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1985:
1992:
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2173:{\displaystyle M=0.18\left({\begin{array}{cc}0&1\\0&1\end{array}}\right)+0.28\left({\begin{array}{cc}1&0\\1&0\end{array}}\right)+0.42\left({\begin{array}{cc}0&1\\1&0\end{array}}\right)+0.12\left({\begin{array}{cc}1&0\\0&1\end{array}}\right).}
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987:) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems.
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1980:{\displaystyle M=0.6\left({\begin{array}{cc}0&1\\1&0\end{array}}\right)+0.3\left({\begin{array}{cc}1&0\\0&1\end{array}}\right)+0.1\left({\begin{array}{cc}1&0\\1&0\end{array}}\right).}
1707:
Reversely, can, and how, a given MC be represented by the compositions of i.i.d. random transformations? Yes, it can, but not unique. The proof for existence is similar with
Birkhoff–von Neumann theorem for
563:
367:
917:
3195:
205:. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.
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for a random dynamical system is not as straightforward to define as in the deterministic case. For technical reasons, it is necessary to "rewind time", as in the definition of a
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749:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ):=\left(C_{0}(\mathbb {R} ;\mathbb {R} ^{d}),{\mathcal {B}}(C_{0}(\mathbb {R} ;\mathbb {R} ^{d})),\gamma \right).}
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480:{\displaystyle \left\{{\begin{matrix}\mathrm {d} X=f(X)\,\mathrm {d} t+\varepsilon \,\mathrm {d} W(t);\\X(0)=x_{0};\end{matrix}}\right.}
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While true in most applications, it is not usually part of the formal definition of a random dynamical system to require that the
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that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state
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3095:{\displaystyle ({\mathcal {B}}(\mathbb {R} )\otimes {\mathcal {F}}\otimes {\mathcal {B}}(X),{\mathcal {B}}(X))}
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consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.
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instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition
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3465:{\displaystyle \varphi (t,\vartheta _{s}(\omega ))\circ \varphi (s,\omega )=\varphi (t+s,\omega ).}
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827:{\displaystyle \varphi :\mathbb {R} \times \Omega \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}}
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evolving according to a succession of maps randomly chosen according to the distribution
125:
into itself that can be thought of as the set of all possible equations of motion, and a
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expressed in terms of deterministic transition matrices. Then a Markov transition matrix
1666:{\displaystyle X_{n}=\alpha _{n}\circ \alpha _{n-1}\circ \dots \circ \alpha _{1}(X_{0})}
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exists for all positive time and some (small) interval of negative time dependent upon
266:
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have an element of randomness to them. Random dynamical systems are characterized by a
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4485:
4307:
4223:
4163:
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2804:; for discrete-time random dynamical systems, one would consider only integer-valued
1831:{\displaystyle M=\left({\begin{array}{cc}0.4&0.6\\0.7&0.3\end{array}}\right)}
3973:
1199:. It is a binary matrix but it has exactly one entry 1 in each row and 0s otherwise.
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2784:. For one-sided random dynamical systems, one would consider only positive indices
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3950:
Bhattacharya, Rabi; Majumdar, Mukul (2003). "Random dynamical systems: a review".
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2010:
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An i.i.d random dynamical system in the discrete space is described by a triplet
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Crauel, Hans; Debussche, Arnaud; Flandoli, Franco (1997). "Random attractors".
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can be represented by the following decomposition by the min-max algorithm,
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2550:{\displaystyle \vartheta _{0}=\mathrm {id} _{\Omega }:\Omega \to \Omega }
3651:{\displaystyle W(t,\vartheta _{s}(\omega ))=W(t+s,\omega )-W(s,\omega )}
4646:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4086:
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2915:
1715:
Here is an example that illustrates the existence and non-uniqueness.
2855:
2428:{\displaystyle \mathbb {P} (E)=\mathbb {P} (\vartheta _{s}^{-1}(E))}
3477:
In the case of random dynamical systems driven by a Wiener process
2670:{\displaystyle \vartheta _{s}\circ \vartheta _{t}=\vartheta _{s+t}}
1579:
is constructed by means of composition of independent random maps,
3006:{\displaystyle \varphi :\mathbb {R} \times \Omega \times X\to X}
2907:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ,\vartheta )}
2282:{\displaystyle \vartheta :\mathbb {R} \times \Omega \to \Omega }
558:{\displaystyle W:\mathbb {R} \times \Omega \to \mathbb {R} ^{d}}
4090:
3989:"Stochastic dynamics: Markov chains and random transformations"
759:
In this context, the Wiener process is the coordinate process.
15:
912:{\displaystyle \varphi (t,\omega ,x_{0}):=X(t,\omega ;x_{0})}
209:
Motivation 1: Solutions to a stochastic differential equation
189:; in this case the distribution Q is typically determined by
3891:. Moreover, the attractor is dependent upon the realisation
3190:{\displaystyle \varphi (0,\omega )=\mathrm {id} _{X}:X\to X}
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4626:
Autoregressive conditional heteroskedasticity (ARCH) model
4154:
Independent and identically distributed random variables
1265:
The discrete random dynamical system comes as follows,
36:
4631:
Autoregressive integrated moving average (ARIMA) model
3987:
Ye, Felix X.-F.; Wang, Yue; Qian, Hong (August 2016).
376:
256:{\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} ^{d}}
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2777:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
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2231:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
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3993:
Discrete and
Continuous Dynamical Systems - Series B
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may be too technical for most readers to understand
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3298:{\displaystyle (t,x)\mapsto \varphi (t,\omega ,x)}
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2740:of measure-preserving transformation of the noise
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3796:seconds with the same noise (as started from the
3549:{\displaystyle \vartheta _{s}:\Omega \to \Omega }
3510:{\displaystyle W:\mathbb {R} \times \Omega \to X}
2349:{\displaystyle \vartheta _{s}:\Omega \to \Omega }
4513:Stochastic chains with memory of variable length
3816:seconds mark) gives the same result as evolving
1989:In the meantime, another decomposition could be
1469:is chosen according to the probability measure
1363:is chosen according to the probability measure
4049:Journal of Dynamics and Differential Equations
4102:
1114:{\displaystyle \{s_{1},s_{2},\cdots ,s_{n}\}}
185:An example of a random dynamical system is a
8:
1746:
1734:
1442:Independently of previous maps, another map
1108:
1063:
980:{\displaystyle (\mathbb {R} ^{d},\varphi )}
4641:Autoregressive–moving-average (ARMA) model
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59:Learn how and when to remove this message
43:, without removing the technical details.
1538:{\displaystyle x_{2}=\alpha _{2}(x_{1})}
1432:{\displaystyle x_{1}=\alpha _{1}(x_{0})}
991:Motivation 2: Connection to Markov Chain
361:to the stochastic differential equation
3942:
3879:Attractors for random dynamical systems
593:). Implicitly, this statement uses the
4947:Doob's martingale convergence theorems
4699:Constant elasticity of variance (CEV)
4689:Chan–Karolyi–Longstaff–Sanders (CKLS)
41:make it understandable to non-experts
7:
2863:measure-preserving dynamical system
2611:{\displaystyle s,t\in \mathbb {R} }
2458:{\displaystyle E\in {\mathcal {F}}}
1759:and the set of the transformations
201:dynamical system on the "physical"
83:is a dynamical system in which the
5186:Skorokhod's representation theorem
4967:Law of large numbers (weak/strong)
3543:
3537:
3498:
3357:{\displaystyle \omega \in \Omega }
3351:
3239:{\displaystyle \omega \in \Omega }
3233:
3165:
3162:
3125:{\displaystyle \omega \in \Omega }
3119:
2988:
2874:
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2568:
2544:
2538:
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2522:
2343:
2337:
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2270:
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1766:
1489:and the system moves to the state
1383:and the system moves to the state
1350:
1248:
1221:is the probability measure of the
1130:
1011:
791:
612:
537:
509:{\displaystyle \omega \in \Omega }
503:
419:
404:
380:
354:{\displaystyle X(t,\omega ;x_{0})}
139:
108:
14:
5385:Stochastic differential equations
5156:Martingale representation theorem
2729:{\displaystyle s\in \mathbb {R} }
2486:{\displaystyle s\in \mathbb {R} }
2310:{\displaystyle s\in \mathbb {R} }
922:(whenever the right hand side is
306:{\displaystyle \varepsilon >0}
5201:Stochastic differential equation
5091:Doob's optional stopping theorem
5086:Doob–Meyer decomposition theorem
3662:This can be read as saying that
187:stochastic differential equation
20:
5071:Convergence of random variables
4957:Fisher–Tippett–Gnedenko theorem
3102:-measurable function such that
1197:deterministic transition matrix
4669:Binomial options pricing model
3875:seconds with that same noise.
3862:
3850:
3682:{\displaystyle \vartheta _{s}}
3645:
3633:
3624:
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3597:
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3588:
3569:
3540:
3501:
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3438:
3429:
3417:
3408:
3405:
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3253:
3181:
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3142:
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3038:
3030:
3020:
2997:
2940:
2928:
2901:
2871:
2844:{\displaystyle \vartheta _{s}}
2771:
2747:
2701:{\displaystyle \vartheta _{s}}
2541:
2422:
2419:
2413:
2392:
2381:
2375:
2340:
2273:
2225:
2201:
1660:
1647:
1532:
1519:
1426:
1413:
1195:matrix representation, called
1162:{\displaystyle S\rightarrow S}
1153:
1020:
1002:
974:
953:
946:(or, more precisely, the pair
906:
881:
872:
847:
809:
729:
726:
703:
690:
677:
654:
633:
609:
540:
451:
445:
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348:
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238:
1:
5136:Kolmogorov continuity theorem
4972:Law of the iterated logarithm
1026:{\displaystyle (S,\Gamma ,Q)}
5141:Kolmogorov extension theorem
4820:Generalized queueing network
4328:Interacting particle systems
2289:as follows: for each "time"
1269:The system is in some state
313:. Suppose that the solution
4273:Continuous-time random walk
3197:, the identity function on
2824:; in these cases, the maps
1462:{\displaystyle \alpha _{2}}
1336:{\displaystyle \alpha _{1}}
5406:
5281:Extreme value theory (EVT)
5081:Doob decomposition theorem
4373:Ornstein–Uhlenbeck process
4144:Chinese restaurant process
3689:"starts the noise at time
5349:
5161:Optional stopping theorem
4962:Large deviation principle
4714:Heath–Jarrow–Morton (HJM)
4651:Moving-average (MA) model
4636:Autoregressive (AR) model
4461:Hidden Markov model (HMM)
4395:Schramm–Loewner evolution
3966:10.1007/s00199-003-0357-4
3776:seconds and then through
1752:{\displaystyle S=\{1,2\}}
1188:{\displaystyle n\times n}
5380:Random dynamical systems
5076:Doléans-Dade exponential
4906:Progressively measurable
4704:Cox–Ingersoll–Ross (CIR)
4023:Random Dynamical Systems
3324:{\displaystyle \varphi }
2356:be a measure-preserving
1710:doubly stochastic matrix
939:{\displaystyle \varphi }
127:probability distribution
5296:Mathematical statistics
5286:Large deviations theory
5116:Infinitesimal generator
4977:Maximal ergodic theorem
4896:Piecewise-deterministic
4498:Random dynamical system
4363:Markov additive process
4020:Arnold, Ludwig (1998).
3904:{\displaystyle \omega }
3749:{\displaystyle \omega }
2574:{\displaystyle \Omega }
2189:random dynamical system
1772:{\displaystyle \Gamma }
1356:{\displaystyle \Gamma }
1254:{\displaystyle \Gamma }
1234:{\displaystyle \sigma }
1143:is a family of maps of
1136:{\displaystyle \Gamma }
145:{\displaystyle \Gamma }
114:{\displaystyle \Gamma }
81:random dynamical system
5131:Karhunen–Loève theorem
5066:Cameron–Martin formula
5030:Burkholder–Davis–Gundy
4425:Variance gamma process
3905:
3869:
3837:
3810:
3790:
3770:
3750:
3730:
3703:
3683:
3652:
3550:
3511:
3466:
3358:
3331:satisfies the (crude)
3325:
3299:
3240:
3211:
3191:
3126:
3096:
3007:
2947:
2908:
2845:
2818:
2798:
2778:
2730:
2702:
2671:
2612:
2575:
2551:
2487:
2459:
2429:
2350:
2311:
2283:
2232:
2174:
1981:
1832:
1773:
1753:
1694:
1667:
1573:
1548:The procedure repeats.
1539:
1483:
1463:
1433:
1377:
1357:
1337:
1310:
1290:
1255:
1235:
1215:
1189:
1169:. Each such map has a
1163:
1137:
1115:
1050:
1027:
981:
940:
913:
828:
750:
579:
559:
510:
481:
355:
307:
277:
257:
172:
171:{\displaystyle X\in S}
146:
115:
5261:Actuarial mathematics
5223:Uniform integrability
5218:Stratonovich integral
5146:Lévy–Prokhorov metric
5050:Marcinkiewicz–Zygmund
4937:Central limit theorem
4539:Gaussian random field
4368:McKean–Vlasov process
4288:Dyson Brownian motion
4149:Galton–Watson process
4006:10.3934/dcdsb.2016050
3906:
3870:
3868:{\displaystyle (t+s)}
3838:
3836:{\displaystyle x_{0}}
3811:
3791:
3771:
3751:
3731:
3729:{\displaystyle x_{0}}
3704:
3684:
3653:
3551:
3512:
3467:
3359:
3326:
3300:
3241:
3212:
3192:
3127:
3097:
3008:
2948:
2946:{\displaystyle (X,d)}
2909:
2846:
2819:
2799:
2779:
2731:
2703:
2672:
2613:
2576:
2552:
2488:
2460:
2430:
2351:
2312:
2284:
2233:
2175:
1982:
1833:
1774:
1754:
1695:
1693:{\displaystyle X_{n}}
1668:
1574:
1572:{\displaystyle X_{n}}
1540:
1484:
1464:
1434:
1378:
1358:
1338:
1311:
1291:
1289:{\displaystyle x_{0}}
1256:
1236:
1216:
1190:
1164:
1138:
1116:
1051:
1028:
982:
941:
914:
829:
751:
580:
560:
511:
482:
356:
308:
278:
258:
197:, the "noise", and a
173:
147:
116:
5390:Stochastic processes
5336:Time series analysis
5291:Mathematical finance
5176:Reflection principle
4503:Regenerative process
4303:Fleming–Viot process
4118:Stochastic processes
3895:
3847:
3820:
3800:
3780:
3760:
3740:
3713:
3693:
3666:
3563:
3521:
3481:
3374:
3342:
3315:
3250:
3224:
3201:
3136:
3110:
3017:
2971:
2925:
2868:
2858:instead of a group.
2828:
2808:
2788:
2744:
2712:
2685:
2622:
2588:
2565:
2504:
2469:
2439:
2367:
2321:
2293:
2253:
2198:
1993:
1842:
1783:
1763:
1725:
1677:
1583:
1556:
1552:The random variable
1493:
1473:
1446:
1387:
1367:
1347:
1320:
1300:
1273:
1245:
1225:
1205:
1173:
1147:
1127:
1060:
1056:is the state space,
1040:
999:
950:
930:
841:
774:
606:
569:
520:
494:
368:
317:
291:
267:
217:
156:
136:
105:
5331:Stochastic analysis
5171:Quadratic variation
5166:Prokhorov's theorem
5101:Feynman–Kac formula
4571:Markov random field
4219:Birth–death process
4061:1997JDDE....9..307C
2412:
2358:measurable function
1721:If the state space
193:. It consists of a
85:equations of motion
5301:Probability theory
5181:Skorokhod integral
5151:Malliavin calculus
4734:Korn-Kreer-Lenssen
4618:Time series models
4581:Pitman–Yor process
4069:10.1007/BF02219225
3931:Stochastic control
3901:
3889:pullback attractor
3865:
3833:
3806:
3786:
3766:
3746:
3726:
3699:
3679:
3648:
3556:would be given by
3546:
3507:
3462:
3354:
3321:
3295:
3236:
3207:
3187:
3122:
3092:
3003:
2943:
2904:
2851:would only form a
2841:
2814:
2794:
2774:
2726:
2698:
2667:
2608:
2571:
2547:
2497:Suppose also that
2483:
2455:
2425:
2395:
2346:
2307:
2279:
2246:space. Define the
2228:
2170:
2161:
2119:
2077:
2035:
1977:
1968:
1926:
1884:
1828:
1822:
1769:
1749:
1690:
1663:
1569:
1535:
1479:
1459:
1429:
1373:
1353:
1333:
1306:
1286:
1251:
1231:
1211:
1185:
1159:
1133:
1111:
1046:
1023:
977:
936:
909:
824:
746:
575:
555:
506:
477:
472:
351:
303:
273:
253:
168:
142:
111:
5367:
5366:
5321:Signal processing
5040:Doob's upcrossing
5035:Doob's martingale
4999:Engelbert–Schmidt
4942:Donsker's theorem
4876:Feller-continuous
4744:Rendleman–Bartter
4534:Dirichlet process
4451:Branching process
4420:Telegraph process
4313:Geometric process
4293:Empirical process
4283:Diffusion process
4139:Branching process
4134:Bernoulli process
3926:Diffusion process
3883:The notion of an
3809:{\displaystyle s}
3789:{\displaystyle t}
3769:{\displaystyle s}
3702:{\displaystyle s}
3220:for (almost) all
3210:{\displaystyle X}
2817:{\displaystyle s}
2797:{\displaystyle s}
2559:identity function
2240:probability space
2183:Formal definition
1482:{\displaystyle Q}
1376:{\displaystyle Q}
1309:{\displaystyle S}
1214:{\displaystyle Q}
1049:{\displaystyle S}
768:solution operator
598:probability space
578:{\displaystyle d}
276:{\displaystyle d}
77:dynamical systems
69:
68:
61:
5397:
5341:Machine learning
5228:Usual hypotheses
5111:Girsanov theorem
5096:Dynkin's formula
4861:Continuous paths
4769:Actuarial models
4709:Garman–Kohlhagen
4679:Black–Karasinski
4674:Black–Derman–Toy
4661:Financial models
4527:Fields and other
4456:Gaussian process
4405:Sigma-martingale
4209:Additive process
4111:
4104:
4097:
4088:
4081:
4080:
4044:
4038:
4037:
4017:
4011:
4010:
4008:
3999:(7): 2337–2361.
3984:
3978:
3977:
3947:
3910:
3908:
3907:
3902:
3874:
3872:
3871:
3866:
3842:
3840:
3839:
3834:
3832:
3831:
3815:
3813:
3812:
3807:
3795:
3793:
3792:
3787:
3775:
3773:
3772:
3767:
3755:
3753:
3752:
3747:
3736:with some noise
3735:
3733:
3732:
3727:
3725:
3724:
3708:
3706:
3705:
3700:
3688:
3686:
3685:
3680:
3678:
3677:
3657:
3655:
3654:
3649:
3587:
3586:
3555:
3553:
3552:
3547:
3533:
3532:
3517:, the base flow
3516:
3514:
3513:
3508:
3494:
3471:
3469:
3468:
3463:
3398:
3397:
3363:
3361:
3360:
3355:
3333:cocycle property
3330:
3328:
3327:
3322:
3304:
3302:
3301:
3296:
3245:
3243:
3242:
3237:
3216:
3214:
3213:
3208:
3196:
3194:
3193:
3188:
3174:
3173:
3168:
3131:
3129:
3128:
3123:
3101:
3099:
3098:
3093:
3079:
3078:
3060:
3059:
3050:
3049:
3037:
3029:
3028:
3012:
3010:
3009:
3004:
2984:
2952:
2950:
2949:
2944:
2913:
2911:
2910:
2905:
2894:
2886:
2885:
2850:
2848:
2847:
2842:
2840:
2839:
2823:
2821:
2820:
2815:
2803:
2801:
2800:
2795:
2783:
2781:
2780:
2775:
2770:
2762:
2761:
2735:
2733:
2732:
2727:
2725:
2707:
2705:
2704:
2699:
2697:
2696:
2676:
2674:
2673:
2668:
2666:
2665:
2647:
2646:
2634:
2633:
2617:
2615:
2614:
2609:
2607:
2580:
2578:
2577:
2572:
2556:
2554:
2553:
2548:
2534:
2533:
2528:
2516:
2515:
2492:
2490:
2489:
2484:
2482:
2464:
2462:
2461:
2456:
2454:
2453:
2434:
2432:
2431:
2426:
2411:
2403:
2391:
2374:
2355:
2353:
2352:
2347:
2333:
2332:
2316:
2314:
2313:
2308:
2306:
2288:
2286:
2285:
2280:
2266:
2237:
2235:
2234:
2229:
2224:
2216:
2215:
2179:
2177:
2176:
2171:
2166:
2162:
2124:
2120:
2082:
2078:
2040:
2036:
1986:
1984:
1983:
1978:
1973:
1969:
1931:
1927:
1889:
1885:
1837:
1835:
1834:
1829:
1827:
1823:
1778:
1776:
1775:
1770:
1758:
1756:
1755:
1750:
1699:
1697:
1696:
1691:
1689:
1688:
1672:
1670:
1669:
1664:
1659:
1658:
1646:
1645:
1627:
1626:
1608:
1607:
1595:
1594:
1578:
1576:
1575:
1570:
1568:
1567:
1544:
1542:
1541:
1536:
1531:
1530:
1518:
1517:
1505:
1504:
1488:
1486:
1485:
1480:
1468:
1466:
1465:
1460:
1458:
1457:
1438:
1436:
1435:
1430:
1425:
1424:
1412:
1411:
1399:
1398:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1342:
1340:
1339:
1334:
1332:
1331:
1315:
1313:
1312:
1307:
1295:
1293:
1292:
1287:
1285:
1284:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1194:
1192:
1191:
1186:
1168:
1166:
1165:
1160:
1142:
1140:
1139:
1134:
1120:
1118:
1117:
1112:
1107:
1106:
1088:
1087:
1075:
1074:
1055:
1053:
1052:
1047:
1032:
1030:
1029:
1024:
986:
984:
983:
978:
967:
966:
961:
945:
943:
942:
937:
918:
916:
915:
910:
905:
904:
871:
870:
833:
831:
830:
825:
823:
822:
817:
808:
807:
802:
787:
755:
753:
752:
747:
742:
738:
725:
724:
719:
710:
702:
701:
689:
688:
676:
675:
670:
661:
653:
652:
632:
624:
623:
595:classical Wiener
584:
582:
581:
576:
564:
562:
561:
556:
554:
553:
548:
533:
515:
513:
512:
507:
486:
484:
483:
478:
476:
473:
466:
465:
422:
407:
383:
360:
358:
357:
352:
347:
346:
312:
310:
309:
304:
282:
280:
279:
274:
262:
260:
259:
254:
252:
251:
246:
237:
236:
231:
177:
175:
174:
169:
151:
149:
148:
143:
120:
118:
117:
112:
64:
57:
53:
50:
44:
24:
23:
16:
5405:
5404:
5400:
5399:
5398:
5396:
5395:
5394:
5370:
5369:
5368:
5363:
5345:
5306:Queueing theory
5249:
5191:Skorokhod space
5054:
5045:Kunita–Watanabe
5016:
4982:Sanov's theorem
4952:Ergodic theorem
4925:
4921:Time-reversible
4839:
4802:Queueing models
4796:
4792:Sparre–Anderson
4782:Cramér–Lundberg
4763:
4749:SABR volatility
4655:
4612:
4564:Boolean network
4522:
4508:Renewal process
4439:
4388:Non-homogeneous
4378:Poisson process
4268:Contact process
4231:Brownian motion
4201:Continuous time
4195:
4189:Maximal entropy
4120:
4115:
4085:
4084:
4046:
4045:
4041:
4034:
4019:
4018:
4014:
3986:
3985:
3981:
3953:Economic Theory
3949:
3948:
3944:
3939:
3917:
3893:
3892:
3881:
3845:
3844:
3823:
3818:
3817:
3798:
3797:
3778:
3777:
3758:
3757:
3738:
3737:
3716:
3711:
3710:
3691:
3690:
3669:
3664:
3663:
3578:
3561:
3560:
3524:
3519:
3518:
3479:
3478:
3389:
3372:
3371:
3340:
3339:
3313:
3312:
3248:
3247:
3222:
3221:
3199:
3198:
3160:
3134:
3133:
3108:
3107:
3015:
3014:
2969:
2968:
2923:
2922:
2866:
2865:
2831:
2826:
2825:
2806:
2805:
2786:
2785:
2742:
2741:
2710:
2709:
2688:
2683:
2682:
2651:
2638:
2625:
2620:
2619:
2586:
2585:
2563:
2562:
2520:
2507:
2502:
2501:
2467:
2466:
2437:
2436:
2365:
2364:
2324:
2319:
2318:
2291:
2290:
2251:
2250:
2196:
2195:
2185:
2160:
2159:
2154:
2148:
2147:
2142:
2131:
2118:
2117:
2112:
2106:
2105:
2100:
2089:
2076:
2075:
2070:
2064:
2063:
2058:
2047:
2034:
2033:
2028:
2022:
2021:
2016:
2005:
1991:
1990:
1967:
1966:
1961:
1955:
1954:
1949:
1938:
1925:
1924:
1919:
1913:
1912:
1907:
1896:
1883:
1882:
1877:
1871:
1870:
1865:
1854:
1840:
1839:
1821:
1820:
1815:
1809:
1808:
1803:
1792:
1781:
1780:
1761:
1760:
1723:
1722:
1680:
1675:
1674:
1650:
1637:
1612:
1599:
1586:
1581:
1580:
1559:
1554:
1553:
1522:
1509:
1496:
1491:
1490:
1471:
1470:
1449:
1444:
1443:
1416:
1403:
1390:
1385:
1384:
1365:
1364:
1345:
1344:
1323:
1318:
1317:
1298:
1297:
1276:
1271:
1270:
1243:
1242:
1223:
1222:
1203:
1202:
1171:
1170:
1145:
1144:
1125:
1124:
1098:
1079:
1066:
1058:
1057:
1038:
1037:
997:
996:
993:
956:
948:
947:
928:
927:
896:
862:
839:
838:
812:
797:
772:
771:
714:
693:
665:
644:
643:
639:
604:
603:
591:Brownian motion
567:
566:
543:
518:
517:
492:
491:
471:
470:
457:
439:
438:
371:
366:
365:
338:
315:
314:
289:
288:
265:
264:
241:
226:
215:
214:
211:
154:
153:
134:
133:
103:
102:
65:
54:
48:
45:
37:help improve it
34:
25:
21:
12:
11:
5:
5403:
5401:
5393:
5392:
5387:
5382:
5372:
5371:
5365:
5364:
5362:
5361:
5356:
5354:List of topics
5350:
5347:
5346:
5344:
5343:
5338:
5333:
5328:
5323:
5318:
5313:
5311:Renewal theory
5308:
5303:
5298:
5293:
5288:
5283:
5278:
5276:Ergodic theory
5273:
5268:
5266:Control theory
5263:
5257:
5255:
5251:
5250:
5248:
5247:
5246:
5245:
5240:
5230:
5225:
5220:
5215:
5210:
5209:
5208:
5198:
5196:Snell envelope
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5123:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5083:
5078:
5073:
5068:
5062:
5060:
5056:
5055:
5053:
5052:
5047:
5042:
5037:
5032:
5026:
5024:
5018:
5017:
5015:
5014:
4995:Borel–Cantelli
4984:
4979:
4974:
4969:
4964:
4959:
4954:
4949:
4944:
4939:
4933:
4931:
4930:Limit theorems
4927:
4926:
4924:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4888:
4883:
4878:
4873:
4868:
4863:
4858:
4853:
4847:
4845:
4841:
4840:
4838:
4837:
4832:
4827:
4822:
4817:
4812:
4806:
4804:
4798:
4797:
4795:
4794:
4789:
4784:
4779:
4773:
4771:
4765:
4764:
4762:
4761:
4756:
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4665:
4663:
4657:
4656:
4654:
4653:
4648:
4643:
4638:
4633:
4628:
4622:
4620:
4614:
4613:
4611:
4610:
4605:
4600:
4599:
4598:
4593:
4583:
4578:
4573:
4568:
4567:
4566:
4561:
4551:
4549:Hopfield model
4546:
4541:
4536:
4530:
4528:
4524:
4523:
4521:
4520:
4515:
4510:
4505:
4500:
4495:
4494:
4493:
4488:
4483:
4478:
4468:
4466:Markov process
4463:
4458:
4453:
4447:
4445:
4441:
4440:
4438:
4437:
4435:Wiener sausage
4432:
4430:Wiener process
4427:
4422:
4417:
4412:
4410:Stable process
4407:
4402:
4400:Semimartingale
4397:
4392:
4391:
4390:
4385:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4343:Jump diffusion
4340:
4335:
4330:
4325:
4320:
4318:Hawkes process
4315:
4310:
4305:
4300:
4298:Feller process
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4263:Cauchy process
4260:
4259:
4258:
4253:
4248:
4243:
4238:
4228:
4227:
4226:
4216:
4214:Bessel process
4211:
4205:
4203:
4197:
4196:
4194:
4193:
4192:
4191:
4186:
4181:
4176:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4130:
4128:
4122:
4121:
4116:
4114:
4113:
4106:
4099:
4091:
4083:
4082:
4055:(2): 307–341.
4039:
4032:
4012:
3979:
3941:
3940:
3938:
3935:
3934:
3933:
3928:
3923:
3916:
3913:
3911:of the noise.
3900:
3880:
3877:
3864:
3861:
3858:
3855:
3852:
3830:
3826:
3805:
3785:
3765:
3745:
3723:
3719:
3698:
3676:
3672:
3660:
3659:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3585:
3581:
3577:
3574:
3571:
3568:
3545:
3542:
3539:
3536:
3531:
3527:
3506:
3503:
3500:
3497:
3493:
3489:
3486:
3475:
3474:
3473:
3472:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3396:
3392:
3388:
3385:
3382:
3379:
3366:
3365:
3353:
3350:
3347:
3320:
3310:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3235:
3232:
3229:
3218:
3206:
3186:
3183:
3180:
3177:
3172:
3167:
3164:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3121:
3118:
3115:
3091:
3088:
3085:
3082:
3077:
3072:
3069:
3066:
3063:
3058:
3053:
3048:
3043:
3040:
3036:
3032:
3027:
3022:
3002:
2999:
2996:
2993:
2990:
2987:
2983:
2979:
2976:
2942:
2939:
2936:
2933:
2930:
2903:
2900:
2897:
2893:
2889:
2884:
2879:
2876:
2873:
2838:
2834:
2813:
2793:
2773:
2769:
2765:
2760:
2755:
2752:
2749:
2724:
2720:
2717:
2695:
2691:
2679:
2678:
2664:
2661:
2658:
2654:
2650:
2645:
2641:
2637:
2632:
2628:
2606:
2602:
2599:
2596:
2593:
2582:
2570:
2546:
2543:
2540:
2537:
2532:
2527:
2524:
2519:
2514:
2510:
2495:
2494:
2481:
2477:
2474:
2452:
2447:
2444:
2424:
2421:
2418:
2415:
2410:
2407:
2402:
2398:
2394:
2390:
2386:
2383:
2380:
2377:
2373:
2345:
2342:
2339:
2336:
2331:
2327:
2305:
2301:
2298:
2278:
2275:
2272:
2269:
2265:
2261:
2258:
2227:
2223:
2219:
2214:
2209:
2206:
2203:
2184:
2181:
2169:
2165:
2158:
2155:
2153:
2150:
2149:
2146:
2143:
2141:
2138:
2137:
2134:
2130:
2127:
2123:
2116:
2113:
2111:
2108:
2107:
2104:
2101:
2099:
2096:
2095:
2092:
2088:
2085:
2081:
2074:
2071:
2069:
2066:
2065:
2062:
2059:
2057:
2054:
2053:
2050:
2046:
2043:
2039:
2032:
2029:
2027:
2024:
2023:
2020:
2017:
2015:
2012:
2011:
2008:
2004:
2001:
1998:
1976:
1972:
1965:
1962:
1960:
1957:
1956:
1953:
1950:
1948:
1945:
1944:
1941:
1937:
1934:
1930:
1923:
1920:
1918:
1915:
1914:
1911:
1908:
1906:
1903:
1902:
1899:
1895:
1892:
1888:
1881:
1878:
1876:
1873:
1872:
1869:
1866:
1864:
1861:
1860:
1857:
1853:
1850:
1847:
1826:
1819:
1816:
1814:
1811:
1810:
1807:
1804:
1802:
1799:
1798:
1795:
1791:
1788:
1768:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1687:
1683:
1662:
1657:
1653:
1649:
1644:
1640:
1636:
1633:
1630:
1625:
1622:
1619:
1615:
1611:
1606:
1602:
1598:
1593:
1589:
1566:
1562:
1550:
1549:
1546:
1534:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1478:
1456:
1452:
1440:
1428:
1423:
1419:
1415:
1410:
1406:
1402:
1397:
1393:
1372:
1352:
1330:
1326:
1305:
1283:
1279:
1263:
1262:
1250:
1230:
1210:
1200:
1184:
1181:
1178:
1158:
1155:
1152:
1132:
1122:
1110:
1105:
1101:
1097:
1094:
1091:
1086:
1082:
1078:
1073:
1069:
1065:
1045:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
992:
989:
976:
973:
970:
965:
960:
955:
935:
920:
919:
908:
903:
899:
895:
892:
889:
886:
883:
880:
877:
874:
869:
865:
861:
858:
855:
852:
849:
846:
821:
816:
811:
806:
801:
796:
793:
790:
786:
782:
779:
757:
756:
745:
741:
737:
734:
731:
728:
723:
718:
713:
709:
705:
700:
696:
692:
687:
682:
679:
674:
669:
664:
660:
656:
651:
647:
642:
638:
635:
631:
627:
622:
617:
614:
611:
587:Wiener process
574:
552:
547:
542:
539:
536:
532:
528:
525:
505:
502:
499:
488:
487:
475:
469:
464:
460:
456:
453:
450:
447:
444:
441:
440:
437:
434:
431:
428:
425:
421:
416:
413:
410:
406:
401:
398:
395:
392:
389:
386:
382:
378:
377:
374:
350:
345:
341:
337:
334:
331:
328:
325:
322:
302:
299:
296:
272:
250:
245:
240:
235:
230:
225:
222:
210:
207:
167:
164:
161:
141:
110:
67:
66:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
5402:
5391:
5388:
5386:
5383:
5381:
5378:
5377:
5375:
5360:
5357:
5355:
5352:
5351:
5348:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5258:
5256:
5252:
5244:
5241:
5239:
5236:
5235:
5234:
5231:
5229:
5226:
5224:
5221:
5219:
5216:
5214:
5213:Stopping time
5211:
5207:
5204:
5203:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5063:
5061:
5057:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5027:
5025:
5023:
5019:
5012:
5008:
5004:
5003:Hewitt–Savage
5000:
4996:
4992:
4988:
4987:Zero–one laws
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4934:
4932:
4928:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4848:
4846:
4842:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4807:
4805:
4803:
4799:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4774:
4772:
4770:
4766:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4685:
4684:Black–Scholes
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4666:
4664:
4662:
4658:
4652:
4649:
4647:
4644:
4642:
4639:
4637:
4634:
4632:
4629:
4627:
4624:
4623:
4621:
4619:
4615:
4609:
4606:
4604:
4601:
4597:
4594:
4592:
4589:
4588:
4587:
4586:Point process
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4565:
4562:
4560:
4557:
4556:
4555:
4552:
4550:
4547:
4545:
4544:Gibbs measure
4542:
4540:
4537:
4535:
4532:
4531:
4529:
4525:
4519:
4516:
4514:
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4473:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4448:
4446:
4442:
4436:
4433:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4389:
4386:
4384:
4381:
4380:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4333:ItĂ´ diffusion
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4308:Gamma process
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4233:
4232:
4229:
4225:
4222:
4221:
4220:
4217:
4215:
4212:
4210:
4207:
4206:
4204:
4202:
4198:
4190:
4187:
4185:
4182:
4180:
4179:Self-avoiding
4177:
4175:
4172:
4171:
4170:
4167:
4165:
4164:Moran process
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4131:
4129:
4127:
4126:Discrete time
4123:
4119:
4112:
4107:
4105:
4100:
4098:
4093:
4092:
4089:
4078:
4074:
4070:
4066:
4062:
4058:
4054:
4050:
4043:
4040:
4035:
4033:9783540637585
4029:
4025:
4024:
4016:
4013:
4007:
4002:
3998:
3994:
3990:
3983:
3980:
3975:
3971:
3967:
3963:
3959:
3955:
3954:
3946:
3943:
3936:
3932:
3929:
3927:
3924:
3922:
3919:
3918:
3914:
3912:
3898:
3890:
3886:
3878:
3876:
3859:
3856:
3853:
3828:
3824:
3803:
3783:
3763:
3743:
3721:
3717:
3696:
3674:
3670:
3642:
3639:
3636:
3630:
3627:
3621:
3618:
3615:
3612:
3609:
3603:
3600:
3591:
3583:
3579:
3575:
3572:
3566:
3559:
3558:
3557:
3534:
3529:
3525:
3504:
3495:
3487:
3484:
3459:
3453:
3450:
3447:
3444:
3441:
3435:
3432:
3426:
3423:
3420:
3414:
3411:
3402:
3394:
3390:
3386:
3383:
3377:
3370:
3369:
3368:
3367:
3348:
3345:
3338:
3334:
3318:
3311:
3308:
3289:
3286:
3283:
3280:
3277:
3271:
3262:
3259:
3256:
3230:
3227:
3219:
3204:
3184:
3178:
3175:
3170:
3157:
3151:
3148:
3145:
3139:
3116:
3113:
3105:
3104:
3103:
3083:
3070:
3064:
3051:
3041:
3000:
2994:
2991:
2985:
2977:
2974:
2966:
2962:
2959:
2956:
2937:
2934:
2931:
2919:
2917:
2898:
2895:
2887:
2877:
2864:
2859:
2857:
2854:
2836:
2832:
2811:
2791:
2763:
2753:
2739:
2718:
2715:
2693:
2689:
2662:
2659:
2656:
2652:
2648:
2643:
2639:
2635:
2630:
2626:
2600:
2597:
2594:
2591:
2583:
2560:
2535:
2517:
2512:
2508:
2500:
2499:
2498:
2475:
2472:
2445:
2442:
2416:
2408:
2405:
2400:
2396:
2384:
2378:
2363:
2362:
2361:
2359:
2334:
2329:
2325:
2299:
2296:
2267:
2259:
2256:
2249:
2245:
2241:
2217:
2207:
2192:
2190:
2182:
2180:
2167:
2163:
2156:
2151:
2144:
2139:
2132:
2128:
2125:
2121:
2114:
2109:
2102:
2097:
2090:
2086:
2083:
2079:
2072:
2067:
2060:
2055:
2048:
2044:
2041:
2037:
2030:
2025:
2018:
2013:
2006:
2002:
1999:
1996:
1987:
1974:
1970:
1963:
1958:
1951:
1946:
1939:
1935:
1932:
1928:
1921:
1916:
1909:
1904:
1897:
1893:
1890:
1886:
1879:
1874:
1867:
1862:
1855:
1851:
1848:
1845:
1824:
1817:
1812:
1805:
1800:
1793:
1789:
1786:
1743:
1740:
1737:
1731:
1728:
1720:
1716:
1713:
1711:
1705:
1703:
1685:
1681:
1655:
1651:
1642:
1638:
1634:
1631:
1628:
1623:
1620:
1617:
1613:
1609:
1604:
1600:
1596:
1591:
1587:
1564:
1560:
1547:
1527:
1523:
1514:
1510:
1506:
1501:
1497:
1476:
1454:
1450:
1441:
1421:
1417:
1408:
1404:
1400:
1395:
1391:
1370:
1328:
1324:
1303:
1281:
1277:
1268:
1267:
1266:
1228:
1208:
1201:
1198:
1182:
1179:
1176:
1156:
1150:
1123:
1103:
1099:
1095:
1092:
1089:
1084:
1080:
1076:
1071:
1067:
1043:
1036:
1035:
1034:
1017:
1014:
1008:
1005:
990:
988:
971:
968:
963:
933:
925:
901:
897:
893:
890:
887:
884:
878:
875:
867:
863:
859:
856:
853:
850:
844:
837:
836:
835:
819:
804:
794:
788:
780:
777:
769:
765:
762:Now define a
760:
743:
739:
735:
732:
721:
711:
698:
694:
680:
672:
662:
649:
645:
640:
636:
625:
615:
602:
601:
600:
599:
596:
592:
588:
585:-dimensional
572:
550:
534:
526:
523:
500:
497:
467:
462:
458:
454:
448:
442:
435:
429:
423:
414:
411:
408:
396:
390:
387:
384:
372:
364:
363:
362:
343:
339:
335:
332:
329:
326:
320:
300:
297:
294:
286:
283:-dimensional
270:
248:
233:
223:
220:
208:
206:
204:
200:
196:
192:
188:
183:
181:
165:
162:
159:
131:
128:
124:
101:
97:
93:
90:
86:
82:
78:
74:
63:
60:
52:
42:
38:
32:
29:This article
27:
18:
17:
5271:Econometrics
5233:Wiener space
5121:ItĂ´ integral
5022:Inequalities
4911:Self-similar
4881:Gauss–Markov
4871:Exchangeable
4851:CĂ dlĂ g paths
4787:Risk process
4739:LIBOR market
4608:Random graph
4603:Random field
4497:
4415:Superprocess
4353:LĂ©vy process
4348:Jump process
4323:Hunt process
4159:Markov chain
4052:
4048:
4042:
4022:
4015:
3996:
3992:
3982:
3960:(1): 13–38.
3957:
3951:
3945:
3921:Chaos theory
3882:
3661:
3476:
3332:
2964:
2961:metric space
2920:
2860:
2680:
2496:
2247:
2243:
2193:
2188:
2187:Formally, a
2186:
1988:
1718:
1717:
1714:
1706:
1702:Markov Chain
1673:. Clearly,
1551:
1264:
1196:
994:
924:well-defined
921:
767:
763:
761:
758:
489:
285:vector field
212:
190:
184:
179:
129:
122:
91:
80:
73:mathematical
70:
55:
49:January 2022
46:
30:
5316:Ruin theory
5254:Disciplines
5126:ItĂ´'s lemma
4901:Predictable
4576:Percolation
4559:Potts model
4554:Ising model
4518:White noise
4476:Differences
4338:ItĂ´ process
4278:Cox process
4174:Loop-erased
4169:Random walk
2965:phase space
2853:commutative
203:phase space
191:noise terms
132:on the set
89:state space
5374:Categories
5326:Statistics
5106:Filtration
5007:Kolmogorov
4991:Blumenthal
4916:Stationary
4856:Continuous
4844:Properties
4729:Hull–White
4471:Martingale
4358:Local time
4246:Fractional
4224:pure birth
3937:References
3337:almost all
3307:continuous
2736:, forms a
1439:in step 1.
1241:-field of
565:denotes a
287:, and let
5238:Classical
4251:Geometric
4241:Excursion
4077:192603977
3899:ω
3885:attractor
3744:ω
3671:ϑ
3643:ω
3628:−
3622:ω
3592:ω
3580:ϑ
3544:Ω
3541:→
3538:Ω
3526:ϑ
3502:→
3499:Ω
3496:×
3454:ω
3436:φ
3427:ω
3415:φ
3412:∘
3403:ω
3391:ϑ
3378:φ
3352:Ω
3349:∈
3346:ω
3319:φ
3284:ω
3272:φ
3269:↦
3234:Ω
3231:∈
3228:ω
3182:→
3152:ω
3140:φ
3120:Ω
3117:∈
3114:ω
3052:⊗
3042:⊗
2998:→
2992:×
2989:Ω
2986:×
2975:φ
2958:separable
2899:ϑ
2875:Ω
2833:ϑ
2751:Ω
2719:∈
2690:ϑ
2681:That is,
2653:ϑ
2640:ϑ
2636:∘
2627:ϑ
2601:∈
2569:Ω
2545:Ω
2542:→
2539:Ω
2531:Ω
2509:ϑ
2476:∈
2446:∈
2406:−
2397:ϑ
2344:Ω
2341:→
2338:Ω
2326:ϑ
2300:∈
2277:Ω
2274:→
2271:Ω
2268:×
2257:ϑ
2248:base flow
2205:Ω
1767:Γ
1639:α
1635:∘
1632:⋯
1629:∘
1621:−
1614:α
1610:∘
1601:α
1511:α
1451:α
1405:α
1351:Γ
1325:α
1249:Γ
1229:σ
1180:×
1154:→
1131:Γ
1093:⋯
1012:Γ
972:φ
934:φ
891:ω
857:ω
845:φ
810:→
795:×
792:Ω
789:×
778:φ
736:γ
613:Ω
541:→
538:Ω
535:×
504:Ω
501:∈
498:ω
415:ε
333:ω
295:ε
239:→
195:base flow
163:∈
140:Γ
109:Γ
75:field of
5359:Category
5243:Abstract
4777:BĂĽhlmann
4383:Compound
3974:15055697
3915:See also
3843:through
3106:for all
2955:complete
2921:Now let
2584:for all
2435:for all
1719:Example:
1316:, a map
926:). Then
764:flow map
516:, where
4866:Ergodic
4754:VašĂÄŤek
4596:Poisson
4256:Meander
4057:Bibcode
2916:ergodic
199:cocycle
71:In the
35:Please
5206:Tanaka
4891:Mixing
4886:Markov
4759:Wilkie
4724:Ho–Lee
4719:Heston
4491:Super-
4236:Bridge
4184:Biased
4075:
4030:
3972:
3335:: for
2967:. Let
2963:, the
2856:monoid
2557:, the
2317:, let
2242:, the
5059:Tools
4835:M/M/c
4830:M/M/1
4825:M/G/1
4815:Fluid
4481:Local
4073:S2CID
3970:S2CID
3013:be a
2953:be a
2738:group
2244:noise
2238:be a
1700:is a
263:be a
121:from
5011:LĂ©vy
4810:Bulk
4694:Chen
4486:Sub-
4444:Both
4028:ISBN
3756:for
2465:and
2194:Let
2129:0.12
2087:0.42
2045:0.28
2003:0.18
766:or (
298:>
213:Let
100:maps
94:, a
79:, a
4591:Cox
4065:doi
4001:doi
3962:doi
3305:is
2914:is
2561:on
1936:0.1
1894:0.3
1852:0.6
1818:0.3
1813:0.7
1806:0.6
1801:0.4
1343:in
1296:in
834:by
98:of
96:set
39:to
5376::
5009:,
5005:,
5001:,
4997:,
4993:,
4071:.
4063:.
4051:.
4026:.
3997:21
3995:.
3991:.
3968:.
3958:23
3956:.
3246:,
3132:,
2918:.
2708:,
2618:,
2360::
1712:.
1704:.
1033:.
876::=
770:)
637::=
182:.
5013:)
4989:(
4110:e
4103:t
4096:v
4079:.
4067::
4059::
4053:9
4036:.
4009:.
4003::
3976:.
3964::
3863:)
3860:s
3857:+
3854:t
3851:(
3829:0
3825:x
3804:s
3784:t
3764:s
3722:0
3718:x
3697:s
3675:s
3658:.
3646:)
3640:,
3637:s
3634:(
3631:W
3625:)
3619:,
3616:s
3613:+
3610:t
3607:(
3604:W
3601:=
3598:)
3595:)
3589:(
3584:s
3576:,
3573:t
3570:(
3567:W
3535::
3530:s
3505:X
3492:R
3488::
3485:W
3460:.
3457:)
3451:,
3448:s
3445:+
3442:t
3439:(
3433:=
3430:)
3424:,
3421:s
3418:(
3409:)
3406:)
3400:(
3395:s
3387:,
3384:t
3381:(
3364:,
3309:;
3293:)
3290:x
3287:,
3281:,
3278:t
3275:(
3266:)
3263:x
3260:,
3257:t
3254:(
3217:;
3205:X
3185:X
3179:X
3176::
3171:X
3166:d
3163:i
3158:=
3155:)
3149:,
3146:0
3143:(
3090:)
3087:)
3084:X
3081:(
3076:B
3071:,
3068:)
3065:X
3062:(
3057:B
3047:F
3039:)
3035:R
3031:(
3026:B
3021:(
3001:X
2995:X
2982:R
2978::
2941:)
2938:d
2935:,
2932:X
2929:(
2902:)
2896:,
2892:P
2888:,
2883:F
2878:,
2872:(
2837:s
2812:s
2792:s
2772:)
2768:P
2764:,
2759:F
2754:,
2748:(
2723:R
2716:s
2694:s
2677:.
2663:t
2660:+
2657:s
2649:=
2644:t
2631:s
2605:R
2598:t
2595:,
2592:s
2581:;
2536::
2526:d
2523:i
2518:=
2513:0
2493:;
2480:R
2473:s
2451:F
2443:E
2423:)
2420:)
2417:E
2414:(
2409:1
2401:s
2393:(
2389:P
2385:=
2382:)
2379:E
2376:(
2372:P
2335::
2330:s
2304:R
2297:s
2264:R
2260::
2226:)
2222:P
2218:,
2213:F
2208:,
2202:(
2168:.
2164:)
2157:1
2152:0
2145:0
2140:1
2133:(
2126:+
2122:)
2115:0
2110:1
2103:1
2098:0
2091:(
2084:+
2080:)
2073:0
2068:1
2061:0
2056:1
2049:(
2042:+
2038:)
2031:1
2026:0
2019:1
2014:0
2007:(
2000:=
1997:M
1975:.
1971:)
1964:0
1959:1
1952:0
1947:1
1940:(
1933:+
1929:)
1922:1
1917:0
1910:0
1905:1
1898:(
1891:+
1887:)
1880:0
1875:1
1868:1
1863:0
1856:(
1849:=
1846:M
1825:)
1794:(
1790:=
1787:M
1747:}
1744:2
1741:,
1738:1
1735:{
1732:=
1729:S
1686:n
1682:X
1661:)
1656:0
1652:X
1648:(
1643:1
1624:1
1618:n
1605:n
1597:=
1592:n
1588:X
1565:n
1561:X
1545:.
1533:)
1528:1
1524:x
1520:(
1515:2
1507:=
1502:2
1498:x
1477:Q
1455:2
1427:)
1422:0
1418:x
1414:(
1409:1
1401:=
1396:1
1392:x
1371:Q
1329:1
1304:S
1282:0
1278:x
1261:.
1209:Q
1183:n
1177:n
1157:S
1151:S
1121:.
1109:}
1104:n
1100:s
1096:,
1090:,
1085:2
1081:s
1077:,
1072:1
1068:s
1064:{
1044:S
1021:)
1018:Q
1015:,
1009:,
1006:S
1003:(
975:)
969:,
964:d
959:R
954:(
907:)
902:0
898:x
894:;
888:,
885:t
882:(
879:X
873:)
868:0
864:x
860:,
854:,
851:t
848:(
820:d
815:R
805:d
800:R
785:R
781::
744:.
740:)
733:,
730:)
727:)
722:d
717:R
712:;
708:R
704:(
699:0
695:C
691:(
686:B
681:,
678:)
673:d
668:R
663:;
659:R
655:(
650:0
646:C
641:(
634:)
630:P
626:,
621:F
616:,
610:(
589:(
573:d
551:d
546:R
531:R
527::
524:W
468:;
463:0
459:x
455:=
452:)
449:0
446:(
443:X
436:;
433:)
430:t
427:(
424:W
420:d
412:+
409:t
405:d
400:)
397:X
394:(
391:f
388:=
385:X
381:d
373:{
349:)
344:0
340:x
336:;
330:,
327:t
324:(
321:X
301:0
271:d
249:d
244:R
234:d
229:R
224::
221:f
180:Q
166:S
160:X
130:Q
123:S
92:S
62:)
56:(
51:)
47:(
33:.
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