:
2589:
2566:
2562:
2552:in some space
2541:
2536:
2533:
2530:
2526:
2522:
2519:
2516:
2511:
2508:
2505:
2501:
2497:
2494:
2491:
2471:
2460:
2455:
2438:
2419:
2414:
2394:
2393:
2392:=1, 2, 3, ...}
2384:
2373:
2359:
2358:
2357:
2356:
2342:
2337:
2333:
2329:
2326:
2321:
2316:
2312:
2308:
2305:
2297:
2292:
2289:
2286:
2282:
2278:
2273:
2269:
2255:
2254:
2248:
2239:
2232:
2204:
2192:
2191:
2190:
2189:
2174:
2169:
2165:
2161:
2154:
2150:
2145:
2141:
2137:
2133:
2130:
2127:
2120:
2116:
2111:
2107:
2103:
2099:
2096:
2093:
2090:
2083:
2079:
2074:
2070:
2066:
2062:
2059:
2056:
2053:
2050:
2048:
2046:
2039:
2035:
2030:
2026:
2022:
2018:
2015:
2012:
2009:
2005:
2001:
1998:
1995:
1992:
1985:
1981:
1976:
1972:
1968:
1964:
1961:
1958:
1955:
1951:
1947:
1944:
1941:
1938:
1935:
1933:
1931:
1926:
1923:
1918:
1914:
1909:
1905:
1901:
1897:
1894:
1891:
1888:
1883:
1880:
1875:
1871:
1866:
1862:
1858:
1854:
1851:
1848:
1845:
1842:
1840:
1838:
1833:
1829:
1825:
1822:
1819:
1814:
1810:
1806:
1801:
1798:
1795:
1791:
1787:
1784:
1781:
1780:
1761:
1750:
1744:
1743:
1742:
1741:
1726:
1722:
1717:
1713:
1709:
1705:
1702:
1699:
1694:
1690:
1676:
1675:
1672:
1671:
1670:
1659:
1656:
1651:
1647:
1643:
1638:
1635:
1632:
1628:
1614:
1613:
1612:of tails. Let
1591:
1578:
1571:
1560:
1553:
1548:
1535:
1526:
1517:
1514:
1499:
1495:
1487:
1483:
1477:
1445:
1444:
1437:
1436:
1435:
1424:
1421:
1416:
1412:
1408:
1403:
1399:
1395:
1389:
1383:
1378:
1373:
1369:
1341:
1336:
1335:
1334:
1323:
1320:
1317:
1313:
1307:
1303:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1269:
1262:
1256:
1241:
1240:
1234:
1233: â ÎŁ
1210:
1209:
1208:
1207:
1196:
1193:
1190:
1187:
1184:
1179:
1175:
1169:
1165:
1161:
1158:
1152:
1146:
1131:
1130:
1113:
1098:
1083:
1074:
1064:
1059:
1042:
1032:
1019:
998:
980:
953:
949:
923:
919:
915:
912:
892:
869:
866:
863:
860:
857:
854:
851:
836:
833:
818:
814:
810:
805:
801:
778:
774:
741:
726:
725:
714:
711:
708:
705:
702:
696:
692:
688:
685:
682:
679:
676:
673:
670:
665:
661:
657:
654:
649:
645:
641:
637:
625:
624:
613:
610:
607:
604:
599:
595:
591:
588:
584:
564:
554:
539:
538:
527:
522:
518:
514:
511:
506:
502:
498:
495:
492:
487:
483:
479:
474:
471:
468:
464:
460:
456:
444:
443:
432:
429:
426:
423:
418:
414:
410:
407:
403:
384:
377:
370:
359:
352:
345:
338:
335:
327:
326:
315:
310:
306:
302:
299:
294:
290:
286:
283:
280:
275:
271:
267:
262:
259:
256:
252:
248:
244:
232:
231:
220:
217:
214:
211:
206:
202:
198:
195:
191:
171:
164:
157:
131:
128:
77:
74:
26:
9:
6:
4:
3:
2:
5471:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5441:
5439:
5424:
5421:
5419:
5416:
5415:
5412:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5322:
5320:
5316:
5308:
5305:
5303:
5300:
5299:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5277:Stopping time
5275:
5271:
5268:
5267:
5266:
5263:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5127:
5125:
5121:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5091:
5089:
5087:
5083:
5076:
5072:
5068:
5067:HewittâSavage
5064:
5060:
5056:
5052:
5051:Zeroâone laws
5049:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4992:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4921:
4918:
4916:
4913:
4912:
4910:
4906:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4871:
4869:
4867:
4863:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4838:
4836:
4834:
4830:
4824:
4821:
4819:
4816:
4814:
4811:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4791:
4789:
4786:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4749:
4748:BlackâScholes
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4730:
4728:
4726:
4722:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4688:
4687:
4685:
4683:
4679:
4673:
4670:
4668:
4665:
4661:
4658:
4656:
4653:
4652:
4651:
4650:Point process
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4629:
4626:
4624:
4621:
4620:
4619:
4616:
4614:
4611:
4609:
4608:Gibbs measure
4606:
4604:
4601:
4599:
4596:
4595:
4593:
4589:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4537:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4512:
4510:
4506:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4453:
4450:
4448:
4445:
4444:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4397:ItĂŽ diffusion
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4372:Gamma process
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4321:
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4297:
4296:
4293:
4289:
4286:
4285:
4284:
4281:
4279:
4276:
4274:
4271:
4270:
4268:
4266:
4262:
4254:
4251:
4249:
4246:
4244:
4243:Self-avoiding
4241:
4239:
4236:
4235:
4234:
4231:
4229:
4228:Moran process
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4195:
4193:
4191:
4190:Discrete time
4187:
4183:
4176:
4171:
4169:
4164:
4162:
4157:
4156:
4153:
4145:
4141:
4137:
4132:
4127:
4123:
4119:
4115:
4110:
4100:on 2018-02-19
4096:
4089:
4084:
4079:
4074:
4069:
4064:
4060:
4056:
4052:
4047:
4043:
4041:981-238-107-4
4037:
4033:
4032:
4027:
4023:
4019:
4013:
4009:
4005:
4001:
3997:
3991:
3987:
3982:
3977:
3973:
3969:
3965:
3961:
3957:
3956:
3951:
3947:
3946:
3933:
3927:
3923:
3916:
3908:
3902:
3898:
3891:
3873:
3869:
3865:
3858:
3851:
3844:
3840:
3834:
3830:
3825:
3824:
3815:
3811:
3800:
3797:
3795:
3794:Normal number
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3726:
3719:
3717:
3712:
3693:
3690:
3687:
3677:
3673:
3668:
3663:
3659:
3636:
3633:
3630:
3620:
3615:
3611:
3587:
3565:
3562:
3559:
3549:
3545:
3532:
3530:
3524:
3520:
3514:
3510:
3506:
3502:
3499: =
3498:
3494:
3493:stopping time
3489:
3487:
3482:
3478:
3471:
3464:
3457:
3453:
3449:
3445:
3438:
3431:
3424:
3420:
3419:stopping time
3414:
3413:Stopping time
3401:
3397:
3394: â
3393:
3386:
3382:
3378:
3373:
3369:
3366:
3362:
3359:
3355:
3354:
3352:
3348:
3345:
3341:
3340:
3331:
3327:
3323:
3319:
3312:
3308:
3305:
3301:
3285:
3282:
3279:
3276:
3267:
3263:
3259:
3250:
3247:
3244:
3241:
3236:
3232:
3225:
3220:
3216:
3209:
3199:
3198:
3196:
3180:
3175:
3171:
3167:
3159:
3155:
3151:
3148:
3145:
3140:
3136:
3132:
3127:
3124:
3121:
3117:
3110:
3100:
3099:
3098:
3097:
3093:
3089:
3088:
3084:
3080:
3076:
3072:
3065:
3061:
3057:
3054:
3050:
3034:
3031:
3028:
3025:
3016:
3012:
3008:
2999:
2996:
2993:
2990:
2985:
2981:
2974:
2969:
2965:
2958:
2948:
2947:
2945:
2929:
2924:
2920:
2916:
2908:
2904:
2900:
2897:
2894:
2889:
2885:
2881:
2876:
2873:
2870:
2866:
2859:
2849:
2848:
2847:
2846:
2842:
2826:
2823:
2818:
2814:
2810:
2805:
2801:
2797:
2792:
2788:
2779:
2778:submartingale
2775:
2774:
2773:
2770:
2766:
2762:
2758:
2753:
2749:
2745:
2741:
2737:
2733:
2729:
2725:
2722: â€
2721:
2716:
2712:
2708:
2704:
2700:
2696:
2668:
2643:
2636:
2632:
2626:
2618:
2614:
2591:
2587:
2564:
2560:
2531:
2524:
2520:
2517:
2514:
2506:
2499:
2492:
2489:
2469:
2461:
2459:
2456:
2454:sample paths.
2453:
2449:
2445:
2442: â
2441:
2434:
2430:
2426:
2422:
2415:
2412:
2408:
2404:
2403:
2398:
2391:
2387:
2380:
2376:
2369:
2365:
2361:
2360:
2335:
2331:
2324:
2314:
2310:
2303:
2295:
2290:
2287:
2284:
2280:
2276:
2271:
2267:
2259:
2258:
2257:
2256:
2251:
2247:
2242:
2238:
2231:
2228:
2227:random sample
2224:
2220:
2216:
2212:
2208:
2205:
2201:
2197:
2194:
2193:
2172:
2167:
2163:
2159:
2152:
2148:
2139:
2135:
2131:
2125:
2118:
2114:
2105:
2101:
2097:
2091:
2088:
2081:
2077:
2068:
2064:
2060:
2054:
2051:
2049:
2037:
2033:
2024:
2020:
2016:
2007:
2003:
1999:
1993:
1990:
1983:
1979:
1970:
1966:
1962:
1953:
1949:
1945:
1939:
1936:
1934:
1924:
1921:
1916:
1912:
1903:
1899:
1895:
1889:
1886:
1881:
1878:
1873:
1869:
1860:
1856:
1852:
1846:
1843:
1841:
1831:
1827:
1823:
1820:
1817:
1812:
1808:
1804:
1799:
1796:
1793:
1789:
1782:
1771:
1770:
1768:
1764:
1757:
1753:
1746:
1745:
1724:
1720:
1711:
1707:
1703:
1697:
1692:
1688:
1680:
1679:
1678:
1677:
1673:
1657:
1654:
1649:
1645:
1641:
1636:
1633:
1630:
1626:
1618:
1617:
1616:
1615:
1611:
1607:
1603:
1599:
1595:
1592:
1589:
1585:
1581:
1574:
1567:
1563:
1556:
1549:
1546:
1542:
1538:
1531:
1530:coin flipping
1527:
1524:
1520:
1519:
1513:
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313:
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18:Submartingale
5335:Econometrics
5297:Wiener space
5185:ItĂŽ integral
5086:Inequalities
4975:Self-similar
4945:GaussâMarkov
4935:Exchangeable
4915:CĂ dlĂ g paths
4851:Risk process
4803:LIBOR market
4672:Random graph
4667:Random field
4534:
4479:Superprocess
4417:LĂ©vy process
4412:Jump process
4387:Hunt process
4223:Markov chain
4121:
4117:
4102:. Retrieved
4095:the original
4078:10419/258120
4058:
4054:
4030:
4007:
3985:
3975:
3971:
3953:
3950:"Martingale"
3921:
3915:
3896:
3890:
3879:. Retrieved
3867:
3863:
3850:
3842:
3822:
3814:
3764:Markov chain
3713:
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3528:
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3325:
3322:greater than
3321:
3314:
3310:
3306:
3091:
3082:
3078:
3074:
3067:
3059:
3058:satisfies Î
3055:
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2760:
2756:
2751:
2747:
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2727:
2723:
2719:
2714:
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2691:
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1576:
1569:
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1551:
1543:tosses of a
1540:
1533:
1521:An unbiased
1465:
1452:
1448:
1446:
1354:
1350:
1345:denotes the
1339:
1235:
1230:
1226:
1222:
1218:
1214:
1126:
1099:
1094:
1089:
1087:lies in the
1081:
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1057:
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937:
882:Banach space
838:
761:
757:
753:
729:
727:
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540:
388:
381:
374:
367:
363:
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340:
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175:
168:
161:
154:
138:
133:
120:Ville (1939)
113:
81:
80:Originally,
79:
42:
36:
5454:Game theory
5380:Ruin theory
5318:Disciplines
5190:ItĂŽ's lemma
4965:Predictable
4640:Percolation
4623:Potts model
4618:Ising model
4582:White noise
4540:Differences
4402:ItĂŽ process
4342:Cox process
4238:Loop-erased
4233:Random walk
3843:martingale.
3651:defined by
3309:satisfies Î
2196:PĂłlya's urn
1588:square root
1523:random walk
1461:ItĆ process
130:Definitions
5438:Categories
5390:Statistics
5170:Filtration
5071:Kolmogorov
5055:Blumenthal
4980:Stationary
4920:Continuous
4908:Properties
4793:HullâWhite
4535:Martingale
4422:Local time
4310:Fractional
4288:pure birth
4140:0021.14601
4104:2010-06-18
3943:References
3881:2011-10-22
3330:from above
3083:from below
2841:integrable
2742:. Given a
2211:statistics
1532:. Suppose
1097:(Ω, Σ
1003:filtration
903:with norm
139:martingale
101:sure thing
83:martingale
43:martingale
5302:Classical
4315:Geometric
4305:Excursion
4061:(2): 31.
3960:EMS Press
3688:τ
3669:τ
3621:τ
3588:τ
3280:≤
3274:∀
3260:≤
3248:≤
3245:τ
3237:τ
3226:∣
3210:
3168:≤
3149:…
3133:∣
3111:
3094:satisfies
3075:less than
3029:≤
3023:∀
3009:≥
2997:≤
2994:τ
2986:τ
2975:∣
2959:
2917:≥
2898:…
2882:∣
2860:
2827:…
2518:…
2281:∏
2200:iteration
1922:−
1821:…
1805:∣
1655:±
1594:de Moivre
1545:fair coin
1482:Σ
1411:Σ
1407:∣
1302:χ
1285:−
1192:∞
1174:‖
1160:‖
1076:for each
1051:index set
952:∗
948:Σ
918:‖
914:⋅
911:‖
865:→
862:Ω
859:×
809:…
707:≤
701:∀
675:≤
672:τ
664:τ
653:∣
612:∞
494:…
478:∣
431:∞
282:…
266:∣
219:∞
145:(i.e., a
116:Paul LĂ©vy
53:(i.e., a
5423:Category
5307:Abstract
4841:BĂŒhlmann
4447:Compound
4028:(2004).
4006:(1991).
3872:Archived
3722:See also
2746:process
2446: :
2423: :
2370:, then {
1765: :
1754: :
1582: :
1229:and all
1213:for all
147:sequence
47:sequence
4930:Ergodic
4818:VaĆĄĂÄek
4660:Poisson
4320:Meander
3962:, 2001
3517:,
3468:,
3461:,
3435:,
3428:,
2388: :
2377: :
2235:, ...,
1129:), i.e.
1125:;
1103:,
1049:in the
1039:adapted
380:,
373:,
355:,
348:,
167:,
160:,
96:gambler
76:History
5270:Tanaka
4955:Mixing
4950:Markov
4823:Wilkie
4788:HoâLee
4783:Heston
4555:Super-
4300:Bridge
4248:Biased
4138:
4038:
4014:
3992:
3928:
3903:
3835:
3064:prefix
2405:In an
1747:Then {
1568:where
1338:where
1063:is a ÎŁ
92:France
5123:Tools
4899:M/M/c
4894:M/M/1
4889:M/G/1
4879:Fluid
4545:Local
4098:(PDF)
4091:(PDF)
4055:Risks
3875:(PDF)
3870:(1).
3860:(PDF)
3806:Notes
2416:If {
2225:. A
1221:with
1092:space
1001:is a
936:is a
558:is a
45:is a
5075:LĂ©vy
4874:Bulk
4758:Chen
4550:Sub-
4508:Both
4036:ISBN
4012:ISBN
3990:ISBN
3926:ISBN
3901:ISBN
3833:ISBN
3634:>
3563:>
3344:both
1550:Let
1217:and
1186:<
967:and
609:<
548:the
428:<
216:<
41:, a
4655:Cox
4136:Zbl
4126:doi
4073:hdl
4063:doi
3829:122
3682:min
3507:of
3503:is
3370:If
3363:If
3356:If
3320:is
3073:is
2839:of
2209:in
1449:and
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3862:.
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