66:
712:
1021:
57:) 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.
865:
1135:
390:
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
385:
2338:
564:
73:
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
2516:
2227:
2411:
451:
2869:
2876:
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.
2080:
519:
2700:
1938:
1365:
1238:
3391:
2148:
548:
1786:
2016:. 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.
1710:
1474:
930:
1994:
1317:
1186:
3926:
2774:
2608:
1901:
1872:
1583:
1554:
898:
1271:
1816:
1843:
1525:
3750:
192:
1734:
1498:
2014:
1961:
742:
287:
161:
141:
1943:
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
4353:
3346:
110:
3883:
3863:
918:
762:
233:
4267:
85:
The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate
259:
212:
770:
1039:
295:
4184:
707:{\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))}
4194:
3868:
3878:
4236:
2233:
164:
3951:
4133:
3003:
Lundberg, F. (1903) Approximerad Framställning av
Sannolikehetsfunktionen, Återförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala.
2920:
4423:
4413:
3936:
3073:
2961:
2417:
4323:
4287:
4240:
4591:
4328:
2159:
1031:
E. Sparre
Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.
3438:
3339:
1788:
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
4393:
3971:
3941:
3160:
2349:
397:
4244:
4228:
4438:
4143:
3363:
2780:
551:
4343:
4308:
4277:
4272:
3911:
3708:
3625:
4282:
3610:
2019:
There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.
3906:
3713:
1390:. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.
3632:
4368:
4248:
4627:
4622:
4596:
4373:
4209:
4108:
4093:
3505:
3421:
3332:
2041:
4383:
4019:
4378:
460:
3981:
2614:
3565:
3510:
3426:
4313:
4303:
3946:
3916:
1906:
1527:
is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation
1382:, which is commonly referred to as Gerber-Shiu function in the ruin literature and named after actuarial scientists
1322:
1195:
4617:
4318:
3483:
3381:
2915:
554:
as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an
4029:
3605:
3386:
4398:
4199:
4113:
4098:
3488:
3102:
4232:
4118:
3540:
2086:
3620:
3595:
3019:
524:
237:
4338:
3921:
3456:
1739:
1016:{\displaystyle \psi (x)={\frac {\lambda \mu }{c}}e^{-\left({\frac {1}{\mu }}-{\frac {\lambda }{c}}\right)x}.}
4533:
4523:
4214:
3996:
3735:
3600:
3411:
3818:
1594:
1403:
4475:
4403:
3662:
2941:
4498:
4480:
4460:
4455:
4174:
4006:
3986:
3833:
3776:
3615:
3525:
1387:
3966:
1966:
1276:
1145:
2711:
2527:
1877:
1848:
1559:
1530:
4573:
4528:
4518:
4259:
4204:
4179:
4148:
4128:
3888:
3873:
3740:
3194:
The ASTIN bulletin: international journal for actuarial studies in non-life insurance and risk theory
873:
4568:
4408:
4333:
4138:
3898:
3808:
3698:
1243:
42:
1240:
are independent and identically distributed random variables. The model furthermore assumes that
4538:
4503:
4418:
4388:
4158:
4153:
3976:
3813:
3478:
3416:
3355:
3282:
3263:
3122:
3114:
3038:
4219:
3090:
1791:
3143:
Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, Volker; Teugels, Jozef (2008). "Risk
Processes".
1821:
1503:
4558:
4363:
4014:
3771:
3688:
3657:
3550:
3530:
3520:
3376:
3371:
3213:
3156:
3069:
2957:
1394:
1375:
170:
38:
4224:
3961:
3175:
Andersen, E. Sparre. "On the collective theory of risk in case of contagion between claims."
1719:
1483:
4578:
4465:
4348:
3718:
3693:
3642:
3570:
3493:
3446:
3255:
3225:
3148:
3106:
3061:
3028:
2986:
2949:
79:
1999:
1946:
720:
265:
119:
65:
4543:
4443:
4428:
4189:
4123:
3801:
3745:
3728:
3473:
1383:
1189:
146:
114:
4358:
3590:
2977:
Delbaen, F.; Haezendonck, J. (1987). "Classical risk theory in an economic environment".
89:
4548:
4513:
4433:
4039:
3786:
3703:
3672:
3667:
3647:
3637:
3580:
3575:
3555:
3535:
3500:
3468:
3451:
2910:
903:
747:
75:
924:. In the case where the claim sizes are exponentially distributed, this simplifies to
218:
4632:
4611:
4450:
3991:
3828:
3823:
3781:
3723:
3545:
3461:
3401:
3229:
2990:
860:{\displaystyle F_{l}(x)={\frac {1}{\mu }}\int _{0}^{x}\left(1-F(u)\right){\text{d}}u}
17:
3267:
3126:
4508:
4470:
4024:
3956:
3845:
3840:
3652:
3585:
3560:
3396:
3259:
244:
197:
31:
4088:
3189:
1130:{\displaystyle X_{t}=x+ct-\sum _{i=1}^{N_{t}}\xi _{i}\quad {\text{ for }}t\geq 0,}
380:{\displaystyle X_{t}=x+ct-\sum _{i=1}^{N_{t}}\xi _{i}\quad {\text{ for t}}\geq 0.}
4072:
4067:
4062:
4052:
3855:
3796:
3791:
3755:
3515:
3406:
3065:
2953:
921:
555:
1378:
and Gerber and Shiu analyzed the behavior of the insurer's surplus through the
4563:
4103:
4047:
3931:
3152:
3110:
3033:
3014:
1996:, and then averaged over the probability distribution of the waiting time to
4057:
2333:{\displaystyle \delta =0,w(x_{1},x_{2})=\mathbb {I} (x_{1}<x,x_{2}<y)}
1585:. The function is called expected discounted cost of insolvency by Powers.
2948:. Stochastic Modelling and Applied Probability. Vol. 33. p. 21.
3058:
Introductory
Lectures on Fluctuations of LĂ©vy Processes with Applications
2900:
Accident probability factor (APF) calculator – risk analysis model (@SBH)
3884:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
3324:
3118:
3042:
2511:{\displaystyle \delta =0,w(x_{1},x_{2})=\mathbb {I} (x_{1}+x_{2}<z)}
1367:
are independent. The model is also known as the renewal risk model.
3246:
Gerber, H. U.; Shiu, E. S. W. (1998). "On the Time Value of Ruin".
2222:{\displaystyle \mathbb {P} ^{x}\{X_{\tau -}<x,X_{\tau }<y\}}
64:
3147:. Wiley Series in Probability and Statistics. pp. 147–204.
1940:
emphasizes that the penalty is exercised only when ruin occurs.
3328:
1963:, the penalty function is multiplied by the discounting factor
3190:
Some comments on the Sparre
Andersen model in the risk theory
2406:{\displaystyle \mathbb {P} ^{x}\{X_{\tau -}-X_{\tau }<z\}}
446:{\displaystyle \psi (x)=\mathbb {P} ^{x}\{\tau <\infty \}}
3177:
3093:(2004). "Ruin Probabilities for Competing Claim Processes".
3056:
Kyprianou, A. E. (2006). "LĂ©vy
Processes and Applications".
3311:. Philadelphia: S.S. Heubner Foundation Monograph Series 8.
2864:{\displaystyle \delta =0,w(x_{1},x_{2})=x_{1}^{j}x_{2}^{k}}
3864:
Autoregressive conditional heteroskedasticity (ARCH) model
2916:
Volterra integral equation § Application: Ruin theory
2522:
Trivariate
Laplace transform of time, surplus and deficit
3392:
Independent and identically distributed random variables
3869:
Autoregressive integrated moving average (ARIMA) model
2154:
Joint (defective) distribution of surplus and deficit
247:
221:
200:
149:
92:
2783:
2714:
2617:
2530:
2420:
2352:
2236:
2162:
2089:
2044:
2002:
1969:
1949:
1909:
1880:
1851:
1824:
1794:
1742:
1722:
1597:
1562:
1533:
1506:
1486:
1406:
1325:
1279:
1246:
1198:
1148:
1042:
933:
906:
876:
773:
750:
723:
567:
527:
463:
400:
298:
268:
240:). So for an insurer who starts with initial surplus
173:
122:
2888:
Compound-Poisson risk model with stochastic interest
2075:{\displaystyle \mathbb {P} ^{x}\{\tau <\infty \}}
4491:
4296:
4258:
4167:
4081:
4038:
4005:
3897:
3854:
3764:
3681:
3437:
3362:
30:"Risk theory" redirects here. For another use, see
27:
Theory in actuarial science and applied probability
2885:Compound-Poisson risk model with constant interest
2863:
2768:
2694:
2602:
2510:
2405:
2332:
2221:
2142:
2074:
2008:
1988:
1955:
1932:
1895:
1866:
1837:
1810:
1780:
1728:
1704:
1577:
1548:
1519:
1492:
1468:
1359:
1311:
1265:
1232:
1180:
1129:
1015:
912:
892:
859:
756:
736:
706:
542:
514:{\displaystyle \tau =\inf\{t>0\,:\,X(t)<0\}}
513:
445:
379:
281:
253:
227:
206:
186:
155:
135:
104:
78:. Lundberg's work was republished in the 1930s by
3216:(1995). "A theory of risk, return and solvency".
2695:{\displaystyle w(x_{1},x_{2})=e^{-sx_{1}-zx_{2}}}
3751:Stochastic chains with memory of variable length
3145:Stochastic Processes for Insurance & Finance
3089:Huzak, Miljenko; Perman, Mihael; Šikić, Hrvoje;
528:
470:
113:from customers and claims arrive according to a
3241:
3239:
1933:{\displaystyle \mathbb {I} (\tau <\infty )}
1588:In Gerber and Shiu's notation, it is given as
1360:{\displaystyle (\xi _{i})_{i\in \mathbb {N} }}
1233:{\displaystyle (\xi _{i})_{i\in \mathbb {N} }}
69:A sample path of compound Poisson risk process
3340:
3060:. Springer Berlin Heidelberg. pp. 1–32.
2344:Defective distribution of claim causing ruin
744:is the transform of the tail distribution of
8:
3320:. Singapore: World Scientific Publishing Co.
2400:
2365:
2216:
2175:
2069:
2057:
508:
473:
440:
428:
3309:An Introduction to Mathematical Risk Theory
3879:Autoregressive–moving-average (ARMA) model
3347:
3333:
3325:
2143:{\displaystyle \delta =0,w(x_{1},x_{2})=1}
3208:
3206:
3204:
3202:
3032:
2855:
2850:
2840:
2835:
2819:
2806:
2782:
2757:
2752:
2742:
2734:
2721:
2717:
2716:
2713:
2684:
2668:
2657:
2641:
2628:
2616:
2589:
2570:
2550:
2537:
2533:
2532:
2529:
2493:
2480:
2469:
2468:
2456:
2443:
2419:
2388:
2372:
2359:
2355:
2354:
2351:
2315:
2296:
2285:
2284:
2272:
2259:
2235:
2204:
2182:
2169:
2165:
2164:
2161:
2125:
2112:
2088:
2051:
2047:
2046:
2043:
2001:
1974:
1968:
1948:
1911:
1910:
1908:
1887:
1883:
1882:
1879:
1858:
1854:
1853:
1850:
1829:
1823:
1799:
1793:
1769:
1753:
1741:
1736:is the discounting force of interest and
1721:
1680:
1679:
1670:
1654:
1632:
1619:
1615:
1614:
1596:
1569:
1565:
1564:
1561:
1540:
1536:
1535:
1532:
1511:
1505:
1485:
1457:
1441:
1428:
1424:
1423:
1405:
1351:
1350:
1343:
1333:
1324:
1297:
1287:
1278:
1251:
1245:
1224:
1223:
1216:
1206:
1197:
1166:
1156:
1147:
1110:
1103:
1091:
1086:
1075:
1047:
1041:
990:
977:
968:
949:
932:
905:
881:
875:
849:
815:
810:
796:
778:
772:
749:
728:
722:
683:
678:
659:
640:
629:
618:
594:
566:
550:. This can be computed exactly using the
543:{\displaystyle \inf \varnothing =\infty }
526:
489:
485:
462:
422:
418:
417:
399:
366:
359:
347:
342:
331:
303:
297:
273:
267:
246:
220:
199:
178:
172:
148:
127:
121:
91:
3138:
3136:
2021:
3290:AFIR Colloquium, Cairns, Australia 1997
2944:; Mikosch, T. (1997). "1 Risk Theory".
2932:
1874:corresponds to the probability measure
1781:{\displaystyle w(X_{\tau -},X_{\tau })}
1556:corresponds to the probability measure
531:
165:independent and identically distributed
4185:Doob's martingale convergence theorems
2921:Chance-constrained portfolio selection
1500:is the discounting force of interest,
3937:Constant elasticity of variance (CEV)
3927:Chan–Karolyi–Longstaff–Sanders (CKLS)
2706:Joint moments of surplus and deficit
1705:{\displaystyle m(x)=\mathbb {E} ^{x}}
1469:{\displaystyle m(x)=\mathbb {E} ^{x}}
7:
3283:"From ruin theory to option pricing"
3218:Insurance: Mathematics and Economics
2979:Insurance: Mathematics and Economics
1380:expected discounted penalty function
1371:Expected discounted penalty function
3281:Gerber, H.U.; Shiu, E.S.W. (1997).
4424:Skorokhod's representation theorem
4205:Law of large numbers (weak/strong)
3316:Asmussen S., Albrecher H. (2010).
2066:
1924:
1693:
630:
537:
437:
25:
4394:Martingale representation theorem
1989:{\displaystyle e^{-\delta \tau }}
1312:{\displaystyle (N_{t})_{t\geq 0}}
1181:{\displaystyle (N_{t})_{t\geq 0}}
4439:Stochastic differential equation
4329:Doob's optional stopping theorem
4324:Doob–Meyer decomposition theorem
3248:North American Actuarial Journal
2769:{\displaystyle \mathbb {E} ^{x}}
2603:{\displaystyle \mathbb {E} ^{x}}
1896:{\displaystyle \mathbb {P} ^{x}}
1867:{\displaystyle \mathbb {E} ^{x}}
1578:{\displaystyle \mathbb {P} ^{x}}
1549:{\displaystyle \mathbb {E} ^{x}}
4309:Convergence of random variables
4195:Fisher–Tippett–Gnedenko theorem
3318:Ruin Probabilities, 2nd Edition
2894:General diffusion-process model
1903:. Here the indicator function
1397:' notation, this is defined as
1142:where the claim number process
1109:
893:{\displaystyle \cdot ^{\ast n}}
365:
3907:Binomial options pricing model
3260:10.1080/10920277.1998.10595671
3095:Journal of Applied Probability
2825:
2799:
2763:
2727:
2647:
2621:
2597:
2543:
2505:
2473:
2462:
2436:
2327:
2289:
2278:
2252:
2131:
2105:
1927:
1915:
1775:
1746:
1699:
1696:
1684:
1676:
1647:
1625:
1607:
1601:
1463:
1434:
1416:
1410:
1340:
1326:
1294:
1280:
1213:
1199:
1163:
1149:
943:
937:
841:
835:
790:
784:
701:
698:
692:
665:
577:
571:
499:
493:
410:
404:
167:non-negative random variables
1:
4374:Kolmogorov continuity theorem
4210:Law of the iterated logarithm
2036:Probability of ultimate ruin
1266:{\displaystyle \xi _{i}>0}
4379:Kolmogorov extension theorem
4058:Generalized queueing network
3566:Interacting particle systems
3230:10.1016/0167-6687(95)00006-E
2991:10.1016/0167-6687(87)90019-9
2028:Mathematical representation
3511:Continuous-time random walk
3066:10.1007/978-3-540-31343-4_1
2954:10.1007/978-3-642-33483-2_2
2897:Markov-modulated risk model
2031:Choice of penalty function
552:Pollaczek–Khinchine formula
4649:
4519:Extreme value theory (EVT)
4319:Doob decomposition theorem
3611:Ornstein–Uhlenbeck process
3382:Chinese restaurant process
2891:Brownian-motion risk model
1811:{\displaystyle X_{\tau -}}
457:where the time of ruin is
29:
4587:
4399:Optional stopping theorem
4200:Large deviation principle
3952:Heath–Jarrow–Morton (HJM)
3889:Moving-average (MA) model
3874:Autoregressive (AR) model
3699:Hidden Markov model (HMM)
3633:Schramm–Loewner evolution
3153:10.1002/9780470317044.ch5
3103:Applied Probability Trust
3015:"Harald Cramer 1893-1985"
2946:Modelling Extremal Events
1838:{\displaystyle X_{\tau }}
1520:{\displaystyle K_{\tau }}
521:with the convention that
4314:Doléans-Dade exponential
4144:Progressively measurable
3942:Cox–Ingersoll–Ross (CIR)
3020:The Annals of Statistics
1818:and the deficit at ruin
238:compound Poisson process
187:{\displaystyle \xi _{i}}
4534:Mathematical statistics
4524:Large deviations theory
4354:Infinitesimal generator
4215:Maximal ergodic theorem
4134:Piecewise-deterministic
3736:Random dynamical system
3601:Markov additive process
1845:), and the expectation
1729:{\displaystyle \delta }
1493:{\displaystyle \delta }
1273:almost surely and that
262:, the aggregate assets
4369:Karhunen–Loève theorem
4304:Cameron–Martin formula
4268:Burkholder–Davis–Gundy
3663:Variance gamma process
3179:. Vol. 2. No. 6. 1957.
3111:10.1239/jap/1091543418
3034:10.1214/aos/1176350596
2865:
2770:
2696:
2604:
2512:
2407:
2334:
2223:
2144:
2076:
2010:
1990:
1957:
1934:
1897:
1868:
1839:
1812:
1782:
1730:
1706:
1579:
1550:
1521:
1494:
1470:
1361:
1313:
1267:
1234:
1182:
1131:
1098:
1017:
914:
894:
861:
758:
738:
708:
634:
544:
515:
447:
381:
354:
283:
255:
229:
208:
188:
157:
137:
106:
70:
55:collective risk theory
4499:Actuarial mathematics
4461:Uniform integrability
4456:Stratonovich integral
4384:Lévy–Prokhorov metric
4288:Marcinkiewicz–Zygmund
4175:Central limit theorem
3777:Gaussian random field
3606:McKean–Vlasov process
3526:Dyson Brownian motion
3387:Galton–Watson process
3307:Gerber, H.U. (1979).
2866:
2771:
2697:
2605:
2513:
2408:
2335:
2224:
2145:
2077:
2011:
2009:{\displaystyle \tau }
1991:
1958:
1956:{\displaystyle \tau }
1935:
1898:
1869:
1840:
1813:
1783:
1731:
1707:
1580:
1551:
1522:
1495:
1471:
1362:
1314:
1268:
1235:
1183:
1132:
1071:
1027:Sparre Andersen model
1018:
915:
895:
862:
759:
739:
737:{\displaystyle F_{l}}
709:
614:
545:
516:
448:
382:
327:
284:
282:{\displaystyle X_{t}}
256:
230:
209:
189:
158:
156:{\textstyle \lambda }
138:
136:{\displaystyle N_{t}}
107:
68:
18:Cramér–Lundberg model
4628:Mathematical finance
4623:Stochastic processes
4574:Time series analysis
4529:Mathematical finance
4414:Reflection principle
3741:Regenerative process
3541:Fleming–Viot process
3356:Stochastic processes
2781:
2712:
2615:
2528:
2418:
2350:
2234:
2160:
2087:
2042:
2000:
1967:
1947:
1907:
1878:
1849:
1822:
1792:
1740:
1720:
1595:
1560:
1531:
1504:
1484:
1404:
1323:
1277:
1244:
1196:
1146:
1040:
931:
904:
874:
771:
748:
721:
565:
525:
461:
398:
296:
266:
245:
219:
198:
171:
147:
120:
90:
4569:Stochastic analysis
4409:Quadratic variation
4404:Prokhorov's theorem
4339:Feynman–Kac formula
3809:Markov random field
3457:Birth–death process
2880:Recent developments
2860:
2845:
2762:
2747:
820:
691:
105:{\textstyle c>0}
43:applied probability
4539:Probability theory
4419:Skorokhod integral
4389:Malliavin calculus
3972:Korn-Kreer-Lenssen
3856:Time series models
3819:Pitman–Yor process
2861:
2846:
2831:
2766:
2748:
2730:
2692:
2600:
2508:
2403:
2330:
2219:
2140:
2072:
2006:
1986:
1953:
1930:
1893:
1864:
1835:
1808:
1778:
1726:
1702:
1575:
1546:
1517:
1490:
1466:
1388:Hans-Ulrich Gerber
1357:
1309:
1263:
1230:
1178:
1127:
1013:
910:
890:
857:
806:
754:
734:
704:
674:
540:
511:
443:
377:
279:
251:
225:
204:
194:with distribution
184:
153:
133:
102:
71:
4618:Actuarial science
4605:
4604:
4559:Signal processing
4278:Doob's upcrossing
4273:Doob's martingale
4237:Engelbert–Schmidt
4180:Donsker's theorem
4114:Feller-continuous
3982:Rendleman–Bartter
3772:Dirichlet process
3689:Branching process
3658:Telegraph process
3551:Geometric process
3531:Empirical process
3521:Diffusion process
3377:Branching process
3372:Bernoulli process
3075:978-3-540-31342-7
3013:Blom, G. (1987).
2963:978-3-540-60931-5
2874:
2873:
1376:Michael R. Powers
1113:
998:
985:
962:
913:{\displaystyle n}
852:
804:
757:{\displaystyle F}
653:
607:
369:
228:{\textstyle \mu }
39:actuarial science
16:(Redirected from
4640:
4579:Machine learning
4466:Usual hypotheses
4349:Girsanov theorem
4334:Dynkin's formula
4099:Continuous paths
4007:Actuarial models
3947:Garman–Kohlhagen
3917:Black–Karasinski
3912:Black–Derman–Toy
3899:Financial models
3765:Fields and other
3694:Gaussian process
3643:Sigma-martingale
3447:Additive process
3349:
3342:
3335:
3326:
3321:
3312:
3294:
3293:
3287:
3278:
3272:
3271:
3243:
3234:
3233:
3210:
3197:
3186:
3180:
3173:
3167:
3166:
3140:
3131:
3130:
3091:VondraÄŤek, Zoran
3086:
3080:
3079:
3053:
3047:
3046:
3036:
3027:(4): 1335–1350.
3010:
3004:
3001:
2995:
2994:
2974:
2968:
2967:
2937:
2870:
2868:
2867:
2862:
2859:
2854:
2844:
2839:
2824:
2823:
2811:
2810:
2775:
2773:
2772:
2767:
2761:
2756:
2746:
2741:
2726:
2725:
2720:
2701:
2699:
2698:
2693:
2691:
2690:
2689:
2688:
2673:
2672:
2646:
2645:
2633:
2632:
2609:
2607:
2606:
2601:
2596:
2595:
2594:
2593:
2578:
2577:
2542:
2541:
2536:
2517:
2515:
2514:
2509:
2498:
2497:
2485:
2484:
2472:
2461:
2460:
2448:
2447:
2412:
2410:
2409:
2404:
2393:
2392:
2380:
2379:
2364:
2363:
2358:
2339:
2337:
2336:
2331:
2320:
2319:
2301:
2300:
2288:
2277:
2276:
2264:
2263:
2228:
2226:
2225:
2220:
2209:
2208:
2190:
2189:
2174:
2173:
2168:
2149:
2147:
2146:
2141:
2130:
2129:
2117:
2116:
2081:
2079:
2078:
2073:
2056:
2055:
2050:
2022:
2015:
2013:
2012:
2007:
1995:
1993:
1992:
1987:
1985:
1984:
1962:
1960:
1959:
1954:
1939:
1937:
1936:
1931:
1914:
1902:
1900:
1899:
1894:
1892:
1891:
1886:
1873:
1871:
1870:
1865:
1863:
1862:
1857:
1844:
1842:
1841:
1836:
1834:
1833:
1817:
1815:
1814:
1809:
1807:
1806:
1787:
1785:
1784:
1779:
1774:
1773:
1761:
1760:
1735:
1733:
1732:
1727:
1711:
1709:
1708:
1703:
1683:
1675:
1674:
1662:
1661:
1643:
1642:
1624:
1623:
1618:
1584:
1582:
1581:
1576:
1574:
1573:
1568:
1555:
1553:
1552:
1547:
1545:
1544:
1539:
1526:
1524:
1523:
1518:
1516:
1515:
1499:
1497:
1496:
1491:
1475:
1473:
1472:
1467:
1462:
1461:
1452:
1451:
1433:
1432:
1427:
1366:
1364:
1363:
1358:
1356:
1355:
1354:
1338:
1337:
1318:
1316:
1315:
1310:
1308:
1307:
1292:
1291:
1272:
1270:
1269:
1264:
1256:
1255:
1239:
1237:
1236:
1231:
1229:
1228:
1227:
1211:
1210:
1187:
1185:
1184:
1179:
1177:
1176:
1161:
1160:
1136:
1134:
1133:
1128:
1114:
1111:
1108:
1107:
1097:
1096:
1095:
1085:
1052:
1051:
1022:
1020:
1019:
1014:
1009:
1008:
1004:
1000:
999:
991:
986:
978:
963:
958:
950:
919:
917:
916:
911:
899:
897:
896:
891:
889:
888:
866:
864:
863:
858:
853:
850:
848:
844:
819:
814:
805:
797:
783:
782:
763:
761:
760:
755:
743:
741:
740:
735:
733:
732:
713:
711:
710:
705:
690:
682:
664:
663:
658:
654:
649:
641:
633:
628:
613:
609:
608:
603:
595:
549:
547:
546:
541:
520:
518:
517:
512:
452:
450:
449:
444:
427:
426:
421:
386:
384:
383:
378:
370:
367:
364:
363:
353:
352:
351:
341:
308:
307:
288:
286:
285:
280:
278:
277:
260:
258:
257:
252:
234:
232:
231:
226:
213:
211:
210:
205:
193:
191:
190:
185:
183:
182:
162:
160:
159:
154:
142:
140:
139:
134:
132:
131:
111:
109:
108:
103:
21:
4648:
4647:
4643:
4642:
4641:
4639:
4638:
4637:
4608:
4607:
4606:
4601:
4583:
4544:Queueing theory
4487:
4429:Skorokhod space
4292:
4283:Kunita–Watanabe
4254:
4220:Sanov's theorem
4190:Ergodic theorem
4163:
4159:Time-reversible
4077:
4040:Queueing models
4034:
4030:Sparre–Anderson
4020:Cramér–Lundberg
4001:
3987:SABR volatility
3893:
3850:
3802:Boolean network
3760:
3746:Renewal process
3677:
3626:Non-homogeneous
3616:Poisson process
3506:Contact process
3469:Brownian motion
3439:Continuous time
3433:
3427:Maximal entropy
3358:
3353:
3315:
3306:
3303:
3301:Further reading
3298:
3297:
3285:
3280:
3279:
3275:
3245:
3244:
3237:
3212:
3211:
3200:
3188:Thorin, Olof. "
3187:
3183:
3174:
3170:
3163:
3142:
3141:
3134:
3088:
3087:
3083:
3076:
3055:
3054:
3050:
3012:
3011:
3007:
3002:
2998:
2976:
2975:
2971:
2964:
2942:KlĂĽppelberg, C.
2940:Embrechts, P.;
2939:
2938:
2934:
2929:
2907:
2882:
2815:
2802:
2779:
2778:
2715:
2710:
2709:
2680:
2664:
2653:
2637:
2624:
2613:
2612:
2585:
2566:
2546:
2531:
2526:
2525:
2489:
2476:
2452:
2439:
2416:
2415:
2384:
2368:
2353:
2348:
2347:
2311:
2292:
2268:
2255:
2232:
2231:
2200:
2178:
2163:
2158:
2157:
2121:
2108:
2085:
2084:
2045:
2040:
2039:
1998:
1997:
1970:
1965:
1964:
1945:
1944:
1905:
1904:
1881:
1876:
1875:
1852:
1847:
1846:
1825:
1820:
1819:
1795:
1790:
1789:
1765:
1749:
1738:
1737:
1718:
1717:
1666:
1650:
1628:
1613:
1593:
1592:
1563:
1558:
1557:
1534:
1529:
1528:
1507:
1502:
1501:
1482:
1481:
1453:
1437:
1422:
1402:
1401:
1384:Elias S.W. Shiu
1373:
1339:
1329:
1321:
1320:
1293:
1283:
1275:
1274:
1247:
1242:
1241:
1212:
1202:
1194:
1193:
1190:renewal process
1162:
1152:
1144:
1143:
1112: for
1099:
1087:
1043:
1038:
1037:
1029:
976:
972:
964:
951:
929:
928:
902:
901:
877:
872:
871:
825:
821:
774:
769:
768:
746:
745:
724:
719:
718:
642:
636:
635:
596:
587:
583:
563:
562:
523:
522:
459:
458:
416:
396:
395:
355:
343:
299:
294:
293:
269:
264:
263:
243:
242:
217:
216:
196:
195:
174:
169:
168:
145:
144:
143:with intensity
123:
118:
117:
115:Poisson process
88:
87:
63:
61:Classical model
35:
28:
23:
22:
15:
12:
11:
5:
4646:
4644:
4636:
4635:
4630:
4625:
4620:
4610:
4609:
4603:
4602:
4600:
4599:
4594:
4592:List of topics
4588:
4585:
4584:
4582:
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4549:Renewal theory
4546:
4541:
4536:
4531:
4526:
4521:
4516:
4514:Ergodic theory
4511:
4506:
4504:Control theory
4501:
4495:
4493:
4489:
4488:
4486:
4485:
4484:
4483:
4478:
4468:
4463:
4458:
4453:
4448:
4447:
4446:
4436:
4434:Snell envelope
4431:
4426:
4421:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4300:
4298:
4294:
4293:
4291:
4290:
4285:
4280:
4275:
4270:
4264:
4262:
4256:
4255:
4253:
4252:
4233:Borel–Cantelli
4222:
4217:
4212:
4207:
4202:
4197:
4192:
4187:
4182:
4177:
4171:
4169:
4168:Limit theorems
4165:
4164:
4162:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4085:
4083:
4079:
4078:
4076:
4075:
4070:
4065:
4060:
4055:
4050:
4044:
4042:
4036:
4035:
4033:
4032:
4027:
4022:
4017:
4011:
4009:
4003:
4002:
4000:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3909:
3903:
3901:
3895:
3894:
3892:
3891:
3886:
3881:
3876:
3871:
3866:
3860:
3858:
3852:
3851:
3849:
3848:
3843:
3838:
3837:
3836:
3831:
3821:
3816:
3811:
3806:
3805:
3804:
3799:
3789:
3787:Hopfield model
3784:
3779:
3774:
3768:
3766:
3762:
3761:
3759:
3758:
3753:
3748:
3743:
3738:
3733:
3732:
3731:
3726:
3721:
3716:
3706:
3704:Markov process
3701:
3696:
3691:
3685:
3683:
3679:
3678:
3676:
3675:
3673:Wiener sausage
3670:
3668:Wiener process
3665:
3660:
3655:
3650:
3648:Stable process
3645:
3640:
3638:Semimartingale
3635:
3630:
3629:
3628:
3623:
3613:
3608:
3603:
3598:
3593:
3588:
3583:
3581:Jump diffusion
3578:
3573:
3568:
3563:
3558:
3556:Hawkes process
3553:
3548:
3543:
3538:
3536:Feller process
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3501:Cauchy process
3498:
3497:
3496:
3491:
3486:
3481:
3476:
3466:
3465:
3464:
3454:
3452:Bessel process
3449:
3443:
3441:
3435:
3434:
3432:
3431:
3430:
3429:
3424:
3419:
3414:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3368:
3366:
3360:
3359:
3354:
3352:
3351:
3344:
3337:
3329:
3323:
3322:
3313:
3302:
3299:
3296:
3295:
3273:
3235:
3224:(2): 101–118.
3198:
3181:
3168:
3161:
3132:
3081:
3074:
3048:
3005:
2996:
2969:
2962:
2931:
2930:
2928:
2925:
2924:
2923:
2918:
2913:
2911:Financial risk
2906:
2903:
2902:
2901:
2898:
2895:
2892:
2889:
2886:
2881:
2878:
2872:
2871:
2858:
2853:
2849:
2843:
2838:
2834:
2830:
2827:
2822:
2818:
2814:
2809:
2805:
2801:
2798:
2795:
2792:
2789:
2786:
2776:
2765:
2760:
2755:
2751:
2745:
2740:
2737:
2733:
2729:
2724:
2719:
2707:
2703:
2702:
2687:
2683:
2679:
2676:
2671:
2667:
2663:
2660:
2656:
2652:
2649:
2644:
2640:
2636:
2631:
2627:
2623:
2620:
2610:
2599:
2592:
2588:
2584:
2581:
2576:
2573:
2569:
2565:
2562:
2559:
2556:
2553:
2549:
2545:
2540:
2535:
2523:
2519:
2518:
2507:
2504:
2501:
2496:
2492:
2488:
2483:
2479:
2475:
2471:
2467:
2464:
2459:
2455:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2426:
2423:
2413:
2402:
2399:
2396:
2391:
2387:
2383:
2378:
2375:
2371:
2367:
2362:
2357:
2345:
2341:
2340:
2329:
2326:
2323:
2318:
2314:
2310:
2307:
2304:
2299:
2295:
2291:
2287:
2283:
2280:
2275:
2271:
2267:
2262:
2258:
2254:
2251:
2248:
2245:
2242:
2239:
2229:
2218:
2215:
2212:
2207:
2203:
2199:
2196:
2193:
2188:
2185:
2181:
2177:
2172:
2167:
2155:
2151:
2150:
2139:
2136:
2133:
2128:
2124:
2120:
2115:
2111:
2107:
2104:
2101:
2098:
2095:
2092:
2082:
2071:
2068:
2065:
2062:
2059:
2054:
2049:
2037:
2033:
2032:
2029:
2026:
2005:
1983:
1980:
1977:
1973:
1952:
1929:
1926:
1923:
1920:
1917:
1913:
1890:
1885:
1861:
1856:
1832:
1828:
1805:
1802:
1798:
1777:
1772:
1768:
1764:
1759:
1756:
1752:
1748:
1745:
1725:
1714:
1713:
1701:
1698:
1695:
1692:
1689:
1686:
1682:
1678:
1673:
1669:
1665:
1660:
1657:
1653:
1649:
1646:
1641:
1638:
1635:
1631:
1627:
1622:
1617:
1612:
1609:
1606:
1603:
1600:
1572:
1567:
1543:
1538:
1514:
1510:
1489:
1478:
1477:
1465:
1460:
1456:
1450:
1447:
1444:
1440:
1436:
1431:
1426:
1421:
1418:
1415:
1412:
1409:
1372:
1369:
1353:
1349:
1346:
1342:
1336:
1332:
1328:
1306:
1303:
1300:
1296:
1290:
1286:
1282:
1262:
1259:
1254:
1250:
1226:
1222:
1219:
1215:
1209:
1205:
1201:
1175:
1172:
1169:
1165:
1159:
1155:
1151:
1140:
1139:
1138:
1137:
1126:
1123:
1120:
1117:
1106:
1102:
1094:
1090:
1084:
1081:
1078:
1074:
1070:
1067:
1064:
1061:
1058:
1055:
1050:
1046:
1028:
1025:
1024:
1023:
1012:
1007:
1003:
997:
994:
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289:are given by:
276:
272:
254:{\textstyle x}
250:
224:
207:{\textstyle F}
203:
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76:Filip Lundberg
62:
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26:
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14:
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2:
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4451:Stopping time
4449:
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4250:
4246:
4242:
4241:Hewitt–Savage
4238:
4234:
4230:
4226:
4225:Zero–one laws
4223:
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4218:
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3922:Black–Scholes
3920:
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3825:
3824:Point process
3822:
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3807:
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3800:
3798:
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3782:Gibbs measure
3780:
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3571:ItĂ´ diffusion
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3546:Gamma process
3544:
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3417:Self-avoiding
3415:
3413:
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3408:
3405:
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3402:Moran process
3400:
3398:
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3375:
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3364:Discrete time
3361:
3357:
3350:
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3338:
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3327:
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3257:
3253:
3249:
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3227:
3223:
3219:
3215:
3214:Powers, M. R.
3209:
3207:
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3203:
3199:
3195:
3191:
3185:
3182:
3178:
3172:
3169:
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3162:9780470317044
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2093:
2090:
2083:
2063:
2060:
2052:
2038:
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2027:
2025:Special case
2024:
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2017:
2003:
1981:
1978:
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1950:
1941:
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1918:
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1026:
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987:
982:
979:
973:
969:
965:
959:
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952:
946:
940:
934:
927:
926:
925:
923:
907:
885:
882:
878:
854:
845:
838:
832:
829:
826:
822:
816:
811:
807:
801:
798:
793:
787:
779:
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767:
766:
765:
751:
729:
725:
695:
687:
684:
679:
675:
671:
668:
660:
655:
650:
646:
643:
637:
625:
622:
619:
615:
610:
604:
600:
597:
591:
588:
584:
580:
574:
568:
561:
560:
559:
557:
553:
534:
505:
502:
496:
490:
486:
482:
479:
476:
467:
464:
434:
431:
423:
413:
407:
401:
394:
393:
392:
374:
371:
360:
356:
348:
344:
338:
335:
332:
328:
324:
321:
318:
315:
312:
309:
304:
300:
292:
291:
290:
274:
270:
261:
248:
239:
236:(they form a
235:
222:
201:
179:
175:
166:
150:
128:
124:
116:
112:
99:
96:
93:
83:
81:
80:Harald Cramér
77:
67:
60:
58:
56:
52:
48:
44:
40:
33:
19:
4553:
4509:Econometrics
4471:Wiener space
4359:ItĂ´ integral
4260:Inequalities
4149:Self-similar
4119:Gauss–Markov
4109:Exchangeable
4089:CĂ dlĂ g paths
4025:Risk process
3977:LIBOR market
3846:Random graph
3841:Random field
3653:Superprocess
3591:LĂ©vy process
3586:Jump process
3561:Hunt process
3397:Markov chain
3317:
3308:
3289:
3276:
3251:
3247:
3221:
3217:
3196:(1974): 104.
3193:
3184:
3176:
3171:
3144:
3098:
3094:
3084:
3057:
3051:
3024:
3018:
3008:
2999:
2982:
2978:
2972:
2945:
2935:
2875:
2018:
1942:
1715:
1587:
1479:
1392:
1379:
1374:
1141:
1030:
900:denotes the
869:
716:
456:
389:
241:
215:
86:
84:
72:
54:
50:
46:
36:
32:Tirpitz Plan
4554:Ruin theory
4492:Disciplines
4364:ItĂ´'s lemma
4139:Predictable
3814:Percolation
3797:Potts model
3792:Ising model
3756:White noise
3714:Differences
3576:ItĂ´ process
3516:Cox process
3412:Loop-erased
3407:Random walk
3105:: 679–690.
922:convolution
556:M/G/1 queue
368: for t
51:risk theory
49:(sometimes
47:ruin theory
4612:Categories
4564:Statistics
4344:Filtration
4245:Kolmogorov
4229:Blumenthal
4154:Stationary
4094:Continuous
4082:Properties
3967:Hull–White
3709:Martingale
3596:Local time
3484:Fractional
3462:pure birth
2927:References
4476:Classical
3489:Geometric
3479:Excursion
3254:: 48–72.
2985:(2): 85.
2785:δ
2754:τ
2739:−
2736:τ
2675:−
2659:−
2591:τ
2580:−
2575:−
2572:τ
2561:−
2558:τ
2555:δ
2552:−
2422:δ
2390:τ
2382:−
2377:−
2374:τ
2238:δ
2206:τ
2187:−
2184:τ
2091:δ
2067:∞
2061:τ
2004:τ
1982:τ
1979:δ
1976:−
1951:τ
1925:∞
1919:τ
1831:τ
1804:−
1801:τ
1771:τ
1758:−
1755:τ
1724:δ
1694:∞
1688:τ
1672:τ
1659:−
1656:τ
1640:τ
1637:δ
1634:−
1513:τ
1488:δ
1459:τ
1449:τ
1446:δ
1443:−
1348:∈
1331:ξ
1302:≥
1249:ξ
1221:∈
1204:ξ
1171:≥
1119:≥
1101:ξ
1073:∑
1069:−
993:λ
988:−
983:μ
970:−
956:μ
953:λ
935:ψ
883:∗
879:⋅
830:−
808:∫
802:μ
685:∗
672:−
647:μ
644:λ
631:∞
616:∑
601:μ
598:λ
592:−
569:ψ
538:∞
532:∅
465:τ
438:∞
432:τ
402:ψ
372:≥
357:ξ
329:∑
325:−
223:μ
214:and mean
176:ξ
151:λ
4597:Category
4481:Abstract
4015:BĂĽhlmann
3621:Compound
3268:59054002
3127:14499808
2905:See also
163:and are
4104:Ergodic
3992:VašĂÄŤek
3834:Poisson
3494:Meander
3119:4141346
3043:2241677
4444:Tanaka
4129:Mixing
4124:Markov
3997:Wilkie
3962:Ho–Lee
3957:Heston
3729:Super-
3474:Bridge
3422:Biased
3266:
3159:
3125:
3117:
3072:
3041:
2960:
1716:where
1480:where
1395:Powers
920:-fold
717:where
4297:Tools
4073:M/M/c
4068:M/M/1
4063:M/G/1
4053:Fluid
3719:Local
3286:(PDF)
3264:S2CID
3123:S2CID
3115:JSTOR
3101:(3).
3039:JSTOR
1319:and
1188:is a
4633:Risk
4249:LĂ©vy
4048:Bulk
3932:Chen
3724:Sub-
3682:Both
3157:ISBN
3070:ISBN
2958:ISBN
2500:<
2395:<
2322:<
2303:<
2211:<
2192:<
2064:<
1922:<
1691:<
1386:and
1258:>
1192:and
870:and
503:<
480:>
435:<
97:>
41:and
3829:Cox
3256:doi
3226:doi
3149:doi
3107:doi
3062:doi
3029:doi
2987:doi
2950:doi
1393:In
529:inf
471:inf
53:or
37:In
4614::
4247:,
4243:,
4239:,
4235:,
4231:,
3288:.
3262:.
3250:.
3238:^
3222:17
3220:.
3201:^
3192:"
3155:.
3135:^
3121:.
3113:.
3099:41
3097:.
3068:.
3037:.
3025:15
3023:.
3017:.
2981:.
2956:.
764:,
558:)
375:0.
82:.
45:,
4251:)
4227:(
3348:e
3341:t
3334:v
3292:.
3270:.
3258::
3252:2
3232:.
3228::
3165:.
3151::
3129:.
3109::
3078:.
3064::
3045:.
3031::
2993:.
2989::
2983:6
2966:.
2952::
2857:k
2852:2
2848:x
2842:j
2837:1
2833:x
2829:=
2826:)
2821:2
2817:x
2813:,
2808:1
2804:x
2800:(
2797:w
2794:,
2791:0
2788:=
2764:]
2759:k
2750:X
2744:j
2732:X
2728:[
2723:x
2718:E
2686:2
2682:x
2678:z
2670:1
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