17:
575:
2675:
1230:
1498:
2284:
425:
2524:
992:
253:
2488:
906:
1305:
1675:
1269:
2139:
2129:
3254:
1958:
2749:
1886:
1600:
960:
807:
108:
3789:
2005:
at a hitting time of the underlying process, whilst theorems of the second kind are in terms of a stopping time at which the field of local times first exceeds a given value.
642:
1235:
A more general form was proven independently by Meyer and Wang; the formula extends ItĂŽ's lemma for twice differentiable functions to a more general class of functions. If
985:
1531:
3070:
3613:
1297:
350:
1715:
276:
698:
377:
4216:
3209:
721:
3746:
3726:
2793:
2769:
1978:
1819:
1795:
1775:
1755:
1735:
1695:
1554:
761:
741:
662:
417:
397:
148:
128:
306:
4130:
570:{\displaystyle L^{x}(t)=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}\int _{0}^{t}1_{\{x-\varepsilon <B_{s}<x+\varepsilon \}}\,d_{s},}
2670:{\displaystyle \left\{L_{T_{a}}^{x}+W_{x}^{2}\colon x\geq 0\right\}{\stackrel {\mathcal {D}}{=}}\left\{(W_{x}+{\sqrt {a}})^{2}\colon x\geq 0\right\}.\,}
4047:
4057:
3731:
3741:
4099:
1225:{\displaystyle L^{x}(t)=|X_{t}-x|-|X_{0}-x|-\int _{0}^{t}\left(1_{(0,\infty )}(X_{s}-x)-1_{(-\infty ,0]}(X_{s}-x)\right)\,dX_{s},\qquad t\geq 0.}
3814:
3996:
4286:
4276:
3799:
3181:
3167:
3152:
2923:
4186:
4150:
1825:
4103:
4454:
4191:
3301:
3202:
4256:
3834:
3804:
156:
4107:
4091:
1493:{\displaystyle F(X_{t})=F(X_{0})+\int _{0}^{t}F'_{-}(X_{s})\,dX_{s}+{\frac {1}{2}}\int _{-\infty }^{\infty }L^{x}(t)\,dF'_{-}(x),}
4301:
4006:
3226:
2803:
Results of RayâKnight type for more general stochastic processes have been intensively studied, and analogue statements of both (
2418:
815:
4206:
4171:
4140:
4135:
3774:
3571:
3488:
3019:
4145:
3473:
1605:
4485:
3769:
3576:
3124:
2997:
2886:
3495:
1238:
4231:
4111:
4480:
4459:
4236:
4072:
3971:
3956:
3368:
3284:
3195:
4246:
3882:
4241:
2907:
2279:{\displaystyle \left\{L^{x}(T)\colon x\in \right\}{\stackrel {\mathcal {D}}{=}}\left\{|W_{x}|^{2}\colon x\in \right\}\,}
313:
3844:
3428:
3373:
3289:
580:
which may be shown to always exist. Note that in the special case of
Brownian motion (or more generally a real-valued
4176:
4166:
3809:
3779:
2074:
4181:
3346:
3244:
1829:
3892:
3468:
3249:
4261:
4062:
3976:
3961:
3351:
4095:
3981:
3403:
1899:
3483:
2696:
4201:
3784:
3319:
52:, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of
44:, that characterizes the amount of time a particle has spent at a given level. Local time appears in various
4396:
4386:
4077:
3859:
3598:
3463:
3274:
3681:
4338:
4266:
3525:
45:
1835:
4361:
4343:
4323:
4318:
4037:
3869:
3849:
3696:
3639:
3478:
3388:
1559:
3829:
919:
766:
67:
587:
4436:
4391:
4381:
4122:
4067:
4042:
4011:
3991:
3751:
3736:
3603:
916:
Tanaka's formula also provides a definition of local time for an arbitrary continuous semimartingale
279:
4431:
4271:
4196:
4001:
3761:
3671:
3561:
2989:
2878:
2824:
965:
309:
49:
4401:
4366:
4281:
4251:
4021:
4016:
3839:
3676:
3341:
3279:
3218:
3089:
2962:
29:
4082:
1980:
is a well studied topic in the area of random fields. RayâKnight type theorems relate the field
4421:
4226:
3877:
3634:
3551:
3520:
3413:
3393:
3383:
3239:
3234:
3177:
3163:
3148:
3120:
3116:
2993:
2919:
2882:
319:
4087:
3824:
1700:
261:
4441:
4328:
4211:
3581:
3556:
3505:
3433:
3356:
3309:
3108:
3079:
3046:
3028:
2981:
2952:
2911:
2870:
1990:
1506:
3042:
667:
355:
4406:
4306:
4291:
4052:
3986:
3664:
3608:
3591:
3336:
3050:
3038:
2829:
41:
4221:
3453:
2982:
2871:
1274:
703:
4411:
4376:
4296:
3902:
3649:
3566:
3535:
3530:
3510:
3500:
3443:
3438:
3418:
3398:
3363:
3331:
3314:
2778:
2772:
2754:
2348:
1963:
1804:
1780:
1760:
1740:
1720:
1680:
1539:
746:
726:
647:
402:
382:
133:
113:
37:
285:
4474:
4313:
3854:
3691:
3686:
3644:
3586:
3408:
3324:
3264:
3109:
2966:
581:
4371:
4333:
3887:
3819:
3708:
3703:
3515:
3448:
3423:
3259:
2834:
53:
16:
3951:
1798:
2915:
4416:
3935:
3930:
3925:
3915:
3718:
3659:
3654:
3618:
3378:
3269:
25:
4426:
3966:
3910:
3794:
352:
is an (appropriately rescaled and time-parametrized) measure of how much time
3033:
3014:
3920:
1996:
In general RayâKnight type theorems of the first kind consider the field
20:
A sample path of an ItĆ process together with its surface of local times.
2902:
Meyer, Paul-Andre (2002) . "Un cours sur les intégrales stochastiques".
3747:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
3187:
3093:
2957:
2941:"Generalized ItĂŽ's formula and additive functionals of Brownian motion"
2940:
3084:
3065:
2945:
3191:
3066:"Random walks and a sojourn density process of Brownian motion"
2509:
be an independent one-dimensional
Brownian motion started from
419:. More rigorously, it may be written as the almost sure limit
248:{\displaystyle L^{x}(t)=\int _{0}^{t}\delta (x-B_{s})\,d_{s},}
2771:) is equal in distribution to the square of a 0-dimensional
2600:
2206:
2483:{\displaystyle T_{a}=\inf\{t\geq 0\colon L_{t}^{0}>a\}.}
2056:
be a standard two-dimensional
Brownian motion started from
901:{\displaystyle L^{x}(t)=\int _{0}^{t}1_{\{x\}}(X_{s})\,ds.}
3727:
Autoregressive conditional heteroskedasticity (ARCH) model
2408:
be the first time at which the local time at zero exceeds
150:
is the stochastic process which is informally defined by
3255:
Independent and identically distributed random variables
1670:{\displaystyle L=(L^{x}(t))_{x\in \mathbb {R} ,t\geq 0}}
3732:
Autoregressive integrated moving average (ARIMA) model
1677:
has a modification which is a.s. Hölder continuous in
1264:{\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} }
3160:
Markov
Processes, Gaussian Processes, and Local Times
2815:) are known for strongly symmetric Markov processes.
2781:
2757:
2699:
2527:
2421:
2142:
2077:
1966:
1902:
1838:
1807:
1783:
1763:
1743:
1723:
1703:
1683:
1608:
1562:
1542:
1509:
1308:
1277:
1241:
995:
968:
922:
818:
769:
749:
729:
706:
670:
650:
590:
428:
405:
385:
358:
322:
288:
264:
159:
136:
116:
70:
3111:
Markov
Processes, Gaussian Processes and Local Times
723:, which explains why it is called the local time of
4354:
4159:
4121:
4030:
3944:
3901:
3868:
3760:
3717:
3627:
3544:
3300:
3225:
2787:
2763:
2743:
2669:
2482:
2278:
2123:
2029:be a one-dimensional Brownian motion started from
1972:
1952:
1880:
1813:
1789:
1769:
1749:
1729:
1709:
1689:
1669:
1594:
1548:
1525:
1492:
1291:
1263:
1224:
979:
954:
900:
801:
755:
735:
715:
692:
656:
636:
569:
411:
391:
371:
344:
300:
270:
247:
142:
122:
102:
3176:, 1st edition, 2010, Cambridge University Press,
3115:. New York: Cambridge University Press. pp.
3071:Transactions of the American Mathematical Society
809:, the local time can be expressed more simply as
3614:Stochastic chains with memory of variable length
3162:, 1st edition, 2006, Cambridge University Press
2435:
2084:
452:
2124:{\displaystyle T=\inf\{t\geq 0\colon B_{t}=0\}}
2372:be a standard one-dimensional Brownian motion
1960:associated to a stochastic process on a space
1777:has a modification that is a.s. continuous in
3203:
2131:. Ray and Knight (independently) showed that
8:
2751:(which is a process in the spatial variable
2474:
2438:
2399:be the associated field of local times. Let
2118:
2087:
867:
861:
539:
502:
64:For a continuous real-valued semimartingale
3742:Autoregressiveâmoving-average (ARMA) model
3210:
3196:
3188:
2854:Karatzas, Ioannis; Shreve, Steven (1991).
1953:{\displaystyle L_{t}=(L_{t}^{x})_{x\in E}}
1271:is absolutely continuous with derivative
3083:
3032:
2956:
2780:
2756:
2744:{\displaystyle (L_{T_{a}}^{x})_{x\geq 0}}
2729:
2719:
2712:
2707:
2698:
2666:
2640:
2629:
2620:
2599:
2598:
2593:
2591:
2590:
2567:
2562:
2549:
2542:
2537:
2526:
2462:
2457:
2426:
2420:
2275:
2240:
2235:
2228:
2219:
2205:
2204:
2199:
2197:
2196:
2152:
2141:
2106:
2076:
1965:
1938:
1928:
1923:
1907:
1901:
1866:
1857:
1851:
1842:
1837:
1806:
1782:
1762:
1742:
1722:
1702:
1682:
1649:
1648:
1641:
1622:
1607:
1581:
1561:
1541:
1514:
1508:
1469:
1461:
1446:
1436:
1428:
1414:
1405:
1397:
1388:
1372:
1362:
1357:
1341:
1319:
1307:
1276:
1257:
1256:
1249:
1248:
1240:
1203:
1195:
1175:
1147:
1125:
1100:
1085:
1080:
1068:
1056:
1047:
1039:
1027:
1018:
1000:
994:
970:
969:
967:
940:
930:
921:
888:
879:
860:
850:
845:
823:
817:
787:
777:
768:
748:
728:
705:
684:
669:
649:
618:
589:
558:
544:
521:
501:
491:
486:
467:
455:
433:
427:
404:
384:
363:
357:
327:
321:
287:
263:
236:
222:
213:
191:
186:
164:
158:
135:
115:
88:
78:
69:
36:is a stochastic process associated with
15:
2856:Brownian Motion and Stochastic Calculus
2846:
1824:Tanaka's formula provides the explicit
4048:Doob's martingale convergence theorems
3145:Introduction to Stochastic Integration
3015:"Sojourn times of a diffusion process"
3800:Constant elasticity of variance (CEV)
3790:ChanâKarolyiâLongstaffâSanders (CKLS)
2334:, and equality is in distribution on
763:. For a discrete state-space process
7:
2518:
2133:
2067:. Define the stopping time at which
1299:which is of bounded variation, then
2904:SĂ©minaire de probabilitĂ©s 1967â1980
1881:{\displaystyle (|B_{s}|)_{s\geq 0}}
1556:is a Brownian motion, then for any
4287:Skorokhod's representation theorem
4068:Law of large numbers (weak/strong)
2910:Vol. 1771. pp. 174â329.
1595:{\displaystyle \alpha \in (0,1/2)}
1437:
1432:
1154:
1110:
14:
4257:Martingale representation theorem
3147:, 2nd edition, 1990, BirkhÀuser,
2984:Foundations of Modern Probability
2873:Foundations of Modern Probability
955:{\displaystyle (X_{s})_{s\geq 0}}
802:{\displaystyle (X_{s})_{s\geq 0}}
103:{\displaystyle (B_{s})_{s\geq 0}}
4302:Stochastic differential equation
4192:Doob's optional stopping theorem
4187:DoobâMeyer decomposition theorem
3143:K. L. Chung and R. J. Williams,
2318:is the field of local times of (
664:is a Brownian motion), the term
637:{\displaystyle dB=b(t,B)\,dt+dW}
4172:Convergence of random variables
4058:FisherâTippettâGnedenko theorem
3020:Illinois Journal of Mathematics
2988:. New York: Springer. pp.
2877:. New York: Springer. pp.
2799:Generalized RayâKnight theorems
1212:
3770:Binomial options pricing model
2726:
2700:
2637:
2613:
2267:
2255:
2236:
2220:
2188:
2176:
2164:
2158:
1935:
1916:
1863:
1858:
1843:
1839:
1638:
1634:
1628:
1615:
1589:
1569:
1484:
1478:
1458:
1452:
1394:
1381:
1347:
1334:
1325:
1312:
1253:
1187:
1168:
1163:
1148:
1137:
1118:
1113:
1101:
1069:
1048:
1040:
1019:
1012:
1006:
937:
923:
885:
872:
835:
829:
784:
770:
681:
674:
615:
603:
555:
548:
459:
445:
439:
339:
333:
295:
289:
233:
226:
219:
200:
176:
170:
85:
71:
1:
4237:Kolmogorov continuity theorem
4073:Law of the iterated logarithm
980:{\displaystyle \mathbb {R} :}
312:. It is a notion invented by
4242:Kolmogorov extension theorem
3921:Generalized queueing network
3429:Interacting particle systems
2916:10.1007/978-3-540-45530-1_11
2795:, and as such is Markovian.
3374:Continuous-time random walk
2811:
2805:
4502:
4382:Extreme value theory (EVT)
4182:Doob decomposition theorem
3474:OrnsteinâUhlenbeck process
3245:Chinese restaurant process
2693:Equivalently, the process
2347:| is known as the squared
1830:reflecting Brownian motion
4450:
4262:Optional stopping theorem
4063:Large deviation principle
3815:HeathâJarrowâMorton (HJM)
3752:Moving-average (MA) model
3737:Autoregressive (AR) model
3562:Hidden Markov model (HMM)
3496:SchrammâLoewner evolution
3172:P. Mörters and Y. Peres,
2355:Second RayâKnight theorem
1896:The field of local times
1602:the field of local times
316:. The basic idea is that
4177:Doléans-Dade exponential
4007:Progressively measurable
3805:CoxâIngersollâRoss (CIR)
3158:M. Marcus and J. Rosen,
2009:First RayâKnight theorem
1828:for the one-dimensional
1826:DoobâMeyer decomposition
1717:, uniformly for bounded
1533:is the left derivative.
345:{\displaystyle L^{x}(t)}
4397:Mathematical statistics
4387:Large deviations theory
4217:Infinitesimal generator
4078:Maximal ergodic theorem
3997:Piecewise-deterministic
3599:Random dynamical system
3464:Markov additive process
2071:first hits the origin,
1710:{\displaystyle \alpha }
911:
271:{\displaystyle \delta }
4232:KarhunenâLoĂšve theorem
4167:CameronâMartin formula
4131:BurkholderâDavisâGundy
3526:Variance gamma process
3107:Marcus; Rosen (2006).
3064:Knight, F. B. (1963).
3034:10.1215/ijm/1255645099
2789:
2765:
2745:
2671:
2484:
2280:
2125:
1974:
1954:
1882:
1815:
1791:
1771:
1751:
1731:
1711:
1691:
1671:
1596:
1550:
1527:
1526:{\displaystyle F'_{-}}
1494:
1293:
1265:
1226:
981:
956:
902:
803:
757:
737:
717:
694:
658:
638:
571:
413:
393:
373:
346:
302:
272:
249:
144:
124:
104:
46:stochastic integration
21:
4486:Statistical mechanics
4362:Actuarial mathematics
4324:Uniform integrability
4319:Stratonovich integral
4247:LĂ©vyâProkhorov metric
4151:MarcinkiewiczâZygmund
4038:Central limit theorem
3640:Gaussian random field
3469:McKeanâVlasov process
3389:Dyson Brownian motion
3250:GaltonâWatson process
2790:
2766:
2746:
2672:
2485:
2281:
2126:
1975:
1955:
1883:
1816:
1792:
1772:
1752:
1732:
1712:
1692:
1672:
1597:
1551:
1528:
1495:
1294:
1266:
1227:
982:
957:
903:
804:
758:
738:
718:
695:
693:{\displaystyle d_{s}}
659:
639:
572:
414:
394:
374:
372:{\displaystyle B_{s}}
347:
303:
273:
250:
145:
125:
105:
19:
4481:Stochastic processes
4437:Time series analysis
4392:Mathematical finance
4277:Reflection principle
3604:Regenerative process
3404:FlemingâViot process
3219:Stochastic processes
2908:Lect. Notes in Math.
2779:
2755:
2697:
2525:
2419:
2140:
2075:
1964:
1900:
1836:
1805:
1781:
1761:
1741:
1721:
1701:
1681:
1606:
1560:
1540:
1507:
1306:
1275:
1239:
993:
966:
920:
816:
767:
747:
727:
704:
668:
648:
588:
426:
403:
383:
356:
320:
286:
280:Dirac delta function
262:
157:
134:
114:
110:, the local time of
68:
30:stochastic processes
4432:Stochastic analysis
4272:Quadratic variation
4267:Prokhorov's theorem
4202:FeynmanâKac formula
3672:Markov random field
3320:Birthâdeath process
2980:Kallenberg (1997).
2869:Kallenberg (1997).
2724:
2572:
2554:
2467:
1933:
1892:RayâKnight theorems
1522:
1477:
1441:
1380:
1367:
1292:{\displaystyle F',}
1090:
855:
496:
310:quadratic variation
196:
4402:Probability theory
4282:Skorokhod integral
4252:Malliavin calculus
3835:Korn-Kreer-Lenssen
3719:Time series models
3682:PitmanâYor process
2958:10.1007/bf00538419
2785:
2761:
2741:
2703:
2667:
2558:
2533:
2480:
2453:
2276:
2121:
1970:
1950:
1919:
1878:
1811:
1787:
1767:
1747:
1727:
1707:
1687:
1667:
1592:
1546:
1523:
1510:
1490:
1465:
1424:
1368:
1353:
1289:
1261:
1222:
1076:
977:
952:
898:
841:
799:
753:
733:
716:{\displaystyle ds}
713:
700:simply reduces to
690:
654:
634:
567:
482:
466:
409:
389:
369:
342:
298:
268:
245:
182:
140:
120:
100:
48:formulas, such as
40:processes such as
22:
4468:
4467:
4422:Signal processing
4141:Doob's upcrossing
4136:Doob's martingale
4100:EngelbertâSchmidt
4043:Donsker's theorem
3977:Feller-continuous
3845:RendlemanâBartter
3635:Dirichlet process
3552:Branching process
3521:Telegraph process
3414:Geometric process
3394:Empirical process
3384:Diffusion process
3240:Branching process
3235:Bernoulli process
3182:978-0-521-76018-8
3168:978-0-521-86300-1
3153:978-0-8176-3386-8
2925:978-3-540-42813-8
2788:{\displaystyle a}
2764:{\displaystyle x}
2691:
2690:
2634:
2605:
2300:
2299:
2211:
1989:to an associated
1973:{\displaystyle E}
1814:{\displaystyle x}
1790:{\displaystyle t}
1770:{\displaystyle L}
1750:{\displaystyle t}
1730:{\displaystyle x}
1690:{\displaystyle x}
1549:{\displaystyle X}
1422:
756:{\displaystyle x}
736:{\displaystyle B}
657:{\displaystyle W}
480:
451:
412:{\displaystyle t}
392:{\displaystyle x}
143:{\displaystyle x}
123:{\displaystyle B}
60:Formal definition
4493:
4442:Machine learning
4329:Usual hypotheses
4212:Girsanov theorem
4197:Dynkin's formula
3962:Continuous paths
3870:Actuarial models
3810:GarmanâKohlhagen
3780:BlackâKarasinski
3775:BlackâDermanâToy
3762:Financial models
3628:Fields and other
3557:Gaussian process
3506:Sigma-martingale
3310:Additive process
3212:
3205:
3198:
3189:
3131:
3130:
3114:
3104:
3098:
3097:
3087:
3061:
3055:
3054:
3036:
3013:Ray, D. (1963).
3010:
3004:
3003:
2987:
2977:
2971:
2970:
2960:
2936:
2930:
2929:
2899:
2893:
2892:
2876:
2866:
2860:
2859:
2851:
2825:Tanaka's formula
2794:
2792:
2791:
2786:
2770:
2768:
2767:
2762:
2750:
2748:
2747:
2742:
2740:
2739:
2723:
2718:
2717:
2716:
2685:
2676:
2674:
2673:
2668:
2662:
2658:
2645:
2644:
2635:
2630:
2625:
2624:
2607:
2606:
2604:
2603:
2597:
2592:
2589:
2585:
2571:
2566:
2553:
2548:
2547:
2546:
2519:
2489:
2487:
2486:
2481:
2466:
2461:
2431:
2430:
2294:
2285:
2283:
2282:
2277:
2274:
2270:
2245:
2244:
2239:
2233:
2232:
2223:
2213:
2212:
2210:
2209:
2203:
2198:
2195:
2191:
2157:
2156:
2134:
2130:
2128:
2127:
2122:
2111:
2110:
1991:Gaussian process
1979:
1977:
1976:
1971:
1959:
1957:
1956:
1951:
1949:
1948:
1932:
1927:
1912:
1911:
1887:
1885:
1884:
1879:
1877:
1876:
1861:
1856:
1855:
1846:
1820:
1818:
1817:
1812:
1796:
1794:
1793:
1788:
1776:
1774:
1773:
1768:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1666:
1665:
1652:
1627:
1626:
1601:
1599:
1598:
1593:
1585:
1555:
1553:
1552:
1547:
1532:
1530:
1529:
1524:
1518:
1499:
1497:
1496:
1491:
1473:
1451:
1450:
1440:
1435:
1423:
1415:
1410:
1409:
1393:
1392:
1376:
1366:
1361:
1346:
1345:
1324:
1323:
1298:
1296:
1295:
1290:
1285:
1270:
1268:
1267:
1262:
1260:
1252:
1231:
1229:
1228:
1223:
1208:
1207:
1194:
1190:
1180:
1179:
1167:
1166:
1130:
1129:
1117:
1116:
1089:
1084:
1072:
1061:
1060:
1051:
1043:
1032:
1031:
1022:
1005:
1004:
986:
984:
983:
978:
973:
961:
959:
958:
953:
951:
950:
935:
934:
912:Tanaka's formula
907:
905:
904:
899:
884:
883:
871:
870:
854:
849:
828:
827:
808:
806:
805:
800:
798:
797:
782:
781:
762:
760:
759:
754:
742:
740:
739:
734:
722:
720:
719:
714:
699:
697:
696:
691:
689:
688:
663:
661:
660:
655:
643:
641:
640:
635:
576:
574:
573:
568:
563:
562:
543:
542:
526:
525:
495:
490:
481:
479:
468:
465:
438:
437:
418:
416:
415:
410:
398:
396:
395:
390:
378:
376:
375:
370:
368:
367:
351:
349:
348:
343:
332:
331:
307:
305:
304:
301:{\displaystyle }
299:
277:
275:
274:
269:
254:
252:
251:
246:
241:
240:
218:
217:
195:
190:
169:
168:
149:
147:
146:
141:
129:
127:
126:
121:
109:
107:
106:
101:
99:
98:
83:
82:
50:Tanaka's formula
4501:
4500:
4496:
4495:
4494:
4492:
4491:
4490:
4471:
4470:
4469:
4464:
4446:
4407:Queueing theory
4350:
4292:Skorokhod space
4155:
4146:KunitaâWatanabe
4117:
4083:Sanov's theorem
4053:Ergodic theorem
4026:
4022:Time-reversible
3940:
3903:Queueing models
3897:
3893:SparreâAnderson
3883:CramĂ©râLundberg
3864:
3850:SABR volatility
3756:
3713:
3665:Boolean network
3623:
3609:Renewal process
3540:
3489:Non-homogeneous
3479:Poisson process
3369:Contact process
3332:Brownian motion
3302:Continuous time
3296:
3290:Maximal entropy
3221:
3216:
3174:Brownian Motion
3140:
3135:
3134:
3127:
3106:
3105:
3101:
3085:10.2307/1993647
3063:
3062:
3058:
3012:
3011:
3007:
3000:
2979:
2978:
2974:
2938:
2937:
2933:
2926:
2901:
2900:
2896:
2889:
2868:
2867:
2863:
2853:
2852:
2848:
2843:
2830:Brownian motion
2821:
2801:
2777:
2776:
2753:
2752:
2725:
2708:
2695:
2694:
2683:
2636:
2616:
2612:
2608:
2538:
2532:
2528:
2523:
2522:
2515:
2508:
2501:
2422:
2417:
2416:
2407:
2398:
2391:
2378:
2371:
2367:
2357:
2346:
2338:. The process |
2333:
2326:
2317:
2310:
2292:
2234:
2224:
2218:
2214:
2148:
2147:
2143:
2138:
2137:
2102:
2073:
2072:
2062:
2055:
2048:
2035:
2028:
2021:
2011:
2004:
1988:
1962:
1961:
1934:
1903:
1898:
1897:
1894:
1862:
1847:
1834:
1833:
1803:
1802:
1779:
1778:
1759:
1758:
1757:. In general,
1739:
1738:
1719:
1718:
1699:
1698:
1679:
1678:
1637:
1618:
1604:
1603:
1558:
1557:
1538:
1537:
1505:
1504:
1442:
1401:
1384:
1337:
1315:
1304:
1303:
1278:
1273:
1272:
1237:
1236:
1199:
1171:
1143:
1121:
1096:
1095:
1091:
1052:
1023:
996:
991:
990:
964:
963:
936:
926:
918:
917:
914:
875:
856:
819:
814:
813:
783:
773:
765:
764:
745:
744:
725:
724:
702:
701:
680:
666:
665:
646:
645:
586:
585:
554:
517:
497:
472:
429:
424:
423:
401:
400:
381:
380:
359:
354:
353:
323:
318:
317:
284:
283:
260:
259:
232:
209:
160:
155:
154:
132:
131:
112:
111:
84:
74:
66:
65:
62:
42:Brownian motion
12:
11:
5:
4499:
4497:
4489:
4488:
4483:
4473:
4472:
4466:
4465:
4463:
4462:
4457:
4455:List of topics
4451:
4448:
4447:
4445:
4444:
4439:
4434:
4429:
4424:
4419:
4414:
4412:Renewal theory
4409:
4404:
4399:
4394:
4389:
4384:
4379:
4377:Ergodic theory
4374:
4369:
4367:Control theory
4364:
4358:
4356:
4352:
4351:
4349:
4348:
4347:
4346:
4341:
4331:
4326:
4321:
4316:
4311:
4310:
4309:
4299:
4297:Snell envelope
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4163:
4161:
4157:
4156:
4154:
4153:
4148:
4143:
4138:
4133:
4127:
4125:
4119:
4118:
4116:
4115:
4096:BorelâCantelli
4085:
4080:
4075:
4070:
4065:
4060:
4055:
4050:
4045:
4040:
4034:
4032:
4031:Limit theorems
4028:
4027:
4025:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3948:
3946:
3942:
3941:
3939:
3938:
3933:
3928:
3923:
3918:
3913:
3907:
3905:
3899:
3898:
3896:
3895:
3890:
3885:
3880:
3874:
3872:
3866:
3865:
3863:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3777:
3772:
3766:
3764:
3758:
3757:
3755:
3754:
3749:
3744:
3739:
3734:
3729:
3723:
3721:
3715:
3714:
3712:
3711:
3706:
3701:
3700:
3699:
3694:
3684:
3679:
3674:
3669:
3668:
3667:
3662:
3652:
3650:Hopfield model
3647:
3642:
3637:
3631:
3629:
3625:
3624:
3622:
3621:
3616:
3611:
3606:
3601:
3596:
3595:
3594:
3589:
3584:
3579:
3569:
3567:Markov process
3564:
3559:
3554:
3548:
3546:
3542:
3541:
3539:
3538:
3536:Wiener sausage
3533:
3531:Wiener process
3528:
3523:
3518:
3513:
3511:Stable process
3508:
3503:
3501:Semimartingale
3498:
3493:
3492:
3491:
3486:
3476:
3471:
3466:
3461:
3456:
3451:
3446:
3444:Jump diffusion
3441:
3436:
3431:
3426:
3421:
3419:Hawkes process
3416:
3411:
3406:
3401:
3399:Feller process
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3364:Cauchy process
3361:
3360:
3359:
3354:
3349:
3344:
3339:
3329:
3328:
3327:
3317:
3315:Bessel process
3312:
3306:
3304:
3298:
3297:
3295:
3294:
3293:
3292:
3287:
3282:
3277:
3267:
3262:
3257:
3252:
3247:
3242:
3237:
3231:
3229:
3223:
3222:
3217:
3215:
3214:
3207:
3200:
3192:
3186:
3185:
3170:
3156:
3139:
3136:
3133:
3132:
3125:
3099:
3056:
3027:(4): 615â630.
3005:
2998:
2972:
2951:(2): 153â159.
2931:
2924:
2894:
2887:
2861:
2845:
2844:
2842:
2839:
2838:
2837:
2832:
2827:
2820:
2817:
2800:
2797:
2784:
2773:Bessel process
2760:
2738:
2735:
2732:
2728:
2722:
2715:
2711:
2706:
2702:
2689:
2688:
2679:
2677:
2665:
2661:
2657:
2654:
2651:
2648:
2643:
2639:
2633:
2628:
2623:
2619:
2615:
2611:
2602:
2596:
2588:
2584:
2581:
2578:
2575:
2570:
2565:
2561:
2557:
2552:
2545:
2541:
2536:
2531:
2513:
2503:
2497:
2491:
2490:
2479:
2476:
2473:
2470:
2465:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2429:
2425:
2403:
2393:
2387:
2376:
2369:
2363:
2356:
2353:
2349:Bessel process
2342:
2328:
2322:
2312:
2306:
2298:
2297:
2288:
2286:
2273:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2243:
2238:
2231:
2227:
2222:
2217:
2208:
2202:
2194:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2151:
2146:
2120:
2117:
2114:
2109:
2105:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2060:
2050:
2044:
2033:
2023:
2017:
2010:
2007:
2000:
1984:
1969:
1947:
1944:
1941:
1937:
1931:
1926:
1922:
1918:
1915:
1910:
1906:
1893:
1890:
1875:
1872:
1869:
1865:
1860:
1854:
1850:
1845:
1841:
1810:
1786:
1766:
1746:
1726:
1706:
1697:with exponent
1686:
1664:
1661:
1658:
1655:
1651:
1647:
1644:
1640:
1636:
1633:
1630:
1625:
1621:
1617:
1614:
1611:
1591:
1588:
1584:
1580:
1577:
1574:
1571:
1568:
1565:
1545:
1521:
1517:
1513:
1501:
1500:
1489:
1486:
1483:
1480:
1476:
1472:
1468:
1464:
1460:
1457:
1454:
1449:
1445:
1439:
1434:
1431:
1427:
1421:
1418:
1413:
1408:
1404:
1400:
1396:
1391:
1387:
1383:
1379:
1375:
1371:
1365:
1360:
1356:
1352:
1349:
1344:
1340:
1336:
1333:
1330:
1327:
1322:
1318:
1314:
1311:
1288:
1284:
1281:
1259:
1255:
1251:
1247:
1244:
1233:
1232:
1221:
1218:
1215:
1211:
1206:
1202:
1198:
1193:
1189:
1186:
1183:
1178:
1174:
1170:
1165:
1162:
1159:
1156:
1153:
1150:
1146:
1142:
1139:
1136:
1133:
1128:
1124:
1120:
1115:
1112:
1109:
1106:
1103:
1099:
1094:
1088:
1083:
1079:
1075:
1071:
1067:
1064:
1059:
1055:
1050:
1046:
1042:
1038:
1035:
1030:
1026:
1021:
1017:
1014:
1011:
1008:
1003:
999:
976:
972:
949:
946:
943:
939:
933:
929:
925:
913:
910:
909:
908:
897:
894:
891:
887:
882:
878:
874:
869:
866:
863:
859:
853:
848:
844:
840:
837:
834:
831:
826:
822:
796:
793:
790:
786:
780:
776:
772:
752:
732:
712:
709:
687:
683:
679:
676:
673:
653:
633:
630:
627:
624:
621:
617:
614:
611:
608:
605:
602:
599:
596:
593:
578:
577:
566:
561:
557:
553:
550:
547:
541:
538:
535:
532:
529:
524:
520:
516:
513:
510:
507:
504:
500:
494:
489:
485:
478:
475:
471:
464:
461:
458:
454:
450:
447:
444:
441:
436:
432:
408:
388:
366:
362:
341:
338:
335:
330:
326:
297:
294:
291:
267:
256:
255:
244:
239:
235:
231:
228:
225:
221:
216:
212:
208:
205:
202:
199:
194:
189:
185:
181:
178:
175:
172:
167:
163:
139:
119:
97:
94:
91:
87:
81:
77:
73:
61:
58:
38:semimartingale
13:
10:
9:
6:
4:
3:
2:
4498:
4487:
4484:
4482:
4479:
4478:
4476:
4461:
4458:
4456:
4453:
4452:
4449:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4359:
4357:
4353:
4345:
4342:
4340:
4337:
4336:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4314:Stopping time
4312:
4308:
4305:
4304:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4164:
4162:
4158:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4132:
4129:
4128:
4126:
4124:
4120:
4113:
4109:
4105:
4104:HewittâSavage
4101:
4097:
4093:
4089:
4088:Zeroâone laws
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4035:
4033:
4029:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3947:
3943:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3912:
3909:
3908:
3906:
3904:
3900:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3875:
3873:
3871:
3867:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3785:BlackâScholes
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3767:
3765:
3763:
3759:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3724:
3722:
3720:
3716:
3710:
3707:
3705:
3702:
3698:
3695:
3693:
3690:
3689:
3688:
3687:Point process
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3666:
3663:
3661:
3658:
3657:
3656:
3653:
3651:
3648:
3646:
3645:Gibbs measure
3643:
3641:
3638:
3636:
3633:
3632:
3630:
3626:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3574:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3549:
3547:
3543:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3490:
3487:
3485:
3482:
3481:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3434:ItĂŽ diffusion
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3409:Gamma process
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3334:
3333:
3330:
3326:
3323:
3322:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3305:
3303:
3299:
3291:
3288:
3286:
3283:
3281:
3280:Self-avoiding
3278:
3276:
3273:
3272:
3271:
3268:
3266:
3265:Moran process
3263:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3232:
3230:
3228:
3227:Discrete time
3224:
3220:
3213:
3208:
3206:
3201:
3199:
3194:
3193:
3190:
3183:
3179:
3175:
3171:
3169:
3165:
3161:
3157:
3154:
3150:
3146:
3142:
3141:
3137:
3128:
3122:
3118:
3113:
3112:
3103:
3100:
3095:
3091:
3086:
3081:
3077:
3073:
3072:
3067:
3060:
3057:
3052:
3048:
3044:
3040:
3035:
3030:
3026:
3022:
3021:
3016:
3009:
3006:
3001:
2995:
2991:
2986:
2985:
2976:
2973:
2968:
2964:
2959:
2954:
2950:
2946:
2942:
2939:Wang (1977).
2935:
2932:
2927:
2921:
2917:
2913:
2909:
2905:
2898:
2895:
2890:
2884:
2880:
2875:
2874:
2865:
2862:
2857:
2850:
2847:
2840:
2836:
2833:
2831:
2828:
2826:
2823:
2822:
2818:
2816:
2814:
2813:
2808:
2807:
2798:
2796:
2782:
2774:
2758:
2736:
2733:
2730:
2720:
2713:
2709:
2704:
2687:
2680:
2678:
2663:
2659:
2655:
2652:
2649:
2646:
2641:
2631:
2626:
2621:
2617:
2609:
2594:
2586:
2582:
2579:
2576:
2573:
2568:
2563:
2559:
2555:
2550:
2543:
2539:
2534:
2529:
2521:
2520:
2517:
2512:
2506:
2500:
2496:
2477:
2471:
2468:
2463:
2458:
2454:
2450:
2447:
2444:
2441:
2432:
2427:
2423:
2415:
2414:
2413:
2411:
2406:
2402:
2396:
2390:
2386:
2382:
2375:
2366:
2362:
2354:
2352:
2350:
2345:
2341:
2337:
2331:
2325:
2321:
2315:
2309:
2305:
2296:
2289:
2287:
2271:
2264:
2261:
2258:
2252:
2249:
2246:
2241:
2229:
2225:
2215:
2200:
2192:
2185:
2182:
2179:
2173:
2170:
2167:
2161:
2153:
2149:
2144:
2136:
2135:
2132:
2115:
2112:
2107:
2103:
2099:
2096:
2093:
2090:
2081:
2078:
2070:
2066:
2059:
2053:
2047:
2043:
2040:> 0, and (
2039:
2032:
2026:
2020:
2016:
2008:
2006:
2003:
1999:
1994:
1992:
1987:
1983:
1967:
1945:
1942:
1939:
1929:
1924:
1920:
1913:
1908:
1904:
1891:
1889:
1873:
1870:
1867:
1852:
1848:
1831:
1827:
1822:
1808:
1800:
1784:
1764:
1744:
1724:
1704:
1684:
1662:
1659:
1656:
1653:
1645:
1642:
1631:
1623:
1619:
1612:
1609:
1586:
1582:
1578:
1575:
1572:
1566:
1563:
1543:
1534:
1519:
1515:
1511:
1487:
1481:
1474:
1470:
1466:
1462:
1455:
1447:
1443:
1429:
1425:
1419:
1416:
1411:
1406:
1402:
1398:
1389:
1385:
1377:
1373:
1369:
1363:
1358:
1354:
1350:
1342:
1338:
1331:
1328:
1320:
1316:
1309:
1302:
1301:
1300:
1286:
1282:
1279:
1245:
1242:
1219:
1216:
1213:
1209:
1204:
1200:
1196:
1191:
1184:
1181:
1176:
1172:
1160:
1157:
1151:
1144:
1140:
1134:
1131:
1126:
1122:
1107:
1104:
1097:
1092:
1086:
1081:
1077:
1073:
1065:
1062:
1057:
1053:
1044:
1036:
1033:
1028:
1024:
1015:
1009:
1001:
997:
989:
988:
987:
974:
947:
944:
941:
931:
927:
895:
892:
889:
880:
876:
864:
857:
851:
846:
842:
838:
832:
824:
820:
812:
811:
810:
794:
791:
788:
778:
774:
750:
730:
710:
707:
685:
677:
671:
651:
631:
628:
625:
622:
619:
612:
609:
606:
600:
597:
594:
591:
583:
564:
559:
551:
545:
536:
533:
530:
527:
522:
518:
514:
511:
508:
505:
498:
492:
487:
483:
476:
473:
469:
462:
456:
448:
442:
434:
430:
422:
421:
420:
406:
386:
379:has spent at
364:
360:
336:
328:
324:
315:
311:
292:
281:
265:
242:
237:
229:
223:
214:
210:
206:
203:
197:
192:
187:
183:
179:
173:
165:
161:
153:
152:
151:
137:
130:at the point
117:
95:
92:
89:
79:
75:
59:
57:
55:
54:random fields
51:
47:
43:
39:
35:
31:
27:
18:
4372:Econometrics
4334:Wiener space
4222:ItĂŽ integral
4123:Inequalities
4012:Self-similar
3982:GaussâMarkov
3972:Exchangeable
3952:CĂ dlĂ g paths
3888:Risk process
3840:LIBOR market
3709:Random graph
3704:Random field
3516:Superprocess
3458:
3454:LĂ©vy process
3449:Jump process
3424:Hunt process
3260:Markov chain
3173:
3159:
3144:
3110:
3102:
3078:(1): 56â86.
3075:
3069:
3059:
3024:
3018:
3008:
2983:
2975:
2948:
2944:
2934:
2903:
2897:
2872:
2864:
2855:
2849:
2835:Random field
2810:
2804:
2802:
2692:
2681:
2510:
2504:
2498:
2494:
2492:
2409:
2404:
2400:
2394:
2388:
2384:
2380:
2373:
2364:
2360:
2358:
2343:
2339:
2335:
2329:
2323:
2319:
2313:
2307:
2303:
2301:
2290:
2068:
2064:
2057:
2051:
2045:
2041:
2037:
2030:
2024:
2018:
2014:
2012:
2001:
1997:
1995:
1985:
1981:
1895:
1823:
1535:
1502:
1234:
915:
584:of the form
579:
257:
63:
33:
26:mathematical
23:
4417:Ruin theory
4355:Disciplines
4227:ItĂŽ's lemma
4002:Predictable
3677:Percolation
3660:Potts model
3655:Ising model
3619:White noise
3577:Differences
3439:ItĂŽ process
3379:Cox process
3275:Loop-erased
3270:Random walk
2858:. Springer.
2775:started at
2383:, and let (
399:up to time
4475:Categories
4427:Statistics
4207:Filtration
4108:Kolmogorov
4092:Blumenthal
4017:Stationary
3957:Continuous
3945:Properties
3830:HullâWhite
3572:Martingale
3459:Local time
3347:Fractional
3325:pure birth
3138:References
3126:0521863007
3051:0118.13403
2999:0387949577
2888:0387949577
2516:= 0, then
34:local time
28:theory of
4339:Classical
3352:Geometric
3342:Excursion
2967:123101077
2734:≥
2653:≥
2647::
2580:≥
2574::
2451::
2445:≥
2253:∈
2247::
2174:∈
2168::
2100::
2094:≥
1943:∈
1871:≥
1705:α
1660:≥
1646:∈
1567:∈
1564:α
1516:−
1471:−
1438:∞
1433:∞
1430:−
1426:∫
1374:−
1355:∫
1254:→
1217:≥
1182:−
1155:∞
1152:−
1141:−
1132:−
1111:∞
1078:∫
1074:−
1063:−
1045:−
1034:−
945:≥
843:∫
792:≥
582:diffusion
537:ε
512:ε
509:−
484:∫
477:ε
460:↓
457:ε
314:Paul LĂ©vy
266:δ
207:−
198:δ
184:∫
93:≥
4460:Category
4344:Abstract
3878:BĂŒhlmann
3484:Compound
2819:See also
1520:′
1475:′
1378:′
1283:′
3967:Ergodic
3855:VaĆĄĂÄek
3697:Poisson
3357:Meander
3094:1993647
3043:0156383
2809:) and (
2412:> 0
2302:where (
308:is the
278:is the
24:In the
4307:Tanaka
3992:Mixing
3987:Markov
3860:Wilkie
3825:HoâLee
3820:Heston
3592:Super-
3337:Bridge
3285:Biased
3180:
3166:
3151:
3123:
3092:
3049:
3041:
2996:
2965:
2922:
2885:
2881:â449.
2379:= 0 â
2063:= 0 â
1799:cĂ dlĂ g
1503:where
644:where
258:where
4160:Tools
3936:M/M/c
3931:M/M/1
3926:M/G/1
3916:Fluid
3582:Local
3119:â56.
3090:JSTOR
2963:S2CID
2841:Notes
2493:Let (
2370:t â„ 0
2359:Let (
2013:Let (
4112:LĂ©vy
3911:Bulk
3795:Chen
3587:Sub-
3545:Both
3178:ISBN
3164:ISBN
3149:ISBN
3121:ISBN
2994:ISBN
2920:ISBN
2883:ISBN
2469:>
1797:and
1737:and
528:<
515:<
282:and
3692:Cox
3080:doi
3076:109
3047:Zbl
3029:doi
2990:370
2953:doi
2912:doi
2879:428
2507:â„ 0
2436:inf
2397:â„ 0
2332:â„ 0
2316:â„ 0
2085:inf
2027:â„ 0
1801:in
1536:If
962:on
743:at
453:lim
4477::
4110:,
4106:,
4102:,
4098:,
4094:,
3117:53
3088:.
3074:.
3068:.
3045:.
3039:MR
3037:.
3023:.
3017:.
2992:.
2961:.
2949:41
2947:.
2943:.
2918:.
2906:.
2351:.
2054:â„0
2036:=
1993:.
1888:.
1832:,
1821:.
1220:0.
56:.
32:,
4114:)
4090:(
3211:e
3204:t
3197:v
3184:.
3155:.
3129:.
3096:.
3082::
3053:.
3031::
3025:7
3002:.
2969:.
2955::
2928:.
2914::
2891:.
2812:2
2806:1
2783:a
2759:x
2737:0
2731:x
2727:)
2721:x
2714:a
2710:T
2705:L
2701:(
2686:)
2684:2
2682:(
2664:.
2660:}
2656:0
2650:x
2642:2
2638:)
2632:a
2627:+
2622:x
2618:W
2614:(
2610:{
2601:D
2595:=
2587:}
2583:0
2577:x
2569:2
2564:x
2560:W
2556:+
2551:x
2544:a
2540:T
2535:L
2530:{
2514:0
2511:W
2505:t
2502:)
2499:t
2495:W
2478:.
2475:}
2472:a
2464:0
2459:t
2455:L
2448:0
2442:t
2439:{
2433:=
2428:a
2424:T
2410:a
2405:a
2401:T
2395:t
2392:)
2389:t
2385:L
2381:R
2377:0
2374:B
2368:)
2365:t
2361:B
2344:x
2340:W
2336:C
2330:t
2327:)
2324:t
2320:B
2314:t
2311:)
2308:t
2304:L
2295:)
2293:1
2291:(
2272:}
2268:]
2265:a
2262:,
2259:0
2256:[
2250:x
2242:2
2237:|
2230:x
2226:W
2221:|
2216:{
2207:D
2201:=
2193:}
2189:]
2186:a
2183:,
2180:0
2177:[
2171:x
2165:)
2162:T
2159:(
2154:x
2150:L
2145:{
2119:}
2116:0
2113:=
2108:t
2104:B
2097:0
2091:t
2088:{
2082:=
2079:T
2069:B
2065:R
2061:0
2058:W
2052:t
2049:)
2046:t
2042:W
2038:a
2034:0
2031:B
2025:t
2022:)
2019:t
2015:B
2002:t
1998:L
1986:t
1982:L
1968:E
1946:E
1940:x
1936:)
1930:x
1925:t
1921:L
1917:(
1914:=
1909:t
1905:L
1874:0
1868:s
1864:)
1859:|
1853:s
1849:B
1844:|
1840:(
1809:x
1785:t
1765:L
1745:t
1725:x
1685:x
1663:0
1657:t
1654:,
1650:R
1643:x
1639:)
1635:)
1632:t
1629:(
1624:x
1620:L
1616:(
1613:=
1610:L
1590:)
1587:2
1583:/
1579:1
1576:,
1573:0
1570:(
1544:X
1512:F
1488:,
1485:)
1482:x
1479:(
1467:F
1463:d
1459:)
1456:t
1453:(
1448:x
1444:L
1420:2
1417:1
1412:+
1407:s
1403:X
1399:d
1395:)
1390:s
1386:X
1382:(
1370:F
1364:t
1359:0
1351:+
1348:)
1343:0
1339:X
1335:(
1332:F
1329:=
1326:)
1321:t
1317:X
1313:(
1310:F
1287:,
1280:F
1258:R
1250:R
1246::
1243:F
1214:t
1210:,
1205:s
1201:X
1197:d
1192:)
1188:)
1185:x
1177:s
1173:X
1169:(
1164:]
1161:0
1158:,
1149:(
1145:1
1138:)
1135:x
1127:s
1123:X
1119:(
1114:)
1108:,
1105:0
1102:(
1098:1
1093:(
1087:t
1082:0
1070:|
1066:x
1058:0
1054:X
1049:|
1041:|
1037:x
1029:t
1025:X
1020:|
1016:=
1013:)
1010:t
1007:(
1002:x
998:L
975::
971:R
948:0
942:s
938:)
932:s
928:X
924:(
896:.
893:s
890:d
886:)
881:s
877:X
873:(
868:}
865:x
862:{
858:1
852:t
847:0
839:=
836:)
833:t
830:(
825:x
821:L
795:0
789:s
785:)
779:s
775:X
771:(
751:x
731:B
711:s
708:d
686:s
682:]
678:B
675:[
672:d
652:W
632:W
629:d
626:+
623:t
620:d
616:)
613:B
610:,
607:t
604:(
601:b
598:=
595:B
592:d
565:,
560:s
556:]
552:B
549:[
546:d
540:}
534:+
531:x
523:s
519:B
506:x
503:{
499:1
493:t
488:0
474:2
470:1
463:0
449:=
446:)
443:t
440:(
435:x
431:L
407:t
387:x
365:s
361:B
340:)
337:t
334:(
329:x
325:L
296:]
293:B
290:[
243:,
238:s
234:]
230:B
227:[
224:d
220:)
215:s
211:B
204:x
201:(
193:t
188:0
180:=
177:)
174:t
171:(
166:x
162:L
138:x
118:B
96:0
90:s
86:)
80:s
76:B
72:(
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