55:
701:
1010:
46:) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
854:
1124:
379:
The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as
374:
2327:
553:
62:
The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or
Poisson risk process) was introduced in 1903 by the Swedish actuary
2505:
2216:
2400:
440:
2858:
2865:
Other finance-related quantities belonging to the class of the expected discounted penalty function include the perpetual
American put option, the contingent claim at optimal exercise time, and more.
2069:
508:
2689:
1927:
1354:
1227:
2137:
537:
1775:
2005:. While Gerber and Shiu applied this function to the classical compound-Poisson model, Powers argued that an insurer's surplus is better modeled by a family of diffusion processes.
1699:
1463:
919:
1983:
1306:
1175:
2763:
2597:
1890:
1861:
1572:
1543:
887:
1260:
1805:
1832:
1514:
181:
1723:
1487:
2003:
1950:
731:
276:
150:
130:
3380:
1932:
It is quite intuitive to interpret the expected discounted penalty function. Since the function measures the actuarial present value of the penalty that occurs at
99:
907:
751:
222:
74:
The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate
248:
201:
1376:
759:
1028:
284:
3915:
696:{\displaystyle \psi (x)=\left(1-{\frac {\lambda \mu }{c}}\right)\sum _{n=0}^{\infty }\left({\frac {\lambda \mu }{c}}\right)^{n}(1-F_{l}^{\ast n}(x))}
3739:
2222:
4342:
3335:
2992:
Lundberg, F. (1903) Approximerad Framställning av
Sannolikehetsfunktionen, Återförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala.
3872:
3852:
2406:
4256:
2148:
1020:
E. Sparre
Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.
1777:
is a penalty function capturing the economic costs to the insurer at the time of ruin (assumed to depend on the surplus prior to ruin
4173:
3857:
2338:
386:
4183:
3867:
2769:
153:
2904:
4225:
4122:
2909:
2008:
There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.
4412:
4402:
4248:
3940:
3925:
3062:
2950:
1379:. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.
4312:
4276:
2030:
4580:
4317:
449:
4229:
3427:
3328:
2603:
4382:
3149:
1895:
1516:
is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation
1371:, which is commonly referred to as Gerber-Shiu function in the ruin literature and named after actuarial scientists
1311:
1184:
4427:
4233:
4217:
4132:
3960:
3930:
3352:
543:
as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an
4332:
4297:
4266:
4261:
3697:
3614:
540:
2075:
4271:
3900:
3895:
3702:
3599:
513:
4616:
4611:
4585:
4362:
4198:
4097:
4082:
3621:
3494:
3410:
3321:
1728:
1005:{\displaystyle \psi (x)={\frac {\lambda \mu }{c}}e^{-\left({\frac {1}{\mu }}-{\frac {\lambda }{c}}\right)x}.}
4357:
4237:
4367:
1583:
1392:
4372:
4008:
3970:
3554:
3499:
3415:
4302:
4606:
4307:
4292:
3935:
3905:
3472:
3370:
1955:
1265:
1134:
2700:
2516:
1866:
1837:
1548:
1519:
4387:
4188:
4102:
4087:
4018:
3594:
3477:
3375:
3183:
The ASTIN bulletin: international journal for actuarial studies in non-life insurance and risk theory
3091:
862:
4221:
4107:
3609:
3584:
3529:
3008:
1232:
226:
1229:
are independent and identically distributed random variables. The model furthermore assumes that
4522:
4512:
4327:
4203:
3985:
3910:
3724:
3589:
3445:
3400:
3271:
1780:
4464:
4392:
3817:
3807:
3651:
3132:
Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, Volker; Teugels, Jozef (2008). "Risk
Processes".
1810:
1492:
2930:
159:
4487:
4469:
4449:
4444:
4163:
3995:
3975:
3822:
3765:
3604:
3514:
3164:
Andersen, E. Sparre. "On the collective theory of risk in case of contagion between claims."
1708:
1472:
4562:
4517:
4507:
4193:
4168:
4137:
4117:
3955:
3877:
3862:
3729:
1988:
1935:
709:
254:
108:
135:
8:
4557:
4397:
4322:
4127:
3887:
3797:
3687:
2966:
Delbaen, F.; Haezendonck, J. (1987). "Classical risk theory in an economic environment".
31:
78:
4527:
4492:
4407:
4377:
4208:
4147:
4142:
3965:
3802:
3467:
3405:
3344:
3252:
3111:
3103:
3027:
892:
736:
3079:
913:. In the case where the claim sizes are exponentially distributed, this simplifies to
207:
4547:
3760:
3677:
3646:
3539:
3519:
3509:
3365:
3360:
3218:
3202:
3145:
3058:
2979:
2946:
1383:
1364:
849:{\displaystyle F_{l}(x)={\frac {1}{\mu }}\int _{0}^{x}\left(1-F(u)\right){\text{d}}u}
27:
4352:
4003:
3256:
3115:
4567:
4454:
4337:
4213:
3950:
3707:
3682:
3631:
3482:
3435:
3248:
3244:
3214:
3137:
3095:
3050:
3017:
2975:
2938:
233:
186:
3559:
3178:
1119:{\displaystyle X_{t}=x+ct-\sum _{i=1}^{N_{t}}\xi _{i}\quad {\text{ for }}t\geq 0,}
369:{\displaystyle X_{t}=x+ct-\sum _{i=1}^{N_{t}}\xi _{i}\quad {\text{ for t}}\geq 0.}
68:
4532:
4432:
4417:
4178:
4112:
3790:
3734:
3717:
3462:
1372:
1178:
103:
4347:
3579:
3054:
2942:
4537:
4502:
4422:
4028:
3775:
3692:
3661:
3656:
3636:
3626:
3569:
3544:
3524:
3489:
3457:
3440:
2899:
1367:
and Gerber and Shiu analyzed the behavior of the insurer's surplus through the
64:
3564:
3141:
4621:
4600:
4439:
3980:
3812:
3770:
3712:
3534:
3450:
3390:
3099:
3022:
3003:
1985:, and then averaged over the probability distribution of the waiting time to
2322:{\displaystyle \delta =0,w(x_{1},x_{2})=\mathbb {I} (x_{1}<x,x_{2}<y)}
1574:. The function is called expected discounted cost of insolvency by Powers.
54:
4497:
4459:
4013:
3945:
3834:
3829:
3641:
3574:
3549:
3385:
20:
2937:. Stochastic Modelling and Applied Probability. Vol. 33. p. 21.
4077:
4061:
4056:
4051:
4041:
3844:
3785:
3780:
3744:
3504:
3395:
3047:
Introductory
Lectures on Fluctuations of LĂ©vy Processes with Applications
2889:
Accident probability factor (APF) calculator – risk analysis model (@SBH)
910:
544:
4552:
4092:
4036:
3920:
3873:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
3313:
3107:
3031:
2500:{\displaystyle \delta =0,w(x_{1},x_{2})=\mathbb {I} (x_{1}+x_{2}<z)}
4046:
1356:
are independent. The model is also known as the renewal risk model.
3235:
Gerber, H. U.; Shiu, E. S. W. (1998). "On the Time Value of Ruin".
2211:{\displaystyle \mathbb {P} ^{x}\{X_{\tau -}<x,X_{\tau }<y\}}
3136:. Wiley Series in Probability and Statistics. pp. 147–204.
1929:
emphasizes that the penalty is exercised only when ruin occurs.
1952:, the penalty function is multiplied by the discounting factor
3179:
Some comments on the Sparre
Andersen model in the risk theory
2395:{\displaystyle \mathbb {P} ^{x}\{X_{\tau -}-X_{\tau }<z\}}
435:{\displaystyle \psi (x)=\mathbb {P} ^{x}\{\tau <\infty \}}
3166:
3131:
3082:(2004). "Ruin Probabilities for Competing Claim Processes".
3045:
Kyprianou, A. E. (2006). "LĂ©vy
Processes and Applications".
3300:. Philadelphia: S.S. Heubner Foundation Monograph Series 8.
2928:
2853:{\displaystyle \delta =0,w(x_{1},x_{2})=x_{1}^{j}x_{2}^{k}}
3853:
Autoregressive conditional heteroskedasticity (ARCH) model
2905:
Volterra integral equation § Application: Ruin theory
2511:
Trivariate
Laplace transform of time, surplus and deficit
3304:
3077:
3381:
Independent and identically distributed random variables
1359:
3858:
Autoregressive integrated moving average (ARIMA) model
2143:
Joint (defective) distribution of surplus and deficit
236:
210:
189:
138:
81:
2772:
2703:
2606:
2519:
2409:
2341:
2225:
2151:
2078:
2033:
1991:
1958:
1938:
1898:
1869:
1840:
1813:
1783:
1731:
1711:
1586:
1551:
1522:
1495:
1475:
1395:
1314:
1268:
1235:
1187:
1137:
1031:
922:
895:
865:
762:
739:
712:
556:
516:
452:
389:
287:
257:
229:). So for an insurer who starts with initial surplus
162:
111:
2877:
Compound-Poisson risk model with stochastic interest
2064:{\displaystyle \mathbb {P} ^{x}\{\tau <\infty \}}
19:"Risk theory" redirects here. For another use, see
16:
Theory in actuarial science and applied probability
2965:
2874:Compound-Poisson risk model with constant interest
2852:
2757:
2683:
2591:
2499:
2394:
2321:
2210:
2131:
2063:
1997:
1977:
1944:
1921:
1884:
1855:
1826:
1799:
1769:
1717:
1693:
1566:
1537:
1508:
1481:
1457:
1348:
1300:
1254:
1221:
1169:
1118:
1004:
901:
881:
848:
745:
725:
695:
531:
503:{\displaystyle \tau =\inf\{t>0\,:\,X(t)<0\}}
502:
434:
368:
270:
242:
216:
195:
175:
144:
124:
93:
67:. Lundberg's work was republished in the 1930s by
3205:(1995). "A theory of risk, return and solvency".
2684:{\displaystyle w(x_{1},x_{2})=e^{-sx_{1}-zx_{2}}}
4598:
3740:Stochastic chains with memory of variable length
3134:Stochastic Processes for Insurance & Finance
3078:Huzak, Miljenko; Perman, Mihael; Šikić, Hrvoje;
517:
459:
102:from customers and claims arrive according to a
3230:
3228:
1922:{\displaystyle \mathbb {I} (\tau <\infty )}
1577:In Gerber and Shiu's notation, it is given as
1349:{\displaystyle (\xi _{i})_{i\in \mathbb {N} }}
1222:{\displaystyle (\xi _{i})_{i\in \mathbb {N} }}
58:A sample path of compound Poisson risk process
3329:
3049:. Springer Berlin Heidelberg. pp. 1–32.
2333:Defective distribution of claim causing ruin
733:is the transform of the tail distribution of
3309:. Singapore: World Scientific Publishing Co.
3295:
2389:
2354:
2205:
2164:
2058:
2046:
497:
462:
429:
417:
3298:An Introduction to Mathematical Risk Theory
3225:
3868:Autoregressive–moving-average (ARMA) model
3336:
3322:
3269:
3234:
2132:{\displaystyle \delta =0,w(x_{1},x_{2})=1}
3197:
3195:
3193:
3191:
3044:
3021:
2706:
2522:
2458:
2344:
2274:
2154:
2036:
1900:
1872:
1843:
1725:is the discounting force of interest and
1669:
1604:
1554:
1525:
1413:
1340:
1213:
539:. This can be computed exactly using the
532:{\displaystyle \inf \varnothing =\infty }
478:
474:
407:
3343:
3127:
3125:
1015:
53:
3279:AFIR Colloquium, Cairns, Australia 1997
2933:; Mikosch, T. (1997). "1 Risk Theory".
1863:corresponds to the probability measure
1770:{\displaystyle w(X_{\tau -},X_{\tau })}
1545:corresponds to the probability measure
154:independent and identically distributed
4599:
4174:Doob's martingale convergence theorems
3201:
3188:
2910:Chance-constrained portfolio selection
2868:
1489:is the discounting force of interest,
3926:Constant elasticity of variance (CEV)
3916:Chan–Karolyi–Longstaff–Sanders (CKLS)
3317:
3122:
2695:Joint moments of surplus and deficit
1694:{\displaystyle m(x)=\mathbb {E} ^{x}}
1458:{\displaystyle m(x)=\mathbb {E} ^{x}}
3272:"From ruin theory to option pricing"
3207:Insurance: Mathematics and Economics
3001:
2968:Insurance: Mathematics and Economics
1369:expected discounted penalty function
1360:Expected discounted penalty function
3270:Gerber, H.U.; Shiu, E.S.W. (1997).
13:
4413:Skorokhod's representation theorem
4194:Law of large numbers (weak/strong)
3305:Asmussen S., Albrecher H. (2010).
3289:
2055:
1913:
1682:
619:
526:
426:
49:
14:
4633:
4383:Martingale representation theorem
1978:{\displaystyle e^{-\delta \tau }}
1301:{\displaystyle (N_{t})_{t\geq 0}}
1170:{\displaystyle (N_{t})_{t\geq 0}}
520:
4428:Stochastic differential equation
4318:Doob's optional stopping theorem
4313:Doob–Meyer decomposition theorem
3237:North American Actuarial Journal
2758:{\displaystyle \mathbb {E} ^{x}}
2592:{\displaystyle \mathbb {E} ^{x}}
1885:{\displaystyle \mathbb {P} ^{x}}
1856:{\displaystyle \mathbb {E} ^{x}}
1567:{\displaystyle \mathbb {P} ^{x}}
1538:{\displaystyle \mathbb {E} ^{x}}
4298:Convergence of random variables
4184:Fisher–Tippett–Gnedenko theorem
3307:Ruin Probabilities, 2nd Edition
3263:
2883:General diffusion-process model
1892:. Here the indicator function
1386:' notation, this is defined as
1131:where the claim number process
1098:
882:{\displaystyle \cdot ^{\ast n}}
354:
3896:Binomial options pricing model
3249:10.1080/10920277.1998.10595671
3171:
3158:
3084:Journal of Applied Probability
3071:
3038:
2995:
2986:
2959:
2922:
2814:
2788:
2752:
2716:
2636:
2610:
2586:
2532:
2494:
2462:
2451:
2425:
2316:
2278:
2267:
2241:
2120:
2094:
1916:
1904:
1764:
1735:
1688:
1685:
1673:
1665:
1636:
1614:
1596:
1590:
1452:
1423:
1405:
1399:
1329:
1315:
1283:
1269:
1202:
1188:
1152:
1138:
932:
926:
830:
824:
779:
773:
690:
687:
681:
654:
566:
560:
488:
482:
399:
393:
156:non-negative random variables
1:
4363:Kolmogorov continuity theorem
4199:Law of the iterated logarithm
2915:
2025:Probability of ultimate ruin
1255:{\displaystyle \xi _{i}>0}
4368:Kolmogorov extension theorem
4047:Generalized queueing network
3555:Interacting particle systems
3219:10.1016/0167-6687(95)00006-E
2980:10.1016/0167-6687(87)90019-9
2017:Mathematical representation
7:
3500:Continuous-time random walk
3055:10.1007/978-3-540-31343-4_1
2943:10.1007/978-3-642-33483-2_2
2893:
2886:Markov-modulated risk model
2020:Choice of penalty function
541:Pollaczek–Khinchine formula
10:
4638:
4508:Extreme value theory (EVT)
4308:Doob decomposition theorem
3600:Ornstein–Uhlenbeck process
3371:Chinese restaurant process
2880:Brownian-motion risk model
1800:{\displaystyle X_{\tau -}}
446:where the time of ruin is
18:
4576:
4480:
4388:Optional stopping theorem
4285:
4247:
4189:Large deviation principle
4156:
4070:
4027:
3994:
3941:Heath–Jarrow–Morton (HJM)
3886:
3878:Moving-average (MA) model
3863:Autoregressive (AR) model
3843:
3753:
3688:Hidden Markov model (HMM)
3670:
3622:Schramm–Loewner evolution
3426:
3351:
3142:10.1002/9780470317044.ch5
3092:Applied Probability Trust
3004:"Harald Cramer 1893-1985"
2935:Modelling Extremal Events
1827:{\displaystyle X_{\tau }}
1509:{\displaystyle K_{\tau }}
510:with the convention that
4303:Doléans-Dade exponential
4133:Progressively measurable
3931:Cox–Ingersoll–Ross (CIR)
3009:The Annals of Statistics
1807:and the deficit at ruin
227:compound Poisson process
176:{\displaystyle \xi _{i}}
4523:Mathematical statistics
4513:Large deviations theory
4343:Infinitesimal generator
4204:Maximal ergodic theorem
4123:Piecewise-deterministic
3725:Random dynamical system
3590:Markov additive process
1834:), and the expectation
1718:{\displaystyle \delta }
1482:{\displaystyle \delta }
1262:almost surely and that
251:, the aggregate assets
4358:Karhunen–Loève theorem
4293:Cameron–Martin formula
4257:Burkholder–Davis–Gundy
3652:Variance gamma process
3168:. Vol. 2. No. 6. 1957.
3100:10.1239/jap/1091543418
3023:10.1214/aos/1176350596
2854:
2759:
2685:
2593:
2501:
2396:
2323:
2212:
2133:
2065:
1999:
1979:
1946:
1923:
1886:
1857:
1828:
1801:
1771:
1719:
1695:
1568:
1539:
1510:
1483:
1459:
1350:
1302:
1256:
1223:
1171:
1120:
1087:
1006:
903:
883:
850:
747:
727:
697:
623:
533:
504:
436:
370:
343:
272:
244:
218:
197:
177:
146:
126:
95:
59:
44:collective risk theory
4488:Actuarial mathematics
4450:Uniform integrability
4445:Stratonovich integral
4373:Lévy–Prokhorov metric
4277:Marcinkiewicz–Zygmund
4164:Central limit theorem
3766:Gaussian random field
3595:McKean–Vlasov process
3515:Dyson Brownian motion
3376:Galton–Watson process
3296:Gerber, H.U. (1979).
2855:
2760:
2686:
2594:
2502:
2397:
2324:
2213:
2134:
2066:
2000:
1998:{\displaystyle \tau }
1980:
1947:
1945:{\displaystyle \tau }
1924:
1887:
1858:
1829:
1802:
1772:
1720:
1696:
1569:
1540:
1511:
1484:
1460:
1351:
1303:
1257:
1224:
1172:
1121:
1060:
1016:Sparre Andersen model
1007:
904:
884:
851:
748:
728:
726:{\displaystyle F_{l}}
698:
603:
534:
505:
437:
371:
316:
273:
271:{\displaystyle X_{t}}
245:
219:
198:
178:
147:
145:{\textstyle \lambda }
127:
125:{\displaystyle N_{t}}
96:
57:
4617:Mathematical finance
4612:Stochastic processes
4563:Time series analysis
4518:Mathematical finance
4403:Reflection principle
3730:Regenerative process
3530:Fleming–Viot process
3345:Stochastic processes
2770:
2701:
2604:
2517:
2407:
2339:
2223:
2149:
2076:
2031:
1989:
1956:
1936:
1896:
1867:
1838:
1811:
1781:
1729:
1709:
1584:
1549:
1520:
1493:
1473:
1393:
1312:
1266:
1233:
1185:
1135:
1029:
920:
893:
863:
760:
737:
710:
554:
514:
450:
387:
285:
255:
234:
208:
187:
160:
136:
109:
79:
4558:Stochastic analysis
4398:Quadratic variation
4393:Prokhorov's theorem
4328:Feynman–Kac formula
3798:Markov random field
3446:Birth–death process
2869:Recent developments
2849:
2834:
2751:
2736:
809:
680:
94:{\textstyle c>0}
32:applied probability
4528:Probability theory
4408:Skorokhod integral
4378:Malliavin calculus
3961:Korn-Kreer-Lenssen
3845:Time series models
3808:Pitman–Yor process
2850:
2835:
2820:
2755:
2737:
2719:
2681:
2589:
2497:
2392:
2319:
2208:
2129:
2061:
1995:
1975:
1942:
1919:
1882:
1853:
1824:
1797:
1767:
1715:
1691:
1564:
1535:
1506:
1479:
1455:
1377:Hans-Ulrich Gerber
1346:
1298:
1252:
1219:
1167:
1116:
1002:
899:
879:
846:
795:
743:
723:
693:
663:
529:
500:
432:
366:
268:
240:
214:
193:
183:with distribution
173:
142:
122:
91:
60:
4607:Actuarial science
4594:
4593:
4548:Signal processing
4267:Doob's upcrossing
4262:Doob's martingale
4226:Engelbert–Schmidt
4169:Donsker's theorem
4103:Feller-continuous
3971:Rendleman–Bartter
3761:Dirichlet process
3678:Branching process
3647:Telegraph process
3540:Geometric process
3520:Empirical process
3510:Diffusion process
3366:Branching process
3361:Bernoulli process
3064:978-3-540-31342-7
3002:Blom, G. (1987).
2952:978-3-540-60931-5
2863:
2862:
1365:Michael R. Powers
1102:
987:
974:
951:
902:{\displaystyle n}
841:
793:
746:{\displaystyle F}
642:
596:
358:
217:{\textstyle \mu }
28:actuarial science
4629:
4568:Machine learning
4455:Usual hypotheses
4338:Girsanov theorem
4323:Dynkin's formula
4088:Continuous paths
3996:Actuarial models
3936:Garman–Kohlhagen
3906:Black–Karasinski
3901:Black–Derman–Toy
3888:Financial models
3754:Fields and other
3683:Gaussian process
3632:Sigma-martingale
3436:Additive process
3338:
3331:
3324:
3315:
3314:
3310:
3301:
3283:
3282:
3276:
3267:
3261:
3260:
3232:
3223:
3222:
3199:
3186:
3175:
3169:
3162:
3156:
3155:
3129:
3120:
3119:
3080:VondraÄŤek, Zoran
3075:
3069:
3068:
3042:
3036:
3035:
3025:
3016:(4): 1335–1350.
2999:
2993:
2990:
2984:
2983:
2963:
2957:
2956:
2926:
2859:
2857:
2856:
2851:
2848:
2843:
2833:
2828:
2813:
2812:
2800:
2799:
2764:
2762:
2761:
2756:
2750:
2745:
2735:
2730:
2715:
2714:
2709:
2690:
2688:
2687:
2682:
2680:
2679:
2678:
2677:
2662:
2661:
2635:
2634:
2622:
2621:
2598:
2596:
2595:
2590:
2585:
2584:
2583:
2582:
2567:
2566:
2531:
2530:
2525:
2506:
2504:
2503:
2498:
2487:
2486:
2474:
2473:
2461:
2450:
2449:
2437:
2436:
2401:
2399:
2398:
2393:
2382:
2381:
2369:
2368:
2353:
2352:
2347:
2328:
2326:
2325:
2320:
2309:
2308:
2290:
2289:
2277:
2266:
2265:
2253:
2252:
2217:
2215:
2214:
2209:
2198:
2197:
2179:
2178:
2163:
2162:
2157:
2138:
2136:
2135:
2130:
2119:
2118:
2106:
2105:
2070:
2068:
2067:
2062:
2045:
2044:
2039:
2011:
2010:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1974:
1973:
1951:
1949:
1948:
1943:
1928:
1926:
1925:
1920:
1903:
1891:
1889:
1888:
1883:
1881:
1880:
1875:
1862:
1860:
1859:
1854:
1852:
1851:
1846:
1833:
1831:
1830:
1825:
1823:
1822:
1806:
1804:
1803:
1798:
1796:
1795:
1776:
1774:
1773:
1768:
1763:
1762:
1750:
1749:
1724:
1722:
1721:
1716:
1700:
1698:
1697:
1692:
1672:
1664:
1663:
1651:
1650:
1632:
1631:
1613:
1612:
1607:
1573:
1571:
1570:
1565:
1563:
1562:
1557:
1544:
1542:
1541:
1536:
1534:
1533:
1528:
1515:
1513:
1512:
1507:
1505:
1504:
1488:
1486:
1485:
1480:
1464:
1462:
1461:
1456:
1451:
1450:
1441:
1440:
1422:
1421:
1416:
1355:
1353:
1352:
1347:
1345:
1344:
1343:
1327:
1326:
1307:
1305:
1304:
1299:
1297:
1296:
1281:
1280:
1261:
1259:
1258:
1253:
1245:
1244:
1228:
1226:
1225:
1220:
1218:
1217:
1216:
1200:
1199:
1176:
1174:
1173:
1168:
1166:
1165:
1150:
1149:
1125:
1123:
1122:
1117:
1103:
1100:
1097:
1096:
1086:
1085:
1084:
1074:
1041:
1040:
1011:
1009:
1008:
1003:
998:
997:
993:
989:
988:
980:
975:
967:
952:
947:
939:
908:
906:
905:
900:
888:
886:
885:
880:
878:
877:
855:
853:
852:
847:
842:
839:
837:
833:
808:
803:
794:
786:
772:
771:
752:
750:
749:
744:
732:
730:
729:
724:
722:
721:
702:
700:
699:
694:
679:
671:
653:
652:
647:
643:
638:
630:
622:
617:
602:
598:
597:
592:
584:
538:
536:
535:
530:
509:
507:
506:
501:
441:
439:
438:
433:
416:
415:
410:
375:
373:
372:
367:
359:
356:
353:
352:
342:
341:
340:
330:
297:
296:
277:
275:
274:
269:
267:
266:
249:
247:
246:
241:
223:
221:
220:
215:
202:
200:
199:
194:
182:
180:
179:
174:
172:
171:
151:
149:
148:
143:
131:
129:
128:
123:
121:
120:
100:
98:
97:
92:
4637:
4636:
4632:
4631:
4630:
4628:
4627:
4626:
4597:
4596:
4595:
4590:
4572:
4533:Queueing theory
4476:
4418:Skorokhod space
4281:
4272:Kunita–Watanabe
4243:
4209:Sanov's theorem
4179:Ergodic theorem
4152:
4148:Time-reversible
4066:
4029:Queueing models
4023:
4019:Sparre–Anderson
4009:Cramér–Lundberg
3990:
3976:SABR volatility
3882:
3839:
3791:Boolean network
3749:
3735:Renewal process
3666:
3615:Non-homogeneous
3605:Poisson process
3495:Contact process
3458:Brownian motion
3428:Continuous time
3422:
3416:Maximal entropy
3347:
3342:
3292:
3290:Further reading
3287:
3286:
3274:
3268:
3264:
3233:
3226:
3200:
3189:
3177:Thorin, Olof. "
3176:
3172:
3163:
3159:
3152:
3130:
3123:
3076:
3072:
3065:
3043:
3039:
3000:
2996:
2991:
2987:
2964:
2960:
2953:
2931:KlĂĽppelberg, C.
2929:Embrechts, P.;
2927:
2923:
2918:
2896:
2871:
2844:
2839:
2829:
2824:
2808:
2804:
2795:
2791:
2771:
2768:
2767:
2746:
2741:
2731:
2723:
2710:
2705:
2704:
2702:
2699:
2698:
2673:
2669:
2657:
2653:
2646:
2642:
2630:
2626:
2617:
2613:
2605:
2602:
2601:
2578:
2574:
2559:
2555:
2539:
2535:
2526:
2521:
2520:
2518:
2515:
2514:
2482:
2478:
2469:
2465:
2457:
2445:
2441:
2432:
2428:
2408:
2405:
2404:
2377:
2373:
2361:
2357:
2348:
2343:
2342:
2340:
2337:
2336:
2304:
2300:
2285:
2281:
2273:
2261:
2257:
2248:
2244:
2224:
2221:
2220:
2193:
2189:
2171:
2167:
2158:
2153:
2152:
2150:
2147:
2146:
2114:
2110:
2101:
2097:
2077:
2074:
2073:
2040:
2035:
2034:
2032:
2029:
2028:
1990:
1987:
1986:
1963:
1959:
1957:
1954:
1953:
1937:
1934:
1933:
1899:
1897:
1894:
1893:
1876:
1871:
1870:
1868:
1865:
1864:
1847:
1842:
1841:
1839:
1836:
1835:
1818:
1814:
1812:
1809:
1808:
1788:
1784:
1782:
1779:
1778:
1758:
1754:
1742:
1738:
1730:
1727:
1726:
1710:
1707:
1706:
1668:
1659:
1655:
1643:
1639:
1621:
1617:
1608:
1603:
1602:
1585:
1582:
1581:
1558:
1553:
1552:
1550:
1547:
1546:
1529:
1524:
1523:
1521:
1518:
1517:
1500:
1496:
1494:
1491:
1490:
1474:
1471:
1470:
1446:
1442:
1430:
1426:
1417:
1412:
1411:
1394:
1391:
1390:
1373:Elias S.W. Shiu
1362:
1339:
1332:
1328:
1322:
1318:
1313:
1310:
1309:
1286:
1282:
1276:
1272:
1267:
1264:
1263:
1240:
1236:
1234:
1231:
1230:
1212:
1205:
1201:
1195:
1191:
1186:
1183:
1182:
1179:renewal process
1155:
1151:
1145:
1141:
1136:
1133:
1132:
1101: for
1099:
1092:
1088:
1080:
1076:
1075:
1064:
1036:
1032:
1030:
1027:
1026:
1018:
979:
966:
965:
961:
957:
953:
940:
938:
921:
918:
917:
894:
891:
890:
870:
866:
864:
861:
860:
838:
814:
810:
804:
799:
785:
767:
763:
761:
758:
757:
738:
735:
734:
717:
713:
711:
708:
707:
672:
667:
648:
631:
629:
625:
624:
618:
607:
585:
583:
576:
572:
555:
552:
551:
515:
512:
511:
451:
448:
447:
411:
406:
405:
388:
385:
384:
355:
348:
344:
336:
332:
331:
320:
292:
288:
286:
283:
282:
262:
258:
256:
253:
252:
235:
232:
231:
209:
206:
205:
188:
185:
184:
167:
163:
161:
158:
157:
137:
134:
133:
132:with intensity
116:
112:
110:
107:
106:
104:Poisson process
80:
77:
76:
52:
50:Classical model
24:
17:
12:
11:
5:
4635:
4625:
4624:
4619:
4614:
4609:
4592:
4591:
4589:
4588:
4583:
4581:List of topics
4577:
4574:
4573:
4571:
4570:
4565:
4560:
4555:
4550:
4545:
4540:
4538:Renewal theory
4535:
4530:
4525:
4520:
4515:
4510:
4505:
4503:Ergodic theory
4500:
4495:
4493:Control theory
4490:
4484:
4482:
4478:
4477:
4475:
4474:
4473:
4472:
4467:
4457:
4452:
4447:
4442:
4437:
4436:
4435:
4425:
4423:Snell envelope
4420:
4415:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4289:
4287:
4283:
4282:
4280:
4279:
4274:
4269:
4264:
4259:
4253:
4251:
4245:
4244:
4242:
4241:
4222:Borel–Cantelli
4211:
4206:
4201:
4196:
4191:
4186:
4181:
4176:
4171:
4166:
4160:
4158:
4157:Limit theorems
4154:
4153:
4151:
4150:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4074:
4072:
4068:
4067:
4065:
4064:
4059:
4054:
4049:
4044:
4039:
4033:
4031:
4025:
4024:
4022:
4021:
4016:
4011:
4006:
4000:
3998:
3992:
3991:
3989:
3988:
3983:
3978:
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3938:
3933:
3928:
3923:
3918:
3913:
3908:
3903:
3898:
3892:
3890:
3884:
3883:
3881:
3880:
3875:
3870:
3865:
3860:
3855:
3849:
3847:
3841:
3840:
3838:
3837:
3832:
3827:
3826:
3825:
3820:
3810:
3805:
3800:
3795:
3794:
3793:
3788:
3778:
3776:Hopfield model
3773:
3768:
3763:
3757:
3755:
3751:
3750:
3748:
3747:
3742:
3737:
3732:
3727:
3722:
3721:
3720:
3715:
3710:
3705:
3695:
3693:Markov process
3690:
3685:
3680:
3674:
3672:
3668:
3667:
3665:
3664:
3662:Wiener sausage
3659:
3657:Wiener process
3654:
3649:
3644:
3639:
3637:Stable process
3634:
3629:
3627:Semimartingale
3624:
3619:
3618:
3617:
3612:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3570:Jump diffusion
3567:
3562:
3557:
3552:
3547:
3545:Hawkes process
3542:
3537:
3532:
3527:
3525:Feller process
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3490:Cauchy process
3487:
3486:
3485:
3480:
3475:
3470:
3465:
3455:
3454:
3453:
3443:
3441:Bessel process
3438:
3432:
3430:
3424:
3423:
3421:
3420:
3419:
3418:
3413:
3408:
3403:
3393:
3388:
3383:
3378:
3373:
3368:
3363:
3357:
3355:
3349:
3348:
3341:
3340:
3333:
3326:
3318:
3312:
3311:
3302:
3291:
3288:
3285:
3284:
3262:
3224:
3213:(2): 101–118.
3187:
3170:
3157:
3150:
3121:
3070:
3063:
3037:
2994:
2985:
2958:
2951:
2920:
2919:
2917:
2914:
2913:
2912:
2907:
2902:
2900:Financial risk
2895:
2892:
2891:
2890:
2887:
2884:
2881:
2878:
2875:
2870:
2867:
2861:
2860:
2847:
2842:
2838:
2832:
2827:
2823:
2819:
2816:
2811:
2807:
2803:
2798:
2794:
2790:
2787:
2784:
2781:
2778:
2775:
2765:
2754:
2749:
2744:
2740:
2734:
2729:
2726:
2722:
2718:
2713:
2708:
2696:
2692:
2691:
2676:
2672:
2668:
2665:
2660:
2656:
2652:
2649:
2645:
2641:
2638:
2633:
2629:
2625:
2620:
2616:
2612:
2609:
2599:
2588:
2581:
2577:
2573:
2570:
2565:
2562:
2558:
2554:
2551:
2548:
2545:
2542:
2538:
2534:
2529:
2524:
2512:
2508:
2507:
2496:
2493:
2490:
2485:
2481:
2477:
2472:
2468:
2464:
2460:
2456:
2453:
2448:
2444:
2440:
2435:
2431:
2427:
2424:
2421:
2418:
2415:
2412:
2402:
2391:
2388:
2385:
2380:
2376:
2372:
2367:
2364:
2360:
2356:
2351:
2346:
2334:
2330:
2329:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2288:
2284:
2280:
2276:
2272:
2269:
2264:
2260:
2256:
2251:
2247:
2243:
2240:
2237:
2234:
2231:
2228:
2218:
2207:
2204:
2201:
2196:
2192:
2188:
2185:
2182:
2177:
2174:
2170:
2166:
2161:
2156:
2144:
2140:
2139:
2128:
2125:
2122:
2117:
2113:
2109:
2104:
2100:
2096:
2093:
2090:
2087:
2084:
2081:
2071:
2060:
2057:
2054:
2051:
2048:
2043:
2038:
2026:
2022:
2021:
2018:
2015:
1994:
1972:
1969:
1966:
1962:
1941:
1918:
1915:
1912:
1909:
1906:
1902:
1879:
1874:
1850:
1845:
1821:
1817:
1794:
1791:
1787:
1766:
1761:
1757:
1753:
1748:
1745:
1741:
1737:
1734:
1714:
1703:
1702:
1690:
1687:
1684:
1681:
1678:
1675:
1671:
1667:
1662:
1658:
1654:
1649:
1646:
1642:
1638:
1635:
1630:
1627:
1624:
1620:
1616:
1611:
1606:
1601:
1598:
1595:
1592:
1589:
1561:
1556:
1532:
1527:
1503:
1499:
1478:
1467:
1466:
1454:
1449:
1445:
1439:
1436:
1433:
1429:
1425:
1420:
1415:
1410:
1407:
1404:
1401:
1398:
1361:
1358:
1342:
1338:
1335:
1331:
1325:
1321:
1317:
1295:
1292:
1289:
1285:
1279:
1275:
1271:
1251:
1248:
1243:
1239:
1215:
1211:
1208:
1204:
1198:
1194:
1190:
1164:
1161:
1158:
1154:
1148:
1144:
1140:
1129:
1128:
1127:
1126:
1115:
1112:
1109:
1106:
1095:
1091:
1083:
1079:
1073:
1070:
1067:
1063:
1059:
1056:
1053:
1050:
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278:are given by:
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243:{\textstyle x}
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196:{\textstyle F}
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65:Filip Lundberg
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4440:Stopping time
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4230:Hewitt–Savage
4227:
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4214:Zero–one laws
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3911:Black–Scholes
3909:
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3813:Point process
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3771:Gibbs measure
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3560:ItĂ´ diffusion
3558:
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3535:Gamma process
3533:
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3406:Self-avoiding
3404:
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3391:Moran process
3389:
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3353:Discrete time
3350:
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3208:
3204:
3203:Powers, M. R.
3198:
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3151:9780470317044
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2019:
2016:
2014:Special case
2013:
2012:
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2006:
1992:
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684:
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471:
468:
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423:
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396:
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363:
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349:
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327:
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321:
317:
313:
310:
307:
304:
301:
298:
293:
289:
281:
280:
279:
263:
259:
250:
237:
228:
225:(they form a
224:
211:
190:
168:
164:
155:
139:
117:
113:
105:
101:
88:
85:
82:
72:
70:
69:Harald Cramér
66:
56:
47:
45:
41:
37:
33:
29:
22:
4542:
4498:Econometrics
4460:Wiener space
4348:ItĂ´ integral
4249:Inequalities
4138:Self-similar
4108:Gauss–Markov
4098:Exchangeable
4078:CĂ dlĂ g paths
4014:Risk process
3966:LIBOR market
3835:Random graph
3830:Random field
3642:Superprocess
3580:LĂ©vy process
3575:Jump process
3550:Hunt process
3386:Markov chain
3306:
3297:
3278:
3265:
3240:
3236:
3210:
3206:
3185:(1974): 104.
3182:
3173:
3165:
3160:
3133:
3087:
3083:
3073:
3046:
3040:
3013:
3007:
2997:
2988:
2971:
2967:
2961:
2934:
2924:
2864:
2007:
1931:
1704:
1576:
1468:
1381:
1368:
1363:
1130:
1019:
889:denotes the
858:
705:
445:
378:
230:
204:
75:
73:
61:
43:
39:
35:
25:
21:Tirpitz Plan
4543:Ruin theory
4481:Disciplines
4353:ItĂ´'s lemma
4128:Predictable
3803:Percolation
3786:Potts model
3781:Ising model
3745:White noise
3703:Differences
3565:ItĂ´ process
3505:Cox process
3401:Loop-erased
3396:Random walk
3094:: 679–690.
911:convolution
545:M/G/1 queue
357: for t
40:risk theory
38:(sometimes
36:ruin theory
4601:Categories
4553:Statistics
4333:Filtration
4234:Kolmogorov
4218:Blumenthal
4143:Stationary
4083:Continuous
4071:Properties
3956:Hull–White
3698:Martingale
3585:Local time
3473:Fractional
3451:pure birth
2916:References
4465:Classical
3478:Geometric
3468:Excursion
3243:: 48–72.
2974:(2): 85.
2774:δ
2743:τ
2728:−
2725:τ
2664:−
2648:−
2580:τ
2569:−
2564:−
2561:τ
2550:−
2547:τ
2544:δ
2541:−
2411:δ
2379:τ
2371:−
2366:−
2363:τ
2227:δ
2195:τ
2176:−
2173:τ
2080:δ
2056:∞
2050:τ
1993:τ
1971:τ
1968:δ
1965:−
1940:τ
1914:∞
1908:τ
1820:τ
1793:−
1790:τ
1760:τ
1747:−
1744:τ
1713:δ
1683:∞
1677:τ
1661:τ
1648:−
1645:τ
1629:τ
1626:δ
1623:−
1502:τ
1477:δ
1448:τ
1438:τ
1435:δ
1432:−
1337:∈
1320:ξ
1291:≥
1238:ξ
1210:∈
1193:ξ
1160:≥
1108:≥
1090:ξ
1062:∑
1058:−
982:λ
977:−
972:μ
959:−
945:μ
942:λ
924:ψ
872:∗
868:⋅
819:−
797:∫
791:μ
674:∗
661:−
636:μ
633:λ
620:∞
605:∑
590:μ
587:λ
581:−
558:ψ
527:∞
521:∅
454:τ
427:∞
421:τ
391:ψ
361:≥
346:ξ
318:∑
314:−
212:μ
203:and mean
165:ξ
140:λ
4586:Category
4470:Abstract
4004:BĂĽhlmann
3610:Compound
3257:59054002
3116:14499808
2894:See also
152:and are
4093:Ergodic
3981:VašĂÄŤek
3823:Poisson
3483:Meander
3108:4141346
3032:2241677
4433:Tanaka
4118:Mixing
4113:Markov
3986:Wilkie
3951:Ho–Lee
3946:Heston
3718:Super-
3463:Bridge
3411:Biased
3255:
3148:
3114:
3106:
3061:
3030:
2949:
1705:where
1469:where
1384:Powers
909:-fold
706:where
4286:Tools
4062:M/M/c
4057:M/M/1
4052:M/G/1
4042:Fluid
3708:Local
3275:(PDF)
3253:S2CID
3112:S2CID
3104:JSTOR
3090:(3).
3028:JSTOR
1308:and
1177:is a
4622:Risk
4238:LĂ©vy
4037:Bulk
3921:Chen
3713:Sub-
3671:Both
3146:ISBN
3059:ISBN
2947:ISBN
2489:<
2384:<
2311:<
2292:<
2200:<
2181:<
2053:<
1911:<
1680:<
1375:and
1247:>
1181:and
859:and
492:<
469:>
424:<
86:>
30:and
3818:Cox
3245:doi
3215:doi
3138:doi
3096:doi
3051:doi
3018:doi
2976:doi
2939:doi
1382:In
518:inf
460:inf
42:or
26:In
4603::
4236:,
4232:,
4228:,
4224:,
4220:,
3277:.
3251:.
3239:.
3227:^
3211:17
3209:.
3190:^
3181:"
3144:.
3124:^
3110:.
3102:.
3088:41
3086:.
3057:.
3026:.
3014:15
3012:.
3006:.
2970:.
2945:.
753:,
547:)
364:0.
71:.
34:,
4240:)
4216:(
3337:e
3330:t
3323:v
3281:.
3259:.
3247::
3241:2
3221:.
3217::
3154:.
3140::
3118:.
3098::
3067:.
3053::
3034:.
3020::
2982:.
2978::
2972:6
2955:.
2941::
2846:k
2841:2
2837:x
2831:j
2826:1
2822:x
2818:=
2815:)
2810:2
2806:x
2802:,
2797:1
2793:x
2789:(
2786:w
2783:,
2780:0
2777:=
2753:]
2748:k
2739:X
2733:j
2721:X
2717:[
2712:x
2707:E
2675:2
2671:x
2667:z
2659:1
2655:x
2651:s
2644:e
2640:=
2637:)
2632:2
2628:x
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