36:
151:
5205:
2517:
5538:
1828:
2947:
3776:
5216:
4222:
3349:
1187:
4961:
7216:
2277:
7627:
5359:
2704:
1476:
2746:
3573:
6275:
4503:
6969:
7968:
4000:
332:
3160:
4626:
8907:
1043:
5200:{\displaystyle {\mathcal {A}}f(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\tfrac {1}{2}}\sum _{i,j}\left(\sigma (x)\sigma (x)^{\top }\right)_{i,j}{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x).}
6495:
1454:
9580:
6779:
2210:
7082:
6683:
9278:
2037:
6130:
7405:
3496:
2512:{\displaystyle Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\tfrac {1}{2}}\sum _{i,j}\left(\sigma (x)\sigma (x)^{\top }\right)_{i,j}{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x),}
8406:
7447:
6398:
4942:
5533:{\displaystyle \Delta _{\mathrm {LB} }={\frac {1}{\sqrt {\det(g)}}}\sum _{i=1}^{m}{\frac {\partial }{\partial x_{i}}}\left({\sqrt {\det(g)}}\sum _{j=1}^{m}g^{ij}{\frac {\partial }{\partial x_{j}}}\right),}
493:
4705:
8181:
2543:
9685:
7059:
6135:
This illustrates one of the connections between stochastic analysis and the study of partial differential equations. Conversely, a given second-order linear partial differential equation of the form Λ
9366:
5952:
5852:
5735:
1823:{\displaystyle {\begin{aligned}\mathbf {E} ^{x}\left&=\mathbf {E} ^{x}\left{\big |}F_{t}\right]\\&=\mathbf {E} ^{x}\left{\big |}F_{t}\right]\\&=\mathbf {E} ^{X_{t}}\left.\end{aligned}}}
8604:
1481:
8795:
754:
2942:{\displaystyle Af(x)={\tfrac {1}{2}}\sum _{i,j}\delta _{ij}{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)={\tfrac {1}{2}}\sum _{i}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}(x)}
4874:
206:
7735:
9119:
3981:
3771:{\displaystyle {\begin{cases}{\dfrac {\partial \rho }{\partial t}}(t,x)=A^{*}\rho (t,x),&t>0,x\in \mathbf {R} ^{n};\\\rho (0,x)=\rho _{0}(x),&x\in \mathbf {R} ^{n}.\end{cases}}}
9871:
6593:
3102:
8266:
6169:
4351:
933:
4821:
6860:
7746:
10406:
4745:
4217:{\displaystyle {\begin{cases}{\dfrac {\partial v}{\partial t}}(t,x)=Av(t,x)-q(x)v(t,x),&t>0,x\in \mathbf {R} ^{n};\\v(0,x)=f(x),&x\in \mathbf {R} ^{n}.\end{cases}}}
5308:
4339:
2979:
The generator is used in the formulation of
Kolmogorov's backward equation. Intuitively, this equation tells us how the expected value of any suitably smooth statistic of
4307:
is a partial differential operator closely related to the generator, but somewhat more general. It is more suited to certain problems, for example in the solution of the
65:
10230:
8913:
8186:
Dynkin's formula can be used to calculate many useful statistics of stopping times. For example, canonical
Brownian motion on the real line starting at 0 exits the
3344:{\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Au(t,x),&t>0,x\in \mathbf {R} ^{n};\\u(0,x)=f(x),&x\in \mathbf {R} ^{n}.\end{cases}}}
10833:
9826:
2048:
594:
233:
10363:
10343:
10747:
7241:
and variance (βκ). The expression for the variance may be interpreted as follows: large values of κ mean that the potential well Ψ has "very steep sides", so
4532:
1182:{\displaystyle \Sigma _{t}=\Sigma _{t}^{B}=\sigma \left\{B_{s}^{-1}(A)\subseteq \Omega \ :\ 0\leq s\leq t,A\subseteq \mathbf {R} ^{n}{\mbox{ Borel}}\right\}.}
8822:
11097:
6404:
6292:
is a scalar potential satisfying suitable smoothness and growth conditions. In this case, the Fokker–Planck equation has a unique stationary solution ρ
10664:
6416:
1306:
10674:
10348:
9448:
6694:
2100:
10358:
7211:{\displaystyle \rho _{\infty }(x)=\left({\frac {\beta \kappa }{2\pi }}\right)^{\frac {n}{2}}\exp \left(-{\frac {\beta \kappa |x-m|^{2}}{2}}\right)}
4831:
The characteristic operator and infinitesimal generator are very closely related, and even agree for a large class of functions. One can show that
10716:
10431:
10613:
6604:
9142:
1899:
10903:
10893:
10416:
614:
In particular, an Itô diffusion is a continuous, strongly
Markovian process such that the domain of its characteristic operator includes all
10803:
10767:
6058:
7309:
6784:
is the negative of the Gibbs-Boltzmann entropy functional. Even when the potential Ψ is not well-behaved enough for the partition function
3410:
7622:{\displaystyle f(X_{t})=f(x)+\int _{0}^{t}Af(X_{s})\,\mathrm {d} s+\int _{0}^{t}\nabla f(X_{s})^{\top }\sigma (X_{s})\,\mathrm {d} B_{s}.}
10720:
8330:
11071:
10808:
9918:
9819:
6318:
4885:
10873:
10451:
10421:
9788:
2699:{\displaystyle Af(x)=b(x)\cdot \nabla _{x}f(x)+{\tfrac {1}{2}}\left(\sigma (x)\sigma (x)^{\top }\right):\nabla _{x}\nabla _{x}f(x).}
379:
87:
6006: ≥ 0. The Fokker–Planck equation offers a way to find such a measure, at least if it has a probability density function ρ
4637:
10724:
10708:
8043:
10918:
10623:
9843:
515:
113:
10823:
10788:
10757:
10752:
10391:
10188:
10105:
9603:
8466:
7276:
6988:
10762:
10090:
2066:
of the diffusion. The generator is very useful in many applications and encodes a great deal of information about the process
10386:
10193:
10112:
9293:
7429:. The proof is quite simple: it follows from the usual expression of the action of the generator on smooth enough functions
5898:
5798:
5614:
10848:
10728:
8520:
11076:
10853:
10689:
10588:
10573:
9985:
9901:
9812:
3147:
2984:
2059:
1026:
759:
This follows from the standard existence and uniqueness theory for strong solutions of stochastic differential equations.
631:
10863:
10499:
10858:
8717:
3363:
2970:
689:
10461:
48:
10045:
9990:
9906:
10793:
10783:
10426:
10396:
4837:
2974:
180:
58:
52:
44:
7654:
10798:
9963:
9861:
9728:
9004:
6844:
3842:
10509:
10085:
9866:
10878:
10679:
10593:
10578:
9968:
6513:
5259:Δ, where Δ denotes the Laplace operator. The characteristic operator is useful in defining Brownian motion on an
772:
646:
573:
569:
10712:
10598:
10020:
6270:{\displaystyle \mathrm {d} X_{t}=-\nabla \Psi (X_{t})\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}
69:
10100:
10075:
6526:
4498:{\displaystyle {\mathcal {A}}f(x)=\lim _{U\downarrow x}{\frac {\mathbf {E} ^{x}\left-f(x)}{\mathbf {E} ^{x}}},}
10818:
10401:
9936:
3787:
3029:
8210:
6964:{\displaystyle \mathrm {d} X_{t}=-\kappa (X_{t}-m)\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}
11013:
11003:
10694:
10476:
10215:
10080:
9891:
4300:
2531:
1276:
866:
10298:
4784:
10955:
10883:
10142:
9740:
8420:
8187:
7963:{\displaystyle \mathbf {E} ^{x}=\mathbf {E} ^{x}\left}{\big |}F_{s}\right]=\mathbf {E} ^{x}{\big }=M_{s},}
6151:
is easy to compute, then that measure's density provides a solution to the partial differential equation.
973:
is varied is addressed by the
Kolmogorov backward equation, the Fokker–Planck equation, etc. (See below.)
587:
10978:
10960:
10940:
10935:
10654:
10486:
10466:
10313:
10256:
10095:
10005:
6505:
10446:
158:(Brownian motion) in three-dimensional space (one sample path shown) is an example of an Itô diffusion.
4721:
11053:
11008:
10998:
10739:
10684:
10659:
10628:
10608:
10368:
10353:
10220:
8685:
5575:
then the resulting operator is invertible. The inverse of this operator can be expressed in terms of
1015:
at time 0. The precise mathematical formulation of this statement requires some additional notation:
370:
105:
5289:
4320:
4009:
3582:
3169:
11048:
10888:
10813:
10618:
10378:
10288:
10178:
9384:
8497:
7996:
7234:
6281:
5264:
3821:
837:
833:
668:
607:
9745:
7252:; similarly, large values of β mean that the system is quite "cold" with little noise, so, again,
1881:
be a bounded, Borel-measurable function. Let τ be a stopping time with respect to the filtration Σ
11018:
10983:
10898:
10868:
10638:
10633:
10456:
10293:
9958:
9896:
9835:
9758:
9590:
8477:
6309:
1023:
209:
10699:
11038:
10843:
10494:
10251:
10168:
10137:
10030:
10010:
10000:
9856:
9851:
9784:
9776:
9696:
8669:
7434:
6809:
5965:
5544:
4308:
3506:
220:
117:
10704:
10441:
9716:. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc.
3792:
The
Feynman–Kac formula is a useful generalization of Kolmogorov's backward equation. Again,
11058:
10945:
10828:
10198:
10173:
10122:
9973:
9926:
9750:
8301:
7069:
5564:
5350:
5235:
times the
Laplace-Beltrami operator. Here it is the Laplace-Beltrami operator on a 2-sphere.
3389:
2958:
841:
530:
327:{\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t},}
9798:
9770:
9721:
11023:
10923:
10908:
10669:
10603:
10281:
10225:
10208:
9953:
9795:
9767:
9718:
2254:
1197:
982:
939:
615:
580:
346:
175:
125:
10838:
10070:
6982:
and β, κ > 0 are given constants. In this case, the potential Ψ is given by
4621:{\displaystyle U_{k+1}\subseteq U_{k}{\mbox{ and }}\bigcap _{k=1}^{\infty }U_{k}=\{x\},}
11028:
10993:
10913:
10519:
10266:
10183:
10152:
10147:
10127:
10117:
10060:
10055:
10035:
10015:
9980:
9948:
9931:
8902:{\displaystyle \varphi (x)=\int _{\partial G}\varphi (y)\,\mathrm {d} \mu _{G}^{x}(y).}
8311:
8006:
5580:
4270:
Kolmogorov's backward equation is the special case of the
Feynman–Kac formula in which
810:
561:
155:
11091:
10930:
10471:
10308:
10303:
10261:
10203:
10025:
9941:
9881:
9709:
8925:
8002:
3519:
2534:
1886:
1843:
1838:
The strong Markov property is a generalization of the Markov property above in which
1280:
139:
136:
9762:
8411:
Sometimes, however, one also wishes to know the distribution of the points at which
10988:
10950:
10504:
10436:
10325:
10320:
10132:
10065:
10040:
9876:
8322:
8280:
540:
522:
10568:
565:, not a diffusion. Itô diffusions have a number of nice properties, which include
11033:
10552:
10547:
10542:
10532:
10335:
10276:
10271:
10235:
9995:
9886:
2527:
150:
101:
8275:
at a fairly general stopping time. For more information on the distribution of
6490:{\displaystyle Z=\int _{\mathbf {R} ^{n}}\exp(-\beta \Psi (x))\,\mathrm {d} x.}
5239:
Above, the generator (and hence characteristic operator) of
Brownian motion on
1449:{\displaystyle \mathbf {E} ^{x}{\big }(\omega )=\mathbf {E} ^{X_{t}(\omega )}.}
11043:
10583:
10527:
10411:
9754:
9575:{\displaystyle f(x)=\mathbf {E} ^{x}\left-\int _{D}Af(y)\,G(x,\mathrm {d} y).}
7438:
6774:{\displaystyle S=\int _{\mathbf {R} ^{n}}\rho (x)\log \rho (x)\,\mathrm {d} x}
2205:{\displaystyle Af(x)=\lim _{t\downarrow 0}{\frac {\mathbf {E} ^{x}-f(x)}{t}}.}
1842:
is replaced by a suitable random time τ : Ω → known as a
650:
17:
10537:
8653:
7632:
Since Itô integrals are martingales with respect to the natural filtration Σ
6825:
3563:, and ρ satisfies the following partial differential equation, known as the
129:
8628:
Returning to the earlier example of
Brownian motion, one can show that if
8673:
6678:{\displaystyle E=\int _{\mathbf {R} ^{n}}\Psi (x)\rho (x)\,\mathrm {d} x}
4513:
3514:
2523:
9273:{\displaystyle \int _{D}f(y)\,G(x,\mathrm {d} y)=\mathbf {E} ^{x}\left.}
2032:{\displaystyle \mathbf {E} ^{x}{\big }=\mathbf {E} ^{X_{\tau }}{\big }.}
10364:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
9804:
6159:
An invariant measure is comparatively easy to compute when the process
5215:
4947:
In particular, the generator and characteristic operator agree for all
3359:
121:
6125:{\displaystyle A^{*}\rho _{\infty }(x)=0,\quad x\in \mathbf {R} ^{n}.}
9731:(1998). "The variational formulation of the Fokker–Planck equation".
8914:
solution of partial differential equations using stochastic processes
7400:{\displaystyle M_{t}=f(X_{t})-\int _{0}^{t}Af(X_{s})\,\mathrm {d} s,}
3491:{\displaystyle \mathbf {P} \left=\int _{S}\rho (t,x)\,\mathrm {d} x.}
9781:
Stochastic
Differential Equations: An Introduction with Applications
8401:{\displaystyle \tau _{H}(\omega )=\inf\{t\geq 0|X_{t}\not \in H\}.}
8306:
In many situations, it is sufficient to know when an Itô diffusion
6139: = 0 may be hard to solve directly, but if Λ =
5214:
2260:(twice differentiable with continuous second derivative) function
133:
1008:, is the same as if the process had been started at the position
555:
and σ do not depend upon time; if they were to depend upon time,
8950:
as its generator. Intuitively, the Green measure of a Borel set
6052:
must solve the (time-independent) partial differential equation
227:) and satisfying a stochastic differential equation of the form
9808:
9712:; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965).
6393:{\displaystyle \rho _{\infty }(x)=Z^{-1}\exp(-\beta \Psi (x)),}
4937:{\displaystyle Af={\mathcal {A}}f{\mbox{ for all }}f\in D_{A}.}
27:
Solution to a specific type of stochastic differential equation
8017:) at a stopping time. Precisely, if τ is a stopping time with
5740:
It can be shown, using the Feller continuity of the diffusion
4259:, and satisfies the above partial differential equation, then
488:{\displaystyle |b(x)-b(y)|+|\sigma (x)-\sigma (y)|\leq C|x-y|}
29:
8271:
Dynkin's formula provides information about the behaviour of
4700:{\displaystyle \tau _{U}=\inf\{t\geq 0\ :\ X_{t}\not \in U\}}
8711:
is any bounded, Borel-measurable function and φ is given by
8176:{\displaystyle \mathbf {E} ^{x}=f(x)+\mathbf {E} ^{x}\left.}
5295:
4967:
4900:
4860:
4790:
4731:
4357:
4326:
5567:. However, if a positive multiple of the identity operator
4210:
3764:
3337:
767:
In addition to being (sample) continuous, an Itô diffusion
10344:
Autoregressive conditional heteroskedasticity (ARCH) model
2058:
Associated to each Itô diffusion, there is a second-order
9680:{\displaystyle f(x)=-\int _{D}Af(y)\,G(x,\mathrm {d} y).}
7418:, is a martingale with respect to the natural filtration
7054:{\displaystyle \Psi (x)={\tfrac {1}{2}}\kappa |x-m|^{2},}
6017:
is indeed distributed according to an invariant measure μ
521:
to the stochastic differential equation given above. The
9872:
Independent and identically distributed random variables
6800: ≥ 0, provided that the initial condition has
2726:, which satisfies the stochastic differential equation d
1846:. So, for example, rather than "restarting" the process
1463:
is also a Markov process with respect to the filtration
5987:
is distributed according to such an invariant measure μ
5789:
with compact support, then, for all α > 0,
1267:
be a bounded, Borel-measurable function. Then, for all
10349:
Autoregressive integrated moving average (ARIMA) model
9361:{\displaystyle G(x,H)=\int _{H}G(x,y)\,\mathrm {d} y,}
8415:
exits the set. For example, canonical Brownian motion
7008:
5947:{\displaystyle (\alpha \mathbf {I} -A)R_{\alpha }g=g.}
5847:{\displaystyle R_{\alpha }(\alpha \mathbf {I} -A)f=f;}
5730:{\displaystyle R_{\alpha }g(x)=\mathbf {E} ^{x}\left.}
5056:
4909:
4566:
2870:
2769:
2606:
2368:
1165:
734:
675:. More accurately, there is a "continuous version" of
9606:
9451:
9296:
9283:
The name "Green measure" comes from the fact that if
9145:
9007:
8825:
8720:
8599:{\displaystyle \mu _{G}^{x}(F)=\mathbf {P} ^{x}\left}
8523:
8333:
8213:
8046:
8009:
of any suitably smooth statistic of an Itô diffusion
7749:
7657:
7450:
7312:
7085:
6991:
6863:
6697:
6607:
6529:
6419:
6321:
6172:
6061:
5901:
5801:
5617:
5362:
5292:
5211:
Application: Brownian motion on a Riemannian manifold
4964:
4888:
4840:
4787:
4724:
4640:
4535:
4354:
4323:
4013:
4003:
3845:
3586:
3576:
3413:
3173:
3163:
3032:
2749:
2546:
2280:
2103:
1902:
1479:
1309:
1046:
869:
692:
382:
236:
183:
7299: : [0, +∞) × Ω →
5219:
The characteristic operator of a Brownian motion is
1289:
and the expectation of the process "restarted" from
215: : [0, +∞) × Ω →
10971:
10776:
10738:
10647:
10561:
10518:
10485:
10377:
10334:
10244:
10161:
9917:
9842:
6508:: it minimizes over all probability densities ρ on
514:; this condition ensures the existence of a unique
9679:
9574:
9360:
9272:
9113:
8901:
8790:{\displaystyle \varphi (x)=\mathbf {E} ^{x}\left,}
8789:
8598:
8400:
8260:
8175:
7962:
7729:
7621:
7399:
7210:
7053:
6963:
6773:
6677:
6587:
6489:
6392:
6269:
6124:
5946:
5846:
5729:
5532:
5302:
5199:
4936:
4868:
4815:
4739:
4699:
4620:
4497:
4333:
4216:
3975:
3770:
3490:
3343:
3096:
2995:are the independent variables. More precisely, if
2941:
2698:
2511:
2204:
2031:
1822:
1448:
1181:
927:
749:{\displaystyle \mathbf {P} =1{\mbox{ for all }}t.}
748:
487:
326:
200:
7248:is unlikely to move far from the minimum of Ψ at
3362:" to the backward equation, and tells us how the
10231:Stochastic chains with memory of variable length
8356:
6854:satisfying the stochastic differential equation
5754:is itself a bounded, continuous function. Also,
5452:
5387:
4654:
4378:
2123:
2082:, which is defined to act on suitable functions
1029:of (Ω, Σ) generated by the Brownian motion
601:
57:but its sources remain unclear because it lacks
8934:be a partial differential operator on a domain
6804: < +∞. The free energy functional
4869:{\displaystyle D_{A}\subseteq D_{\mathcal {A}}}
201:{\displaystyle {\boldsymbol {\textbf {R}}}^{n}}
8912:The mean value property is very useful in the
8695:The harmonic measure satisfies an interesting
7730:{\displaystyle \mathbf {E} ^{x}{\big }=M_{s}.}
6280:where β > 0 plays the role of an
2049:Infinitesimal generator (stochastic processes)
9820:
9114:{\displaystyle G(x,H)=\mathbf {E} ^{x}\left,}
7939:
7922:
7905:
7868:
7861:
7844:
7827:
7706:
7689:
7672:
3976:{\displaystyle v(t,x)=\mathbf {E} ^{x}\left.}
2021:
1995:
1966:
1949:
1917:
1854: = 1, one could "restart" whenever
1735:
1638:
1616:
1528:
1373:
1356:
1324:
8:
8392:
8359:
6688:plays the role of an energy functional, and
4694:
4657:
4612:
4606:
3994:satisfies the partial differential equation
128:of a particle subjected to a potential in a
771:satisfies the stronger requirement to be a
132:fluid. Itô diffusions are named after the
10359:Autoregressive–moving-average (ARMA) model
9827:
9813:
9805:
6163:is a stochastic gradient flow of the form
5596:, acting on bounded, continuous functions
3354:The Fokker–Planck equation (also known as
2965:The Kolmogorov and Fokker–Planck equations
1004:, given what has happened up to some time
9744:
9663:
9650:
9629:
9605:
9558:
9545:
9524:
9500:
9495:
9473:
9468:
9450:
9347:
9346:
9322:
9295:
9254:
9253:
9244:
9226:
9221:
9216:
9201:
9196:
9181:
9168:
9150:
9144:
9095:
9094:
9085:
9072:
9060:
9055:
9050:
9035:
9030:
9006:
8881:
8876:
8867:
8866:
8845:
8824:
8768:
8763:
8742:
8737:
8719:
8577:
8572:
8557:
8552:
8533:
8528:
8522:
8419:on the real line starting at 0 exits the
8380:
8371:
8338:
8332:
8249:
8233:
8220:
8215:
8212:
8157:
8156:
8147:
8128:
8123:
8108:
8103:
8072:
8053:
8048:
8045:
7951:
7938:
7937:
7931:
7921:
7920:
7914:
7904:
7903:
7897:
7892:
7877:
7867:
7866:
7860:
7859:
7853:
7843:
7842:
7836:
7826:
7825:
7819:
7814:
7802:
7797:
7784:
7775:
7769:
7756:
7751:
7748:
7718:
7705:
7704:
7698:
7688:
7687:
7681:
7671:
7670:
7664:
7659:
7656:
7610:
7601:
7600:
7591:
7575:
7565:
7546:
7541:
7526:
7525:
7516:
7497:
7492:
7461:
7449:
7386:
7385:
7376:
7357:
7352:
7336:
7317:
7311:
7191:
7186:
7171:
7162:
7137:
7113:
7090:
7084:
7042:
7037:
7022:
7007:
6990:
6952:
6943:
6942:
6931:
6922:
6911:
6910:
6895:
6873:
6864:
6862:
6763:
6762:
6724:
6719:
6717:
6696:
6667:
6666:
6634:
6629:
6627:
6606:
6588:{\displaystyle F=E+{\frac {1}{\beta }}S,}
6560:
6528:
6476:
6475:
6437:
6432:
6430:
6418:
6348:
6326:
6320:
6258:
6249:
6248:
6237:
6228:
6217:
6216:
6207:
6182:
6173:
6171:
6113:
6108:
6076:
6066:
6060:
5976:that does not change under the "flow" of
5926:
5908:
5900:
5818:
5806:
5800:
5711:
5710:
5701:
5679:
5669:
5664:
5649:
5644:
5622:
5616:
5513:
5500:
5491:
5481:
5470:
5450:
5436:
5423:
5417:
5406:
5381:
5368:
5367:
5361:
5294:
5293:
5291:
5176:
5168:
5162:
5144:
5137:
5125:
5114:
5071:
5055:
5034:
5016:
5001:
4991:
4966:
4965:
4963:
4925:
4908:
4899:
4898:
4887:
4859:
4858:
4845:
4839:
4789:
4788:
4786:
4730:
4729:
4723:
4682:
4645:
4639:
4597:
4587:
4576:
4565:
4559:
4540:
4534:
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4467:
4462:
4428:
4423:
4402:
4397:
4393:
4381:
4356:
4355:
4353:
4325:
4324:
4322:
4198:
4193:
4135:
4130:
4012:
4004:
4002:
3956:
3933:
3932:
3923:
3907:
3902:
3873:
3868:
3844:
3824:that is bounded below. Define a function
3752:
3747:
3720:
3682:
3677:
3629:
3585:
3577:
3575:
3522:). Then, given that the initial position
3477:
3476:
3452:
3428:
3414:
3412:
3325:
3320:
3262:
3257:
3172:
3164:
3162:
3079:
3060:
3055:
3031:
2983:evolves in time: it must solve a certain
2921:
2916:
2898:
2891:
2885:
2869:
2848:
2840:
2834:
2816:
2809:
2800:
2784:
2768:
2748:
2675:
2665:
2647:
2605:
2584:
2545:
2488:
2480:
2474:
2456:
2449:
2437:
2426:
2383:
2367:
2346:
2328:
2313:
2303:
2279:
2166:
2147:
2142:
2138:
2126:
2102:
2020:
2019:
2010:
1994:
1993:
1985:
1980:
1975:
1965:
1964:
1958:
1948:
1947:
1932:
1916:
1915:
1909:
1904:
1901:
1799:
1776:
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1744:
1734:
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1696:
1691:
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1647:
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1636:
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1614:
1599:
1578:
1573:
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1556:
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1527:
1526:
1511:
1490:
1485:
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1478:
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1401:
1396:
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1372:
1371:
1365:
1355:
1354:
1339:
1323:
1322:
1316:
1311:
1308:
1164:
1158:
1153:
1095:
1090:
1069:
1064:
1051:
1045:
910:
891:
886:
868:
733:
718:
705:
693:
691:
480:
466:
455:
423:
415:
383:
381:
315:
306:
305:
296:
275:
274:
265:
246:
237:
235:
192:
186:
185:
182:
88:Learn how and when to remove this message
3097:{\displaystyle u(t,x)=\mathbf {E} ^{x},}
1217:-measurable), so the natural filtration
149:
8261:{\displaystyle \mathbf {E} ^{0}=R^{2}.}
2219:for which this limit exists at a point
622:in the sense defined by Dynkin (1965).
10665:Doob's martingale convergence theorems
9727:Jordan, Richard; Kinderlehrer, David;
4231: : [0, +∞) ×
3828: : [0, +∞) ×
3015: : [0, +∞) ×
2957: = Δ/2, where Δ denotes the
928:{\displaystyle u(x)=\mathbf {E} ^{x}.}
10417:Constant elasticity of variance (CEV)
10407:Chan–Karolyi–Longstaff–Sanders (CKLS)
9124:or for bounded, continuous functions
8990:, ·), is defined for Borel sets
6155:Invariant measures for gradient flows
5964:Sometimes it is necessary to find an
4816:{\displaystyle {\mathcal {A}}f(x)=0.}
3396:, i.e., for any Borel-measurable set
7:
9783:(Sixth ed.). Berlin: Springer.
8954:is the expected length of time that
8423:(−1, 1) at −1 with probability
7295:) with compact support, the process
5543:where = in the sense of
4751:for which this limit exists for all
3541:) is differentiable with respect to
3119:) is differentiable with respect to
996:has the important property of being
112:is a solution to a specific type of
8920:The Green measure and Green formula
8321:. That is, one wishes to study the
2245:for which the limit exists for all
1858:first reaches some specified point
187:
10904:Skorokhod's representation theorem
10685:Law of large numbers (weak/strong)
9664:
9559:
9348:
9255:
9182:
9096:
8868:
8846:
8684:and coincides with the normalized
8158:
7850:
7695:
7602:
7576:
7552:
7527:
7387:
7259:is unlikely to move far away from
7091:
6992:
6944:
6912:
6865:
6764:
6668:
6642:
6477:
6460:
6372:
6327:
6250:
6218:
6197:
6194:
6174:
6077:
5998:is also distributed according to μ
5712:
5670:
5506:
5502:
5429:
5425:
5372:
5369:
5364:
5169:
5155:
5141:
5115:
5027:
5019:
4770: = +∞ for all open sets
4588:
4024:
4016:
3934:
3597:
3589:
3478:
3184:
3176:
2909:
2895:
2841:
2827:
2813:
2672:
2662:
2648:
2581:
2481:
2467:
2453:
2427:
2339:
2331:
1955:
1622:
1362:
1116:
1061:
1048:
307:
276:
238:
116:. That equation is similar to the
25:
11098:Stochastic differential equations
10874:Martingale representation theorem
9585:In particular, if the support of
7064:and so the invariant measure for
616:twice-continuously differentiable
10919:Stochastic differential equation
10809:Doob's optional stopping theorem
10804:Doob–Meyer decomposition theorem
9469:
9197:
9031:
8738:
8553:
8216:
8104:
8049:
7893:
7815:
7798:
7752:
7660:
6812:for the Fokker–Planck equation:
6796:still makes sense for each time
6720:
6630:
6433:
6300:has a unique invariant measure μ
6109:
5909:
5889:and, for all α > 0,
5872:is bounded and continuous, then
5819:
5769:are mutually inverse operators:
5645:
5282:is defined to be a diffusion on
4740:{\displaystyle D_{\mathcal {A}}}
4463:
4398:
4194:
4131:
3869:
3748:
3678:
3415:
3321:
3258:
3056:
2143:
1976:
1905:
1767:
1687:
1670:
1574:
1557:
1486:
1392:
1312:
1154:
957:is bounded and continuous, then
887:
694:
559:would be referred to only as an
114:stochastic differential equation
34:
10789:Convergence of random variables
10675:Fisher–Tippett–Gnedenko theorem
8660:, then the harmonic measure of
6792:to be defined, the free energy
6147:, and an invariant measure for
6100:
4827:Relationship with the generator
3808:) and has compact support, and
946:is lower semi-continuous, then
551:. It is important to note that
10387:Binomial options pricing model
9671:
9654:
9647:
9641:
9616:
9610:
9566:
9549:
9542:
9536:
9461:
9455:
9343:
9331:
9312:
9300:
9250:
9237:
9189:
9172:
9165:
9159:
9091:
9078:
9023:
9011:
8893:
8887:
8863:
8857:
8835:
8829:
8776:
8756:
8730:
8724:
8545:
8539:
8372:
8350:
8344:
8239:
8226:
8153:
8140:
8096:
8090:
8081:
8078:
8065:
8059:
8001:Dynkin's formula, named after
7790:
7776:
7762:
7597:
7584:
7572:
7558:
7522:
7509:
7482:
7476:
7467:
7454:
7382:
7369:
7342:
7329:
7187:
7172:
7102:
7096:
7038:
7023:
7001:
6995:
6907:
6888:
6759:
6753:
6741:
6735:
6707:
6701:
6663:
6657:
6651:
6645:
6617:
6611:
6579:
6573:
6554:
6548:
6539:
6533:
6472:
6469:
6463:
6451:
6384:
6381:
6375:
6363:
6338:
6332:
6213:
6200:
6088:
6082:
5919:
5902:
5829:
5812:
5707:
5694:
5637:
5631:
5545:the inverse of a square matrix
5461:
5455:
5396:
5390:
5353:given in local coordinates by
5303:{\displaystyle {\mathcal {A}}}
5286:whose characteristic operator
5191:
5185:
5111:
5104:
5098:
5092:
5049:
5043:
5013:
5007:
4981:
4975:
4804:
4798:
4486:
4473:
4456:
4450:
4436:
4416:
4385:
4371:
4365:
4334:{\displaystyle {\mathcal {A}}}
4178:
4172:
4163:
4151:
4103:
4091:
4085:
4079:
4070:
4058:
4046:
4034:
3962:
3949:
3929:
3916:
3861:
3849:
3732:
3726:
3710:
3698:
3650:
3638:
3619:
3607:
3473:
3461:
3305:
3299:
3290:
3278:
3230:
3218:
3206:
3194:
3152:Kolmogorov's backward equation
3088:
3085:
3072:
3066:
3048:
3036:
2936:
2930:
2863:
2857:
2762:
2756:
2690:
2684:
2644:
2637:
2631:
2625:
2599:
2593:
2574:
2568:
2559:
2553:
2503:
2497:
2423:
2416:
2410:
2404:
2361:
2355:
2325:
2319:
2293:
2287:
2190:
2184:
2175:
2172:
2159:
2153:
2130:
2116:
2110:
2016:
2003:
1944:
1925:
1805:
1792:
1725:
1712:
1611:
1592:
1523:
1504:
1440:
1437:
1424:
1418:
1413:
1407:
1384:
1378:
1351:
1332:
1110:
1104:
965:The behaviour of the function
919:
916:
903:
897:
879:
873:
724:
698:
481:
467:
456:
452:
446:
437:
431:
424:
416:
412:
406:
397:
391:
384:
302:
289:
271:
258:
1:
10854:Kolmogorov continuity theorem
10690:Law of the iterated logarithm
9714:Markov processes. Vols. I, II
7271:In general, an Itô diffusion
3364:probability density functions
3356:Kolmogorov's forward equation
3148:partial differential equation
2985:partial differential equation
2722:-dimensional Brownian motion
2060:partial differential operator
632:Continuous stochastic process
10859:Kolmogorov extension theorem
10538:Generalized queueing network
10046:Interacting particle systems
8962:before it leaves the domain
5586:For α > 0, the
4710:is the first exit time from
3381:, ·) be the density of
2971:Kolmogorov backward equation
1231:of (Ω, Σ) generated by
9991:Continuous-time random walk
9414: < +∞ for all
8037:with compact support, then
4522:that decrease to the point
4290:The characteristic operator
11114:
10999:Extreme value theory (EVT)
10799:Doob decomposition theorem
10091:Ornstein–Uhlenbeck process
9862:Chinese restaurant process
8923:
8439:and at 1 with probability
8299:
7994:
6845:Ornstein-Uhlenbeck process
5555:In general, the generator
3785:
3529:has a prescribed density ρ
3011:) has compact support and
2968:
2046:
1834:The strong Markov property
1470:, as the following shows:
1000:: the future behaviour of
980:
629:
11067:
10879:Optional stopping theorem
10680:Large deviation principle
10432:Heath–Jarrow–Morton (HJM)
10369:Moving-average (MA) model
10354:Autoregressive (AR) model
10179:Hidden Markov model (HMM)
10113:Schramm–Loewner evolution
9801:(See Sections 7, 8 and 9)
9755:10.1137/S0036141096303359
9287:is Brownian motion, then
8946:be an Itô diffusion with
8800:then, for all Borel sets
8469:on the set {−1, 1}.
7222:Heuristically, for large
6308:) and it is given by the
5351:Laplace-Beltrami operator
2991:and the initial position
2215:The set of all functions
1885:with τ < +∞
950:is lower semi-continuous.
773:Feller-continuous process
647:sample continuous process
10794:Doléans-Dade exponential
10624:Progressively measurable
10422:Cox–Ingersoll–Ross (CIR)
9739:(1): 1–17 (electronic).
9442:) with compact support:
8632:is a Brownian motion in
8021: < +∞, and
4278:) = 0 for all
3358:) is in some sense the "
3146:satisfies the following
2253:. One can show that any
1192:It is easy to show that
43:This article includes a
11014:Mathematical statistics
11004:Large deviations theory
10834:Infinitesimal generator
10695:Maximal ergodic theorem
10614:Piecewise-deterministic
10216:Random dynamical system
10081:Markov additive process
7990:
7267:The martingale property
6788:and the Gibbs measure μ
6500:Moreover, the density ρ
6143:for some Itô diffusion
6044:, ·) = ρ
4747:denotes the set of all
4316:characteristic operator
4301:characteristic operator
3782:The Feynman–Kac formula
2241:denotes the set of all
2072:infinitesimal generator
1277:conditional expectation
679:, a continuous process
671:of the time parameter,
602:characteristic operator
595:infinitesimal generator
72:more precise citations.
10849:Karhunen–Loève theorem
10784:Cameron–Martin formula
10748:Burkholder–Davis–Gundy
10143:Variance gamma process
9681:
9576:
9362:
9274:
9115:
8903:
8791:
8600:
8402:
8262:
8177:
7964:
7731:
7623:
7401:
7212:
7055:
6965:
6775:
6679:
6589:
6491:
6394:
6271:
6126:
5948:
5848:
5731:
5551:The resolvent operator
5534:
5486:
5422:
5304:
5236:
5201:
4938:
4870:
4817:
4741:
4701:
4622:
4592:
4499:
4335:
4218:
3977:
3772:
3565:Fokker–Planck equation
3549:, ·) ∈
3492:
3345:
3131:, ·) ∈
3098:
2975:Fokker–Planck equation
2943:
2700:
2513:
2206:
2033:
1824:
1450:
1183:
953:Feller continuity: if
929:
844:and define, for fixed
750:
618:functions, so it is a
588:strong Markov property
489:
328:
202:
159:
10979:Actuarial mathematics
10941:Uniform integrability
10936:Stratonovich integral
10864:Lévy–Prokhorov metric
10768:Marcinkiewicz–Zygmund
10655:Central limit theorem
10257:Gaussian random field
10086:McKean–Vlasov process
10006:Dyson Brownian motion
9867:Galton–Watson process
9682:
9577:
9363:
9275:
9116:
8904:
8792:
8601:
8467:uniformly distributed
8403:
8263:
8178:
7965:
7732:
7624:
7402:
7213:
7056:
6966:
6776:
6680:
6590:
6506:variational principle
6492:
6395:
6272:
6127:
6036:does not change with
6025:, then the density ρ(
5968:for an Itô diffusion
5949:
5849:
5732:
5535:
5466:
5402:
5310:in local coordinates
5305:
5243:was calculated to be
5218:
5202:
4939:
4871:
4818:
4742:
4702:
4623:
4572:
4500:
4336:
4247:in space, bounded on
4219:
3978:
3773:
3513:(with respect to the
3493:
3346:
3099:
2944:
2701:
2514:
2207:
2034:
1825:
1451:
1184:
940:Lower semi-continuity
930:
751:
545:diffusion coefficient
490:
329:
203:
153:
11054:Time series analysis
11009:Mathematical finance
10894:Reflection principle
10221:Regenerative process
10021:Fleming–Viot process
9836:Stochastic processes
9604:
9449:
9294:
9143:
9005:
8823:
8718:
8521:
8490:hitting distribution
8331:
8296:The harmonic measure
8283:, one can study the
8211:
8204:with expected value
8198:) at a random time τ
8044:
7747:
7740:Hence, as required,
7655:
7448:
7414:is the generator of
7310:
7235:normally distributed
7083:
6989:
6861:
6695:
6605:
6527:
6417:
6319:
6170:
6059:
5972:, i.e. a measure on
5899:
5799:
5615:
5559:of an Itô diffusion
5360:
5290:
4962:
4886:
4838:
4785:
4722:
4638:
4533:
4352:
4341:of an Itô diffusion
4321:
4303:of an Itô diffusion
4001:
3843:
3574:
3411:
3161:
3030:
2747:
2544:
2522:or, in terms of the
2278:
2101:
2074:of an Itô diffusion
1900:
1477:
1307:
1275: ≥ 0, the
1044:
969:above when the time
867:
794:given initial datum
690:
593:the existence of an
380:
371:Lipschitz continuity
234:
181:
11049:Stochastic analysis
10889:Quadratic variation
10884:Prokhorov's theorem
10819:Feynman–Kac formula
10289:Markov random field
9937:Birth–death process
9233:
9067:
8886:
8697:mean value property
8538:
8310:will first leave a
8291:Associated measures
8133:
7551:
7502:
7362:
7279:. However, for any
6282:inverse temperature
5674:
5571:is subtracted from
5265:Riemannian manifold
4911: for all
4759:and all sequences {
4512:form a sequence of
3988:Feynman–Kac formula
3912:
3822:continuous function
3788:Feynman–Kac formula
2926:
2255:compactly-supported
1279:conditioned on the
1103:
1074:
988:The Markov property
977:The Markov property
838:measurable function
736: for all
669:continuous function
600:the existence of a
106:stochastic analysis
104:– specifically, in
11019:Probability theory
10899:Skorokhod integral
10869:Malliavin calculus
10452:Korn-Kreer-Lenssen
10336:Time series models
10299:Pitman–Yor process
9777:Øksendal, Bernt K.
9733:SIAM J. Math. Anal
9677:
9591:compactly embedded
9572:
9358:
9270:
9212:
9111:
9046:
8899:
8872:
8787:
8596:
8524:
8478:compactly embedded
8398:
8258:
8173:
8119:
7960:
7727:
7636:of (Ω, Σ) by
7619:
7537:
7488:
7425:of (Ω, Σ) by
7397:
7348:
7208:
7051:
7017:
6961:
6771:
6675:
6585:
6487:
6405:partition function
6390:
6310:Gibbs distribution
6284:and Ψ :
6267:
6122:
5960:Invariant measures
5944:
5844:
5727:
5660:
5588:resolvent operator
5530:
5300:
5277:Brownian motion on
5237:
5197:
5082:
5065:
4996:
4934:
4913:
4866:
4813:
4737:
4697:
4618:
4570:
4526:in the sense that
4495:
4392:
4331:
4267:as defined above.
4214:
4209:
4032:
3973:
3898:
3768:
3763:
3605:
3488:
3341:
3336:
3192:
3094:
2939:
2912:
2890:
2879:
2795:
2778:
2696:
2615:
2509:
2394:
2377:
2308:
2202:
2137:
2029:
1820:
1818:
1446:
1179:
1169:
1086:
1060:
925:
790:denote the law of
746:
738:
660:(ω) of the noise,
543:σ is known as the
498:for some constant
485:
369:satisfy the usual
361:and σ :
324:
198:
160:
45:list of references
11085:
11084:
11039:Signal processing
10758:Doob's upcrossing
10753:Doob's martingale
10717:Engelbert–Schmidt
10660:Donsker's theorem
10594:Feller-continuous
10462:Rendleman–Bartter
10252:Dirichlet process
10169:Branching process
10138:Telegraph process
10031:Geometric process
10011:Empirical process
10001:Diffusion process
9857:Branching process
9852:Bernoulli process
9710:Dynkin, Eugene B.
9697:Diffusion process
9387:for the operator
7233:is approximately
7201:
7145:
7131:
7016:
6940:
6820:increases. Thus,
6816:must decrease as
6810:Lyapunov function
6568:
6246:
5966:invariant measure
5579:itself using the
5520:
5464:
5443:
5400:
5399:
5183:
5067:
5064:
5041:
4987:
4912:
4677:
4671:
4569:
4490:
4377:
4309:Dirichlet problem
4031:
3820:is taken to be a
3604:
3507:Hermitian adjoint
3373:evolve with time
3191:
2928:
2881:
2878:
2855:
2780:
2777:
2614:
2495:
2379:
2376:
2353:
2299:
2197:
2122:
1168:
1127:
1121:
992:An Itô diffusion
763:Feller continuity
737:
641:An Itô diffusion
637:Sample continuity
574:Feller continuity
223:(Ω, Σ,
221:probability space
189:
118:Langevin equation
98:
97:
90:
16:(Redirected from
11105:
11059:Machine learning
10946:Usual hypotheses
10829:Girsanov theorem
10814:Dynkin's formula
10579:Continuous paths
10487:Actuarial models
10427:Garman–Kohlhagen
10397:Black–Karasinski
10392:Black–Derman–Toy
10379:Financial models
10245:Fields and other
10174:Gaussian process
10123:Sigma-martingale
9927:Additive process
9829:
9822:
9815:
9806:
9794:
9766:
9748:
9717:
9686:
9684:
9683:
9678:
9667:
9634:
9633:
9581:
9579:
9578:
9573:
9562:
9529:
9528:
9516:
9512:
9511:
9507:
9506:
9505:
9504:
9478:
9477:
9472:
9403:Δ on the domain
9402:
9400:
9399:
9396:
9393:
9385:Green's function
9367:
9365:
9364:
9359:
9351:
9327:
9326:
9279:
9277:
9276:
9271:
9266:
9262:
9258:
9249:
9248:
9232:
9231:
9230:
9220:
9206:
9205:
9200:
9185:
9155:
9154:
9120:
9118:
9117:
9112:
9107:
9103:
9099:
9090:
9089:
9077:
9076:
9066:
9065:
9064:
9054:
9040:
9039:
9034:
8974:with respect to
8908:
8906:
8905:
8900:
8885:
8880:
8871:
8853:
8852:
8796:
8794:
8793:
8788:
8783:
8779:
8775:
8774:
8773:
8772:
8747:
8746:
8741:
8605:
8603:
8602:
8597:
8595:
8591:
8584:
8583:
8582:
8581:
8562:
8561:
8556:
8537:
8532:
8508:is the measure μ
8486:harmonic measure
8454:
8452:
8451:
8448:
8445:
8438:
8436:
8435:
8432:
8429:
8407:
8405:
8404:
8399:
8385:
8384:
8375:
8343:
8342:
8302:Harmonic measure
8287:of the process.
8285:harmonic measure
8267:
8265:
8264:
8259:
8254:
8253:
8238:
8237:
8225:
8224:
8219:
8182:
8180:
8179:
8174:
8169:
8165:
8161:
8152:
8151:
8132:
8127:
8113:
8112:
8107:
8077:
8076:
8058:
8057:
8052:
8013:(with generator
7997:Dynkin's formula
7991:Dynkin's formula
7969:
7967:
7966:
7961:
7956:
7955:
7943:
7942:
7936:
7935:
7926:
7925:
7919:
7918:
7909:
7908:
7902:
7901:
7896:
7887:
7883:
7882:
7881:
7872:
7871:
7865:
7864:
7858:
7857:
7848:
7847:
7841:
7840:
7831:
7830:
7824:
7823:
7818:
7807:
7806:
7801:
7789:
7788:
7779:
7774:
7773:
7761:
7760:
7755:
7736:
7734:
7733:
7728:
7723:
7722:
7710:
7709:
7703:
7702:
7693:
7692:
7686:
7685:
7676:
7675:
7669:
7668:
7663:
7644: >
7628:
7626:
7625:
7620:
7615:
7614:
7605:
7596:
7595:
7580:
7579:
7570:
7569:
7550:
7545:
7530:
7521:
7520:
7501:
7496:
7466:
7465:
7437:(the stochastic
7406:
7404:
7403:
7398:
7390:
7381:
7380:
7361:
7356:
7341:
7340:
7322:
7321:
7217:
7215:
7214:
7209:
7207:
7203:
7202:
7197:
7196:
7195:
7190:
7175:
7163:
7147:
7146:
7138:
7136:
7132:
7130:
7122:
7114:
7095:
7094:
7070:Gaussian measure
7060:
7058:
7057:
7052:
7047:
7046:
7041:
7026:
7018:
7009:
6970:
6968:
6967:
6962:
6957:
6956:
6947:
6941:
6939:
6938:
6923:
6915:
6900:
6899:
6878:
6877:
6868:
6780:
6778:
6777:
6772:
6767:
6731:
6730:
6729:
6728:
6723:
6684:
6682:
6681:
6676:
6671:
6641:
6640:
6639:
6638:
6633:
6594:
6592:
6591:
6586:
6569:
6561:
6496:
6494:
6493:
6488:
6480:
6444:
6443:
6442:
6441:
6436:
6399:
6397:
6396:
6391:
6356:
6355:
6331:
6330:
6276:
6274:
6273:
6268:
6263:
6262:
6253:
6247:
6245:
6244:
6229:
6221:
6212:
6211:
6187:
6186:
6177:
6131:
6129:
6128:
6123:
6118:
6117:
6112:
6081:
6080:
6071:
6070:
5953:
5951:
5950:
5945:
5931:
5930:
5912:
5853:
5851:
5850:
5845:
5822:
5811:
5810:
5736:
5734:
5733:
5728:
5723:
5719:
5715:
5706:
5705:
5690:
5689:
5673:
5668:
5654:
5653:
5648:
5627:
5626:
5608:, is defined by
5565:bounded operator
5539:
5537:
5536:
5531:
5526:
5522:
5521:
5519:
5518:
5517:
5501:
5499:
5498:
5485:
5480:
5465:
5451:
5444:
5442:
5441:
5440:
5424:
5421:
5416:
5401:
5386:
5382:
5377:
5376:
5375:
5340:
5338:
5337:
5334:
5331:
5317:, 1 ≤
5309:
5307:
5306:
5301:
5299:
5298:
5258:
5256:
5255:
5252:
5249:
5234:
5232:
5231:
5228:
5225:
5206:
5204:
5203:
5198:
5184:
5182:
5181:
5180:
5167:
5166:
5153:
5149:
5148:
5138:
5136:
5135:
5124:
5120:
5119:
5118:
5081:
5066:
5057:
5042:
5040:
5039:
5038:
5025:
5017:
5006:
5005:
4995:
4971:
4970:
4955:, in which case
4943:
4941:
4940:
4935:
4930:
4929:
4914:
4910:
4904:
4903:
4875:
4873:
4872:
4867:
4865:
4864:
4863:
4850:
4849:
4822:
4820:
4819:
4814:
4794:
4793:
4746:
4744:
4743:
4738:
4736:
4735:
4734:
4706:
4704:
4703:
4698:
4687:
4686:
4675:
4669:
4650:
4649:
4627:
4625:
4624:
4619:
4602:
4601:
4591:
4586:
4571:
4567:
4564:
4563:
4551:
4550:
4504:
4502:
4501:
4496:
4491:
4489:
4485:
4484:
4472:
4471:
4466:
4459:
4443:
4439:
4435:
4434:
4433:
4432:
4407:
4406:
4401:
4394:
4391:
4361:
4360:
4340:
4338:
4337:
4332:
4330:
4329:
4255:for all compact
4223:
4221:
4220:
4215:
4213:
4212:
4203:
4202:
4197:
4140:
4139:
4134:
4033:
4030:
4022:
4014:
3982:
3980:
3979:
3974:
3969:
3965:
3961:
3960:
3945:
3941:
3937:
3928:
3927:
3911:
3906:
3878:
3877:
3872:
3777:
3775:
3774:
3769:
3767:
3766:
3757:
3756:
3751:
3725:
3724:
3687:
3686:
3681:
3634:
3633:
3606:
3603:
3595:
3587:
3497:
3495:
3494:
3489:
3481:
3457:
3456:
3444:
3440:
3433:
3432:
3418:
3390:Lebesgue measure
3388:with respect to
3350:
3348:
3347:
3342:
3340:
3339:
3330:
3329:
3324:
3267:
3266:
3261:
3193:
3190:
3182:
3174:
3103:
3101:
3100:
3095:
3084:
3083:
3065:
3064:
3059:
2959:Laplace operator
2948:
2946:
2945:
2940:
2929:
2927:
2925:
2920:
2907:
2903:
2902:
2892:
2889:
2880:
2871:
2856:
2854:
2853:
2852:
2839:
2838:
2825:
2821:
2820:
2810:
2808:
2807:
2794:
2779:
2770:
2705:
2703:
2702:
2697:
2680:
2679:
2670:
2669:
2657:
2653:
2652:
2651:
2616:
2607:
2589:
2588:
2518:
2516:
2515:
2510:
2496:
2494:
2493:
2492:
2479:
2478:
2465:
2461:
2460:
2450:
2448:
2447:
2436:
2432:
2431:
2430:
2393:
2378:
2369:
2354:
2352:
2351:
2350:
2337:
2329:
2318:
2317:
2307:
2211:
2209:
2208:
2203:
2198:
2193:
2171:
2170:
2152:
2151:
2146:
2139:
2136:
2078:is the operator
2070:. Formally, the
2038:
2036:
2035:
2030:
2025:
2024:
2015:
2014:
1999:
1998:
1992:
1991:
1990:
1989:
1979:
1970:
1969:
1963:
1962:
1953:
1952:
1943:
1942:
1921:
1920:
1914:
1913:
1908:
1893: ≥ 0,
1889:. Then, for all
1829:
1827:
1826:
1821:
1819:
1812:
1808:
1804:
1803:
1783:
1782:
1781:
1780:
1770:
1758:
1754:
1750:
1749:
1748:
1739:
1738:
1732:
1728:
1724:
1723:
1703:
1702:
1701:
1700:
1690:
1679:
1678:
1673:
1661:
1657:
1653:
1652:
1651:
1642:
1641:
1635:
1631:
1630:
1629:
1620:
1619:
1610:
1609:
1583:
1582:
1577:
1566:
1565:
1560:
1547:
1543:
1542:
1541:
1532:
1531:
1522:
1521:
1495:
1494:
1489:
1455:
1453:
1452:
1447:
1436:
1435:
1417:
1416:
1406:
1405:
1395:
1377:
1376:
1370:
1369:
1360:
1359:
1350:
1349:
1328:
1327:
1321:
1320:
1315:
1252: ≥ 0.
1188:
1186:
1185:
1180:
1175:
1171:
1170:
1166:
1163:
1162:
1157:
1125:
1119:
1102:
1094:
1073:
1068:
1056:
1055:
1037: ≥ 0,
934:
932:
931:
926:
915:
914:
896:
895:
890:
848: ≥ 0,
813:with respect to
755:
753:
752:
747:
739:
735:
723:
722:
710:
709:
697:
608:Dynkin's formula
528:is known as the
494:
492:
491:
486:
484:
470:
459:
427:
419:
387:
333:
331:
330:
325:
320:
319:
310:
301:
300:
279:
270:
269:
251:
250:
241:
207:
205:
204:
199:
197:
196:
191:
190:
164:time-homogeneous
124:to describe the
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
11113:
11112:
11108:
11107:
11106:
11104:
11103:
11102:
11088:
11087:
11086:
11081:
11063:
11024:Queueing theory
10967:
10909:Skorokhod space
10772:
10763:Kunita–Watanabe
10734:
10700:Sanov's theorem
10670:Ergodic theorem
10643:
10639:Time-reversible
10557:
10520:Queueing models
10514:
10510:Sparre–Anderson
10500:Cramér–Lundberg
10481:
10467:SABR volatility
10373:
10330:
10282:Boolean network
10240:
10226:Renewal process
10157:
10106:Non-homogeneous
10096:Poisson process
9986:Contact process
9949:Brownian motion
9919:Continuous time
9913:
9907:Maximal entropy
9838:
9833:
9791:
9775:
9726:
9708:
9705:
9693:
9625:
9602:
9601:
9520:
9496:
9491:
9487:
9483:
9479:
9467:
9447:
9446:
9397:
9394:
9391:
9390:
9388:
9318:
9292:
9291:
9240:
9222:
9211:
9207:
9195:
9146:
9141:
9140:
9081:
9068:
9056:
9045:
9041:
9029:
9003:
9002:
8966:. That is, the
8928:
8922:
8841:
8821:
8820:
8764:
8759:
8752:
8748:
8736:
8716:
8715:
8686:surface measure
8573:
8568:
8567:
8563:
8551:
8519:
8518:
8513:
8472:In general, if
8464:
8463:
8449:
8446:
8443:
8442:
8440:
8433:
8430:
8427:
8426:
8424:
8376:
8334:
8329:
8328:
8323:first exit time
8304:
8298:
8293:
8245:
8229:
8214:
8209:
8208:
8203:
8143:
8118:
8114:
8102:
8068:
8047:
8042:
8041:
7999:
7993:
7985:
7978:
7947:
7927:
7910:
7891:
7873:
7849:
7832:
7813:
7812:
7808:
7796:
7780:
7765:
7750:
7745:
7744:
7714:
7694:
7677:
7658:
7653:
7652:
7635:
7606:
7587:
7571:
7561:
7512:
7457:
7446:
7445:
7424:
7372:
7332:
7313:
7308:
7307:
7269:
7257:
7246:
7231:
7185:
7164:
7158:
7154:
7123:
7115:
7109:
7108:
7086:
7081:
7080:
7075:
7036:
6987:
6986:
6948:
6927:
6891:
6869:
6859:
6858:
6841:
6808:is, in fact, a
6791:
6718:
6713:
6693:
6692:
6628:
6623:
6603:
6602:
6525:
6524:
6503:
6431:
6426:
6415:
6414:
6344:
6322:
6317:
6316:
6307:
6303:
6295:
6254:
6233:
6203:
6178:
6168:
6167:
6157:
6107:
6072:
6062:
6057:
6056:
6051:
6047:
6034:
6024:
6020:
6016:
6009:
6001:
5996:
5990:
5986:
5962:
5922:
5897:
5896:
5887:
5878:
5802:
5797:
5796:
5760:
5750:
5697:
5675:
5659:
5655:
5643:
5618:
5613:
5612:
5595:
5553:
5509:
5505:
5487:
5449:
5445:
5432:
5428:
5363:
5358:
5357:
5348:
5344:
5335:
5332:
5329:
5328:
5326:
5315:
5288:
5287:
5253:
5250:
5247:
5246:
5244:
5229:
5226:
5223:
5222:
5220:
5213:
5172:
5158:
5154:
5140:
5139:
5110:
5088:
5084:
5083:
5030:
5026:
5018:
4997:
4960:
4959:
4921:
4884:
4883:
4854:
4841:
4836:
4835:
4829:
4783:
4782:
4764:
4725:
4720:
4719:
4678:
4641:
4636:
4635:
4593:
4568: and
4555:
4536:
4531:
4530:
4520:
4508:where the sets
4476:
4461:
4460:
4424:
4419:
4412:
4408:
4396:
4395:
4350:
4349:
4319:
4318:
4297:
4292:
4208:
4207:
4192:
4184:
4145:
4144:
4129:
4109:
4023:
4015:
4005:
3999:
3998:
3952:
3919:
3894:
3890:
3883:
3879:
3867:
3841:
3840:
3790:
3784:
3762:
3761:
3746:
3738:
3716:
3692:
3691:
3676:
3656:
3625:
3596:
3588:
3578:
3572:
3571:
3558:
3554:
3532:
3528:
3448:
3424:
3423:
3419:
3409:
3408:
3386:
3371:
3335:
3334:
3319:
3311:
3272:
3271:
3256:
3236:
3183:
3175:
3165:
3159:
3158:
3136:
3075:
3054:
3028:
3027:
2977:
2969:Main articles:
2967:
2908:
2894:
2893:
2844:
2830:
2826:
2812:
2811:
2796:
2745:
2744:
2738:
2731:
2712:
2671:
2661:
2643:
2621:
2617:
2580:
2542:
2541:
2484:
2470:
2466:
2452:
2451:
2422:
2400:
2396:
2395:
2342:
2338:
2330:
2309:
2276:
2275:
2269:
2239:
2228:
2162:
2141:
2140:
2099:
2098:
2056:
2051:
2045:
2006:
1981:
1974:
1954:
1928:
1903:
1898:
1897:
1884:
1869:As before, let
1836:
1817:
1816:
1795:
1788:
1784:
1772:
1765:
1756:
1755:
1740:
1715:
1708:
1704:
1692:
1685:
1684:
1680:
1668:
1659:
1658:
1643:
1621:
1595:
1588:
1584:
1572:
1571:
1567:
1555:
1548:
1533:
1507:
1500:
1496:
1484:
1475:
1474:
1469:
1427:
1397:
1390:
1361:
1335:
1310:
1305:
1304:
1298:Markov property
1294:
1288:
1247:
1240:
1230:
1223:
1216:
1209:
1203:
1152:
1085:
1081:
1047:
1042:
1041:
1021:
1013:
990:
985:
983:Markov property
979:
906:
885:
865:
864:
800:
765:
714:
701:
688:
687:
665:
658:
639:
634:
628:
581:Markov property
516:strong solution
378:
377:
347:Brownian motion
311:
292:
261:
242:
232:
231:
184:
179:
178:
176:Euclidean space
148:
126:Brownian motion
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
11111:
11109:
11101:
11100:
11090:
11089:
11083:
11082:
11080:
11079:
11074:
11072:List of topics
11068:
11065:
11064:
11062:
11061:
11056:
11051:
11046:
11041:
11036:
11031:
11029:Renewal theory
11026:
11021:
11016:
11011:
11006:
11001:
10996:
10994:Ergodic theory
10991:
10986:
10984:Control theory
10981:
10975:
10973:
10969:
10968:
10966:
10965:
10964:
10963:
10958:
10948:
10943:
10938:
10933:
10928:
10927:
10926:
10916:
10914:Snell envelope
10911:
10906:
10901:
10896:
10891:
10886:
10881:
10876:
10871:
10866:
10861:
10856:
10851:
10846:
10841:
10836:
10831:
10826:
10821:
10816:
10811:
10806:
10801:
10796:
10791:
10786:
10780:
10778:
10774:
10773:
10771:
10770:
10765:
10760:
10755:
10750:
10744:
10742:
10736:
10735:
10733:
10732:
10713:Borel–Cantelli
10702:
10697:
10692:
10687:
10682:
10677:
10672:
10667:
10662:
10657:
10651:
10649:
10648:Limit theorems
10645:
10644:
10642:
10641:
10636:
10631:
10626:
10621:
10616:
10611:
10606:
10601:
10596:
10591:
10586:
10581:
10576:
10571:
10565:
10563:
10559:
10558:
10556:
10555:
10550:
10545:
10540:
10535:
10530:
10524:
10522:
10516:
10515:
10513:
10512:
10507:
10502:
10497:
10491:
10489:
10483:
10482:
10480:
10479:
10474:
10469:
10464:
10459:
10454:
10449:
10444:
10439:
10434:
10429:
10424:
10419:
10414:
10409:
10404:
10399:
10394:
10389:
10383:
10381:
10375:
10374:
10372:
10371:
10366:
10361:
10356:
10351:
10346:
10340:
10338:
10332:
10331:
10329:
10328:
10323:
10318:
10317:
10316:
10311:
10301:
10296:
10291:
10286:
10285:
10284:
10279:
10269:
10267:Hopfield model
10264:
10259:
10254:
10248:
10246:
10242:
10241:
10239:
10238:
10233:
10228:
10223:
10218:
10213:
10212:
10211:
10206:
10201:
10196:
10186:
10184:Markov process
10181:
10176:
10171:
10165:
10163:
10159:
10158:
10156:
10155:
10153:Wiener sausage
10150:
10148:Wiener process
10145:
10140:
10135:
10130:
10128:Stable process
10125:
10120:
10118:Semimartingale
10115:
10110:
10109:
10108:
10103:
10093:
10088:
10083:
10078:
10073:
10068:
10063:
10061:Jump diffusion
10058:
10053:
10048:
10043:
10038:
10036:Hawkes process
10033:
10028:
10023:
10018:
10016:Feller process
10013:
10008:
10003:
9998:
9993:
9988:
9983:
9981:Cauchy process
9978:
9977:
9976:
9971:
9966:
9961:
9956:
9946:
9945:
9944:
9934:
9932:Bessel process
9929:
9923:
9921:
9915:
9914:
9912:
9911:
9910:
9909:
9904:
9899:
9894:
9884:
9879:
9874:
9869:
9864:
9859:
9854:
9848:
9846:
9840:
9839:
9834:
9832:
9831:
9824:
9817:
9809:
9803:
9802:
9789:
9773:
9724:
9704:
9701:
9700:
9699:
9692:
9689:
9688:
9687:
9676:
9673:
9670:
9666:
9662:
9659:
9656:
9653:
9649:
9646:
9643:
9640:
9637:
9632:
9628:
9624:
9621:
9618:
9615:
9612:
9609:
9583:
9582:
9571:
9568:
9565:
9561:
9557:
9554:
9551:
9548:
9544:
9541:
9538:
9535:
9532:
9527:
9523:
9519:
9515:
9510:
9503:
9499:
9494:
9490:
9486:
9482:
9476:
9471:
9466:
9463:
9460:
9457:
9454:
9426:holds for all
9369:
9368:
9357:
9354:
9350:
9345:
9342:
9339:
9336:
9333:
9330:
9325:
9321:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9281:
9280:
9269:
9265:
9261:
9257:
9252:
9247:
9243:
9239:
9236:
9229:
9225:
9219:
9215:
9210:
9204:
9199:
9194:
9191:
9188:
9184:
9180:
9177:
9174:
9171:
9167:
9164:
9161:
9158:
9153:
9149:
9122:
9121:
9110:
9106:
9102:
9098:
9093:
9088:
9084:
9080:
9075:
9071:
9063:
9059:
9053:
9049:
9044:
9038:
9033:
9028:
9025:
9022:
9019:
9016:
9013:
9010:
8924:Main article:
8921:
8918:
8910:
8909:
8898:
8895:
8892:
8889:
8884:
8879:
8875:
8870:
8865:
8862:
8859:
8856:
8851:
8848:
8844:
8840:
8837:
8834:
8831:
8828:
8804: ⊂⊂
8798:
8797:
8786:
8782:
8778:
8771:
8767:
8762:
8758:
8755:
8751:
8745:
8740:
8735:
8732:
8729:
8726:
8723:
8621: ⊆ ∂
8607:
8606:
8594:
8590:
8587:
8580:
8576:
8571:
8566:
8560:
8555:
8550:
8547:
8544:
8541:
8536:
8531:
8527:
8509:
8461:
8459:
8409:
8408:
8397:
8394:
8391:
8388:
8383:
8379:
8374:
8370:
8367:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8341:
8337:
8312:measurable set
8300:Main article:
8297:
8294:
8292:
8289:
8269:
8268:
8257:
8252:
8248:
8244:
8241:
8236:
8232:
8228:
8223:
8218:
8199:
8184:
8183:
8172:
8168:
8164:
8160:
8155:
8150:
8146:
8142:
8139:
8136:
8131:
8126:
8122:
8117:
8111:
8106:
8101:
8098:
8095:
8092:
8089:
8086:
8083:
8080:
8075:
8071:
8067:
8064:
8061:
8056:
8051:
8007:expected value
7995:Main article:
7992:
7989:
7983:
7976:
7971:
7970:
7959:
7954:
7950:
7946:
7941:
7934:
7930:
7924:
7917:
7913:
7907:
7900:
7895:
7890:
7886:
7880:
7876:
7870:
7863:
7856:
7852:
7846:
7839:
7835:
7829:
7822:
7817:
7811:
7805:
7800:
7795:
7792:
7787:
7783:
7778:
7772:
7768:
7764:
7759:
7754:
7738:
7737:
7726:
7721:
7717:
7713:
7708:
7701:
7697:
7691:
7684:
7680:
7674:
7667:
7662:
7633:
7630:
7629:
7618:
7613:
7609:
7604:
7599:
7594:
7590:
7586:
7583:
7578:
7574:
7568:
7564:
7560:
7557:
7554:
7549:
7544:
7540:
7536:
7533:
7529:
7524:
7519:
7515:
7511:
7508:
7505:
7500:
7495:
7491:
7487:
7484:
7481:
7478:
7475:
7472:
7469:
7464:
7460:
7456:
7453:
7422:
7408:
7407:
7396:
7393:
7389:
7384:
7379:
7375:
7371:
7368:
7365:
7360:
7355:
7351:
7347:
7344:
7339:
7335:
7331:
7328:
7325:
7320:
7316:
7268:
7265:
7255:
7244:
7229:
7220:
7219:
7206:
7200:
7194:
7189:
7184:
7181:
7178:
7174:
7170:
7167:
7161:
7157:
7153:
7150:
7144:
7141:
7135:
7129:
7126:
7121:
7118:
7112:
7107:
7104:
7101:
7098:
7093:
7089:
7073:
7072:with density ρ
7062:
7061:
7050:
7045:
7040:
7035:
7032:
7029:
7025:
7021:
7015:
7012:
7006:
7003:
7000:
6997:
6994:
6972:
6971:
6960:
6955:
6951:
6946:
6937:
6934:
6930:
6926:
6921:
6918:
6914:
6909:
6906:
6903:
6898:
6894:
6890:
6887:
6884:
6881:
6876:
6872:
6867:
6840:
6837:
6789:
6782:
6781:
6770:
6766:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6727:
6722:
6716:
6712:
6709:
6706:
6703:
6700:
6686:
6685:
6674:
6670:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6637:
6632:
6626:
6622:
6619:
6616:
6613:
6610:
6596:
6595:
6584:
6581:
6578:
6575:
6572:
6567:
6564:
6559:
6556:
6553:
6550:
6547:
6544:
6541:
6538:
6535:
6532:
6501:
6498:
6497:
6486:
6483:
6479:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6453:
6450:
6447:
6440:
6435:
6429:
6425:
6422:
6401:
6400:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6354:
6351:
6347:
6343:
6340:
6337:
6334:
6329:
6325:
6305:
6304:with density ρ
6301:
6293:
6278:
6277:
6266:
6261:
6257:
6252:
6243:
6240:
6236:
6232:
6227:
6224:
6220:
6215:
6210:
6206:
6202:
6199:
6196:
6193:
6190:
6185:
6181:
6176:
6156:
6153:
6133:
6132:
6121:
6116:
6111:
6106:
6103:
6099:
6096:
6093:
6090:
6087:
6084:
6079:
6075:
6069:
6065:
6049:
6045:
6032:
6022:
6021:with density ρ
6018:
6014:
6007:
5999:
5994:
5988:
5984:
5961:
5958:
5957:
5956:
5955:
5954:
5943:
5940:
5937:
5934:
5929:
5925:
5921:
5918:
5915:
5911:
5907:
5904:
5891:
5890:
5885:
5876:
5857:
5856:
5855:
5854:
5843:
5840:
5837:
5834:
5831:
5828:
5825:
5821:
5817:
5814:
5809:
5805:
5791:
5790:
5758:
5748:
5738:
5737:
5726:
5722:
5718:
5714:
5709:
5704:
5700:
5696:
5693:
5688:
5685:
5682:
5678:
5672:
5667:
5663:
5658:
5652:
5647:
5642:
5639:
5636:
5633:
5630:
5625:
5621:
5593:
5552:
5549:
5541:
5540:
5529:
5525:
5516:
5512:
5508:
5504:
5497:
5494:
5490:
5484:
5479:
5476:
5473:
5469:
5463:
5460:
5457:
5454:
5448:
5439:
5435:
5431:
5427:
5420:
5415:
5412:
5409:
5405:
5398:
5395:
5392:
5389:
5385:
5380:
5374:
5371:
5366:
5346:
5342:
5325:, is given by
5313:
5297:
5212:
5209:
5208:
5207:
5196:
5193:
5190:
5187:
5179:
5175:
5171:
5165:
5161:
5157:
5152:
5147:
5143:
5134:
5131:
5128:
5123:
5117:
5113:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5087:
5080:
5077:
5074:
5070:
5063:
5060:
5054:
5051:
5048:
5045:
5037:
5033:
5029:
5024:
5021:
5015:
5012:
5009:
5004:
5000:
4994:
4990:
4986:
4983:
4980:
4977:
4974:
4969:
4945:
4944:
4933:
4928:
4924:
4920:
4917:
4907:
4902:
4897:
4894:
4891:
4877:
4876:
4862:
4857:
4853:
4848:
4844:
4828:
4825:
4824:
4823:
4812:
4809:
4806:
4803:
4800:
4797:
4792:
4762:
4733:
4728:
4708:
4707:
4696:
4693:
4690:
4685:
4681:
4674:
4668:
4665:
4662:
4659:
4656:
4653:
4648:
4644:
4629:
4628:
4617:
4614:
4611:
4608:
4605:
4600:
4596:
4590:
4585:
4582:
4579:
4575:
4562:
4558:
4554:
4549:
4546:
4543:
4539:
4518:
4506:
4505:
4494:
4488:
4483:
4479:
4475:
4470:
4465:
4458:
4455:
4452:
4449:
4446:
4442:
4438:
4431:
4427:
4422:
4418:
4415:
4411:
4405:
4400:
4390:
4387:
4384:
4380:
4376:
4373:
4370:
4367:
4364:
4359:
4345:is defined by
4328:
4296:
4293:
4291:
4288:
4225:
4224:
4211:
4206:
4201:
4196:
4191:
4188:
4185:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4146:
4143:
4138:
4133:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4029:
4026:
4021:
4018:
4011:
4010:
4008:
3984:
3983:
3972:
3968:
3964:
3959:
3955:
3951:
3948:
3944:
3940:
3936:
3931:
3926:
3922:
3918:
3915:
3910:
3905:
3901:
3897:
3893:
3889:
3886:
3882:
3876:
3871:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3786:Main article:
3783:
3780:
3779:
3778:
3765:
3760:
3755:
3750:
3745:
3742:
3739:
3737:
3734:
3731:
3728:
3723:
3719:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3693:
3690:
3685:
3680:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3632:
3628:
3624:
3621:
3618:
3615:
3612:
3609:
3602:
3599:
3594:
3591:
3584:
3583:
3581:
3556:
3552:
3530:
3526:
3499:
3498:
3487:
3484:
3480:
3475:
3472:
3469:
3466:
3463:
3460:
3455:
3451:
3447:
3443:
3439:
3436:
3431:
3427:
3422:
3417:
3384:
3369:
3352:
3351:
3338:
3333:
3328:
3323:
3318:
3315:
3312:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3273:
3270:
3265:
3260:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3189:
3186:
3181:
3178:
3171:
3170:
3168:
3134:
3105:
3104:
3093:
3090:
3087:
3082:
3078:
3074:
3071:
3068:
3063:
3058:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3023:is defined by
2987:in which time
2966:
2963:
2951:
2950:
2938:
2935:
2932:
2924:
2919:
2915:
2911:
2906:
2901:
2897:
2888:
2884:
2877:
2874:
2868:
2865:
2862:
2859:
2851:
2847:
2843:
2837:
2833:
2829:
2824:
2819:
2815:
2806:
2803:
2799:
2793:
2790:
2787:
2783:
2776:
2773:
2767:
2764:
2761:
2758:
2755:
2752:
2740:, is given by
2736:
2733: = d
2729:
2714:The generator
2711:
2708:
2707:
2706:
2695:
2692:
2689:
2686:
2683:
2678:
2674:
2668:
2664:
2660:
2656:
2650:
2646:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2620:
2613:
2610:
2604:
2601:
2598:
2595:
2592:
2587:
2583:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2535:inner products
2520:
2519:
2508:
2505:
2502:
2499:
2491:
2487:
2483:
2477:
2473:
2469:
2464:
2459:
2455:
2446:
2443:
2440:
2435:
2429:
2425:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2399:
2392:
2389:
2386:
2382:
2375:
2372:
2366:
2363:
2360:
2357:
2349:
2345:
2341:
2336:
2333:
2327:
2324:
2321:
2316:
2312:
2306:
2302:
2298:
2295:
2292:
2289:
2286:
2283:
2267:
2237:
2226:
2213:
2212:
2201:
2196:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2161:
2158:
2155:
2150:
2145:
2135:
2132:
2129:
2125:
2121:
2118:
2115:
2112:
2109:
2106:
2055:
2052:
2047:Main article:
2044:
2041:
2040:
2039:
2028:
2023:
2018:
2013:
2009:
2005:
2002:
1997:
1988:
1984:
1978:
1973:
1968:
1961:
1957:
1951:
1946:
1941:
1938:
1935:
1931:
1927:
1924:
1919:
1912:
1907:
1882:
1835:
1832:
1831:
1830:
1815:
1811:
1807:
1802:
1798:
1794:
1791:
1787:
1779:
1775:
1769:
1764:
1761:
1759:
1757:
1753:
1747:
1743:
1737:
1731:
1727:
1722:
1718:
1714:
1711:
1707:
1699:
1695:
1689:
1683:
1677:
1672:
1667:
1664:
1662:
1660:
1656:
1650:
1646:
1640:
1634:
1628:
1624:
1618:
1613:
1608:
1605:
1602:
1598:
1594:
1591:
1587:
1581:
1576:
1570:
1564:
1559:
1554:
1551:
1549:
1546:
1540:
1536:
1530:
1525:
1520:
1517:
1514:
1510:
1506:
1503:
1499:
1493:
1488:
1483:
1482:
1467:
1457:
1456:
1445:
1442:
1439:
1434:
1430:
1426:
1423:
1420:
1415:
1412:
1409:
1404:
1400:
1394:
1389:
1386:
1383:
1380:
1375:
1368:
1364:
1358:
1353:
1348:
1345:
1342:
1338:
1334:
1331:
1326:
1319:
1314:
1292:
1284:
1243:
1242: ⊆ Σ
1238:
1228:
1221:
1212:
1207:
1201:
1190:
1189:
1178:
1174:
1161:
1156:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1124:
1118:
1115:
1112:
1109:
1106:
1101:
1098:
1093:
1089:
1084:
1080:
1077:
1072:
1067:
1063:
1059:
1054:
1050:
1019:
1011:
989:
986:
981:Main article:
978:
975:
963:
962:
961:is continuous.
951:
936:
935:
924:
921:
918:
913:
909:
905:
902:
899:
894:
889:
884:
881:
878:
875:
872:
798:
764:
761:
757:
756:
745:
742:
732:
729:
726:
721:
717:
713:
708:
704:
700:
696:
663:
656:
638:
635:
630:Main article:
627:
624:
612:
611:
605:
598:
591:
584:
577:
496:
495:
483:
479:
476:
473:
469:
465:
462:
458:
454:
451:
448:
445:
442:
439:
436:
433:
430:
426:
422:
418:
414:
411:
408:
405:
402:
399:
396:
393:
390:
386:
335:
334:
323:
318:
314:
309:
304:
299:
295:
291:
288:
285:
282:
278:
273:
268:
264:
260:
257:
254:
249:
245:
240:
195:
156:Wiener process
147:
144:
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
11110:
11099:
11096:
11095:
11093:
11078:
11075:
11073:
11070:
11069:
11066:
11060:
11057:
11055:
11052:
11050:
11047:
11045:
11042:
11040:
11037:
11035:
11032:
11030:
11027:
11025:
11022:
11020:
11017:
11015:
11012:
11010:
11007:
11005:
11002:
11000:
10997:
10995:
10992:
10990:
10987:
10985:
10982:
10980:
10977:
10976:
10974:
10970:
10962:
10959:
10957:
10954:
10953:
10952:
10949:
10947:
10944:
10942:
10939:
10937:
10934:
10932:
10931:Stopping time
10929:
10925:
10922:
10921:
10920:
10917:
10915:
10912:
10910:
10907:
10905:
10902:
10900:
10897:
10895:
10892:
10890:
10887:
10885:
10882:
10880:
10877:
10875:
10872:
10870:
10867:
10865:
10862:
10860:
10857:
10855:
10852:
10850:
10847:
10845:
10842:
10840:
10837:
10835:
10832:
10830:
10827:
10825:
10822:
10820:
10817:
10815:
10812:
10810:
10807:
10805:
10802:
10800:
10797:
10795:
10792:
10790:
10787:
10785:
10782:
10781:
10779:
10775:
10769:
10766:
10764:
10761:
10759:
10756:
10754:
10751:
10749:
10746:
10745:
10743:
10741:
10737:
10730:
10726:
10722:
10721:Hewitt–Savage
10718:
10714:
10710:
10706:
10705:Zero–one laws
10703:
10701:
10698:
10696:
10693:
10691:
10688:
10686:
10683:
10681:
10678:
10676:
10673:
10671:
10668:
10666:
10663:
10661:
10658:
10656:
10653:
10652:
10650:
10646:
10640:
10637:
10635:
10632:
10630:
10627:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10605:
10602:
10600:
10597:
10595:
10592:
10590:
10587:
10585:
10582:
10580:
10577:
10575:
10572:
10570:
10567:
10566:
10564:
10560:
10554:
10551:
10549:
10546:
10544:
10541:
10539:
10536:
10534:
10531:
10529:
10526:
10525:
10523:
10521:
10517:
10511:
10508:
10506:
10503:
10501:
10498:
10496:
10493:
10492:
10490:
10488:
10484:
10478:
10475:
10473:
10470:
10468:
10465:
10463:
10460:
10458:
10455:
10453:
10450:
10448:
10445:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10425:
10423:
10420:
10418:
10415:
10413:
10410:
10408:
10405:
10403:
10402:Black–Scholes
10400:
10398:
10395:
10393:
10390:
10388:
10385:
10384:
10382:
10380:
10376:
10370:
10367:
10365:
10362:
10360:
10357:
10355:
10352:
10350:
10347:
10345:
10342:
10341:
10339:
10337:
10333:
10327:
10324:
10322:
10319:
10315:
10312:
10310:
10307:
10306:
10305:
10304:Point process
10302:
10300:
10297:
10295:
10292:
10290:
10287:
10283:
10280:
10278:
10275:
10274:
10273:
10270:
10268:
10265:
10263:
10262:Gibbs measure
10260:
10258:
10255:
10253:
10250:
10249:
10247:
10243:
10237:
10234:
10232:
10229:
10227:
10224:
10222:
10219:
10217:
10214:
10210:
10207:
10205:
10202:
10200:
10197:
10195:
10192:
10191:
10190:
10187:
10185:
10182:
10180:
10177:
10175:
10172:
10170:
10167:
10166:
10164:
10160:
10154:
10151:
10149:
10146:
10144:
10141:
10139:
10136:
10134:
10131:
10129:
10126:
10124:
10121:
10119:
10116:
10114:
10111:
10107:
10104:
10102:
10099:
10098:
10097:
10094:
10092:
10089:
10087:
10084:
10082:
10079:
10077:
10074:
10072:
10069:
10067:
10064:
10062:
10059:
10057:
10054:
10052:
10051:Itô diffusion
10049:
10047:
10044:
10042:
10039:
10037:
10034:
10032:
10029:
10027:
10026:Gamma process
10024:
10022:
10019:
10017:
10014:
10012:
10009:
10007:
10004:
10002:
9999:
9997:
9994:
9992:
9989:
9987:
9984:
9982:
9979:
9975:
9972:
9970:
9967:
9965:
9962:
9960:
9957:
9955:
9952:
9951:
9950:
9947:
9943:
9940:
9939:
9938:
9935:
9933:
9930:
9928:
9925:
9924:
9922:
9920:
9916:
9908:
9905:
9903:
9900:
9898:
9897:Self-avoiding
9895:
9893:
9890:
9889:
9888:
9885:
9883:
9882:Moran process
9880:
9878:
9875:
9873:
9870:
9868:
9865:
9863:
9860:
9858:
9855:
9853:
9850:
9849:
9847:
9845:
9844:Discrete time
9841:
9837:
9830:
9825:
9823:
9818:
9816:
9811:
9810:
9807:
9800:
9797:
9792:
9790:3-540-04758-1
9786:
9782:
9778:
9774:
9772:
9769:
9764:
9760:
9756:
9752:
9747:
9746:10.1.1.6.8815
9742:
9738:
9734:
9730:
9725:
9723:
9720:
9715:
9711:
9707:
9706:
9702:
9698:
9695:
9694:
9690:
9674:
9668:
9660:
9657:
9651:
9644:
9638:
9635:
9630:
9626:
9622:
9619:
9613:
9607:
9600:
9599:
9598:
9596:
9592:
9588:
9569:
9563:
9555:
9552:
9546:
9539:
9533:
9530:
9525:
9521:
9517:
9513:
9508:
9501:
9497:
9492:
9488:
9484:
9480:
9474:
9464:
9458:
9452:
9445:
9444:
9443:
9441:
9437:
9433:
9430: ∈
9429:
9425:
9424:Green formula
9421:
9418: ∈
9417:
9413:
9410:Suppose that
9408:
9406:
9386:
9382:
9378:
9374:
9355:
9352:
9340:
9337:
9334:
9328:
9323:
9319:
9315:
9309:
9306:
9303:
9297:
9290:
9289:
9288:
9286:
9267:
9263:
9259:
9245:
9241:
9234:
9227:
9223:
9217:
9213:
9208:
9202:
9192:
9186:
9178:
9175:
9169:
9162:
9156:
9151:
9147:
9139:
9138:
9137:
9135:
9132: →
9131:
9128: :
9127:
9108:
9104:
9100:
9086:
9082:
9073:
9069:
9061:
9057:
9051:
9047:
9042:
9036:
9026:
9020:
9017:
9014:
9008:
9001:
9000:
8999:
8997:
8994: ⊆
8993:
8989:
8985:
8981:
8977:
8973:
8969:
8968:Green measure
8965:
8961:
8957:
8953:
8949:
8945:
8941:
8938: ⊆
8937:
8933:
8927:
8926:Green measure
8919:
8917:
8915:
8896:
8890:
8882:
8877:
8873:
8860:
8854:
8849:
8842:
8838:
8832:
8826:
8819:
8818:
8817:
8815:
8812: ∈
8811:
8807:
8803:
8784:
8780:
8769:
8765:
8760:
8753:
8749:
8743:
8733:
8727:
8721:
8714:
8713:
8712:
8710:
8707: →
8706:
8703: :
8702:
8698:
8693:
8691:
8687:
8683:
8679:
8675:
8671:
8667:
8663:
8659:
8655:
8651:
8648: ⊂
8647:
8643:
8640: ∈
8639:
8635:
8631:
8626:
8624:
8620:
8616:
8613: ∈
8612:
8592:
8588:
8585:
8578:
8574:
8569:
8564:
8558:
8548:
8542:
8534:
8529:
8525:
8517:
8516:
8515:
8512:
8507:
8503:
8499:
8495:
8491:
8487:
8483:
8479:
8475:
8470:
8468:
8458:
8422:
8418:
8414:
8395:
8389:
8386:
8381:
8377:
8368:
8365:
8362:
8353:
8347:
8339:
8335:
8327:
8326:
8325:
8324:
8320:
8317: ⊆
8316:
8313:
8309:
8303:
8295:
8290:
8288:
8286:
8282:
8278:
8274:
8255:
8250:
8246:
8242:
8234:
8230:
8221:
8207:
8206:
8205:
8202:
8197:
8193:
8189:
8170:
8166:
8162:
8148:
8144:
8137:
8134:
8129:
8124:
8120:
8115:
8109:
8099:
8093:
8087:
8084:
8073:
8069:
8062:
8054:
8040:
8039:
8038:
8036:
8032:
8029: →
8028:
8025: :
8024:
8020:
8016:
8012:
8008:
8004:
8003:Eugene Dynkin
7998:
7988:
7987:-measurable.
7986:
7979:
7957:
7952:
7948:
7944:
7932:
7928:
7915:
7911:
7898:
7888:
7884:
7878:
7874:
7854:
7837:
7833:
7820:
7809:
7803:
7793:
7785:
7781:
7770:
7766:
7757:
7743:
7742:
7741:
7724:
7719:
7715:
7711:
7699:
7682:
7678:
7665:
7651:
7650:
7649:
7647:
7643:
7639:
7616:
7611:
7607:
7592:
7588:
7581:
7566:
7562:
7555:
7547:
7542:
7538:
7534:
7531:
7517:
7513:
7506:
7503:
7498:
7493:
7489:
7485:
7479:
7473:
7470:
7462:
7458:
7451:
7444:
7443:
7442:
7440:
7436:
7432:
7428:
7421:
7417:
7413:
7394:
7391:
7377:
7373:
7366:
7363:
7358:
7353:
7349:
7345:
7337:
7333:
7326:
7323:
7318:
7314:
7306:
7305:
7304:
7302:
7298:
7294:
7290:
7286:
7283: ∈
7282:
7278:
7274:
7266:
7264:
7262:
7258:
7251:
7247:
7240:
7236:
7232:
7225:
7204:
7198:
7192:
7182:
7179:
7176:
7168:
7165:
7159:
7155:
7151:
7148:
7142:
7139:
7133:
7127:
7124:
7119:
7116:
7110:
7105:
7099:
7087:
7079:
7078:
7077:
7071:
7067:
7048:
7043:
7033:
7030:
7027:
7019:
7013:
7010:
7004:
6998:
6985:
6984:
6983:
6981:
6978: ∈
6977:
6958:
6953:
6949:
6935:
6932:
6928:
6924:
6919:
6916:
6904:
6901:
6896:
6892:
6885:
6882:
6879:
6874:
6870:
6857:
6856:
6855:
6853:
6849:
6846:
6843:Consider the
6838:
6836:
6834:
6830:
6828:
6823:
6819:
6815:
6811:
6807:
6803:
6799:
6795:
6787:
6768:
6756:
6750:
6747:
6744:
6738:
6732:
6725:
6714:
6710:
6704:
6698:
6691:
6690:
6689:
6672:
6660:
6654:
6648:
6635:
6624:
6620:
6614:
6608:
6601:
6600:
6599:
6582:
6576:
6570:
6565:
6562:
6557:
6551:
6545:
6542:
6536:
6530:
6523:
6522:
6521:
6519:
6515:
6511:
6507:
6484:
6481:
6466:
6457:
6454:
6448:
6445:
6438:
6427:
6423:
6420:
6413:
6412:
6411:
6409:
6406:
6387:
6378:
6369:
6366:
6360:
6357:
6352:
6349:
6345:
6341:
6335:
6323:
6315:
6314:
6313:
6311:
6299:
6291:
6288: →
6287:
6283:
6264:
6259:
6255:
6241:
6238:
6234:
6230:
6225:
6222:
6208:
6204:
6191:
6188:
6183:
6179:
6166:
6165:
6164:
6162:
6154:
6152:
6150:
6146:
6142:
6138:
6119:
6114:
6104:
6101:
6097:
6094:
6091:
6085:
6073:
6067:
6063:
6055:
6054:
6053:
6043:
6039:
6035:
6029:, ·) of
6028:
6013:
6005:
5997:
5983:
5979:
5975:
5971:
5967:
5959:
5941:
5938:
5935:
5932:
5927:
5923:
5916:
5913:
5905:
5895:
5894:
5893:
5892:
5888:
5881:
5875:
5871:
5868: →
5867:
5864: :
5863:
5859:
5858:
5841:
5838:
5835:
5832:
5826:
5823:
5815:
5807:
5803:
5795:
5794:
5793:
5792:
5788:
5784:
5781: →
5780:
5777: :
5776:
5772:
5771:
5770:
5768:
5765: −
5764:
5757:
5753:
5747:
5743:
5724:
5720:
5716:
5702:
5698:
5691:
5686:
5683:
5680:
5676:
5665:
5661:
5656:
5650:
5640:
5634:
5628:
5623:
5619:
5611:
5610:
5609:
5607:
5604: →
5603:
5600: :
5599:
5592:
5589:
5584:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5550:
5548:
5546:
5527:
5523:
5514:
5510:
5495:
5492:
5488:
5482:
5477:
5474:
5471:
5467:
5458:
5446:
5437:
5433:
5418:
5413:
5410:
5407:
5403:
5393:
5383:
5378:
5356:
5355:
5354:
5352:
5324:
5321: ≤
5320:
5316:
5285:
5281:
5278:
5274:
5270:
5266:
5263:-dimensional
5262:
5242:
5217:
5210:
5194:
5188:
5177:
5173:
5163:
5159:
5150:
5145:
5132:
5129:
5126:
5121:
5107:
5101:
5095:
5089:
5085:
5078:
5075:
5072:
5068:
5061:
5058:
5052:
5046:
5035:
5031:
5022:
5010:
5002:
4998:
4992:
4988:
4984:
4978:
4972:
4958:
4957:
4956:
4954:
4950:
4931:
4926:
4922:
4918:
4915:
4905:
4895:
4892:
4889:
4882:
4881:
4880:
4855:
4851:
4846:
4842:
4834:
4833:
4832:
4826:
4810:
4807:
4801:
4795:
4781:
4780:
4779:
4777:
4773:
4769:
4765:
4758:
4755: ∈
4754:
4750:
4726:
4717:
4713:
4691:
4688:
4683:
4679:
4672:
4666:
4663:
4660:
4651:
4646:
4642:
4634:
4633:
4632:
4615:
4609:
4603:
4598:
4594:
4583:
4580:
4577:
4573:
4560:
4556:
4552:
4547:
4544:
4541:
4537:
4529:
4528:
4527:
4525:
4521:
4515:
4511:
4492:
4481:
4477:
4468:
4453:
4447:
4444:
4440:
4429:
4425:
4420:
4413:
4409:
4403:
4388:
4382:
4374:
4368:
4362:
4348:
4347:
4346:
4344:
4317:
4312:
4310:
4306:
4302:
4294:
4289:
4287:
4285:
4282: ∈
4281:
4277:
4273:
4268:
4266:
4262:
4258:
4254:
4251: ×
4250:
4246:
4242:
4238:
4235: →
4234:
4230:
4227:Moreover, if
4204:
4199:
4189:
4186:
4181:
4175:
4169:
4166:
4160:
4157:
4154:
4148:
4141:
4136:
4126:
4123:
4120:
4117:
4114:
4111:
4106:
4100:
4097:
4094:
4088:
4082:
4076:
4073:
4067:
4064:
4061:
4055:
4052:
4049:
4043:
4040:
4037:
4027:
4019:
4006:
3997:
3996:
3995:
3993:
3989:
3970:
3966:
3957:
3953:
3946:
3942:
3938:
3924:
3920:
3913:
3908:
3903:
3899:
3895:
3891:
3887:
3884:
3880:
3874:
3864:
3858:
3855:
3852:
3846:
3839:
3838:
3837:
3835:
3832: →
3831:
3827:
3823:
3819:
3816: →
3815:
3812: :
3811:
3807:
3803:
3799:
3795:
3789:
3781:
3758:
3753:
3743:
3740:
3735:
3729:
3721:
3717:
3713:
3707:
3704:
3701:
3695:
3688:
3683:
3673:
3670:
3667:
3664:
3661:
3658:
3653:
3647:
3644:
3641:
3635:
3630:
3626:
3622:
3616:
3613:
3610:
3600:
3592:
3579:
3570:
3569:
3568:
3566:
3562:
3555:
3548:
3544:
3540:
3536:
3525:
3521:
3520:inner product
3518:
3517:
3512:
3508:
3504:
3485:
3482:
3470:
3467:
3464:
3458:
3453:
3449:
3445:
3441:
3437:
3434:
3429:
3425:
3420:
3407:
3406:
3405:
3403:
3400: ⊆
3399:
3395:
3391:
3387:
3380:
3376:
3372:
3365:
3361:
3357:
3331:
3326:
3316:
3313:
3308:
3302:
3296:
3293:
3287:
3284:
3281:
3275:
3268:
3263:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3227:
3224:
3221:
3215:
3212:
3209:
3203:
3200:
3197:
3187:
3179:
3166:
3157:
3156:
3155:
3153:
3149:
3145:
3141:
3137:
3130:
3126:
3122:
3118:
3114:
3110:
3091:
3080:
3076:
3069:
3061:
3051:
3045:
3042:
3039:
3033:
3026:
3025:
3024:
3022:
3019: →
3018:
3014:
3010:
3006:
3002:
2999: ∈
2998:
2994:
2990:
2986:
2982:
2976:
2972:
2964:
2962:
2960:
2956:
2933:
2922:
2917:
2913:
2904:
2899:
2886:
2882:
2875:
2872:
2866:
2860:
2849:
2845:
2835:
2831:
2822:
2817:
2804:
2801:
2797:
2791:
2788:
2785:
2781:
2774:
2771:
2765:
2759:
2753:
2750:
2743:
2742:
2741:
2739:
2732:
2725:
2721:
2718:for standard
2717:
2709:
2693:
2687:
2681:
2676:
2666:
2658:
2654:
2640:
2634:
2628:
2622:
2618:
2611:
2608:
2602:
2596:
2590:
2585:
2577:
2571:
2565:
2562:
2556:
2550:
2547:
2540:
2539:
2538:
2536:
2533:
2529:
2525:
2506:
2500:
2489:
2485:
2475:
2471:
2462:
2457:
2444:
2441:
2438:
2433:
2419:
2413:
2407:
2401:
2397:
2390:
2387:
2384:
2380:
2373:
2370:
2364:
2358:
2347:
2343:
2334:
2322:
2314:
2310:
2304:
2300:
2296:
2290:
2284:
2281:
2274:
2273:
2272:
2270:
2263:
2259:
2256:
2252:
2249: ∈
2248:
2244:
2240:
2233:
2229:
2222:
2218:
2199:
2194:
2187:
2181:
2178:
2167:
2163:
2156:
2148:
2133:
2127:
2119:
2113:
2107:
2104:
2097:
2096:
2095:
2093:
2090: →
2089:
2086: :
2085:
2081:
2077:
2073:
2069:
2065:
2062:known as the
2061:
2053:
2050:
2043:The generator
2042:
2026:
2011:
2007:
2000:
1986:
1982:
1971:
1959:
1939:
1936:
1933:
1929:
1922:
1910:
1896:
1895:
1894:
1892:
1888:
1887:almost surely
1880:
1877: →
1876:
1873: :
1872:
1867:
1865:
1861:
1857:
1853:
1849:
1845:
1844:stopping time
1841:
1833:
1813:
1809:
1800:
1796:
1789:
1785:
1777:
1773:
1762:
1760:
1751:
1745:
1741:
1729:
1720:
1716:
1709:
1705:
1697:
1693:
1681:
1675:
1665:
1663:
1654:
1648:
1644:
1632:
1626:
1606:
1603:
1600:
1596:
1589:
1585:
1579:
1568:
1562:
1552:
1550:
1544:
1538:
1534:
1518:
1515:
1512:
1508:
1501:
1497:
1491:
1473:
1472:
1471:
1466:
1462:
1443:
1432:
1428:
1421:
1410:
1402:
1398:
1387:
1381:
1366:
1346:
1343:
1340:
1336:
1329:
1317:
1303:
1302:
1301:
1299:
1295:
1287:
1282:
1278:
1274:
1270:
1266:
1263: →
1262:
1259: :
1258:
1253:
1251:
1246:
1241:
1234:
1227:
1224: =
1220:
1215:
1210:
1199:
1195:
1176:
1172:
1159:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1122:
1113:
1107:
1099:
1096:
1091:
1087:
1082:
1078:
1075:
1070:
1065:
1057:
1052:
1040:
1039:
1038:
1036:
1032:
1028:
1025:
1016:
1014:
1007:
1003:
999:
995:
987:
984:
976:
974:
972:
968:
960:
956:
952:
949:
945:
941:
938:
937:
922:
911:
907:
900:
892:
882:
876:
870:
863:
862:
861:
859:
856: →
855:
852: :
851:
847:
843:
842:bounded below
839:
835:
831:
828: →
827:
824: :
823:
818:
816:
812:
808:
804:
801: =
797:
793:
789:
785:
782: ∈
781:
776:
774:
770:
762:
760:
743:
740:
730:
727:
719:
715:
711:
706:
702:
686:
685:
684:
682:
678:
674:
670:
666:
659:
653:realisations
652:
648:
644:
636:
633:
625:
623:
621:
617:
609:
606:
603:
599:
596:
592:
589:
585:
582:
578:
575:
571:
568:
567:
566:
564:
563:
558:
554:
550:
546:
542:
538:
534:
532:
527:
524:
520:
517:
513:
509:
505:
501:
477:
474:
471:
463:
460:
449:
443:
440:
434:
428:
420:
409:
403:
400:
394:
388:
376:
375:
374:
372:
368:
365: →
364:
360:
357: →
356:
353: :
352:
348:
345:-dimensional
344:
340:
321:
316:
312:
297:
293:
286:
283:
280:
266:
262:
255:
252:
247:
243:
230:
229:
228:
226:
222:
219:defined on a
218:
214:
211:
193:
177:
174:-dimensional
173:
169:
168:Itô diffusion
165:
157:
152:
145:
143:
141:
138:
137:mathematician
135:
131:
127:
123:
119:
115:
111:
110:Itô diffusion
107:
103:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
18:Ito diffusion
10989:Econometrics
10951:Wiener space
10839:Itô integral
10740:Inequalities
10629:Self-similar
10599:Gauss–Markov
10589:Exchangeable
10569:Càdlàg paths
10505:Risk process
10457:LIBOR market
10326:Random graph
10321:Random field
10133:Superprocess
10071:Lévy process
10066:Jump process
10050:
10041:Hunt process
9877:Markov chain
9780:
9736:
9732:
9713:
9594:
9586:
9584:
9439:
9435:
9431:
9427:
9423:
9419:
9415:
9411:
9409:
9404:
9380:
9376:
9372:
9370:
9284:
9282:
9133:
9129:
9125:
9123:
8995:
8991:
8987:
8983:
8979:
8975:
8971:
8967:
8963:
8959:
8955:
8951:
8947:
8943:
8939:
8935:
8931:
8929:
8911:
8813:
8809:
8805:
8801:
8799:
8708:
8704:
8700:
8696:
8694:
8689:
8681:
8677:
8665:
8661:
8657:
8649:
8645:
8641:
8637:
8636:starting at
8633:
8629:
8627:
8622:
8618:
8614:
8610:
8608:
8510:
8505:
8501:
8493:
8489:
8485:
8481:
8473:
8471:
8462:(−1, 1)
8456:
8416:
8412:
8410:
8318:
8314:
8307:
8305:
8284:
8281:hitting time
8276:
8272:
8270:
8200:
8195:
8191:
8185:
8034:
8030:
8026:
8022:
8018:
8014:
8010:
8005:, gives the
8000:
7981:
7974:
7972:
7739:
7645:
7641:
7637:
7631:
7430:
7426:
7419:
7415:
7411:
7409:
7300:
7296:
7292:
7288:
7284:
7280:
7272:
7270:
7260:
7253:
7249:
7242:
7238:
7227:
7223:
7221:
7065:
7063:
6979:
6975:
6973:
6851:
6847:
6842:
6832:
6826:
6821:
6817:
6813:
6805:
6801:
6797:
6793:
6785:
6783:
6687:
6597:
6517:
6509:
6504:satisfies a
6499:
6410:is given by
6407:
6402:
6297:
6289:
6285:
6279:
6160:
6158:
6148:
6144:
6140:
6136:
6134:
6041:
6037:
6030:
6026:
6011:
6003:
5992:
5981:
5977:
5973:
5969:
5963:
5883:
5879:
5873:
5869:
5865:
5861:
5786:
5782:
5778:
5774:
5766:
5762:
5755:
5751:
5745:
5741:
5739:
5605:
5601:
5597:
5590:
5587:
5585:
5576:
5572:
5568:
5560:
5556:
5554:
5542:
5322:
5318:
5311:
5283:
5279:
5276:
5272:
5268:
5260:
5240:
5238:
4952:
4948:
4946:
4878:
4830:
4775:
4771:
4767:
4760:
4756:
4752:
4748:
4715:
4711:
4709:
4630:
4523:
4516:
4509:
4507:
4342:
4315:
4313:
4304:
4298:
4283:
4279:
4275:
4271:
4269:
4264:
4260:
4256:
4252:
4248:
4244:
4240:
4236:
4232:
4228:
4226:
3991:
3990:states that
3987:
3985:
3833:
3829:
3825:
3817:
3813:
3809:
3805:
3801:
3797:
3793:
3791:
3564:
3560:
3550:
3546:
3542:
3538:
3534:
3523:
3515:
3510:
3502:
3500:
3401:
3397:
3393:
3382:
3378:
3374:
3367:
3355:
3353:
3151:
3143:
3139:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3106:
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2980:
2978:
2954:
2952:
2734:
2727:
2723:
2719:
2715:
2713:
2521:
2265:
2261:
2257:
2250:
2246:
2242:
2235:
2231:
2224:
2220:
2216:
2214:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2057:
1890:
1878:
1874:
1870:
1868:
1863:
1859:
1855:
1851:
1847:
1839:
1837:
1464:
1460:
1458:
1297:
1296:satisfy the
1290:
1285:
1272:
1268:
1264:
1260:
1256:
1254:
1249:
1244:
1236:
1232:
1225:
1218:
1213:
1205:
1193:
1191:
1034:
1030:
1017:
1009:
1005:
1001:
997:
993:
991:
970:
966:
964:
958:
954:
947:
943:
857:
853:
849:
845:
829:
825:
821:
819:
814:
806:
802:
795:
791:
787:
783:
779:
778:For a point
777:
768:
766:
758:
680:
676:
672:
661:
654:
649:, i.e., for
642:
640:
619:
613:
560:
556:
552:
548:
544:
541:matrix field
536:
529:
525:
523:vector field
518:
511:
507:
503:
499:
497:
366:
362:
358:
354:
350:
342:
338:
336:
224:
216:
212:
171:
167:
163:
161:
109:
99:
84:
75:
64:Please help
56:
11034:Ruin theory
10972:Disciplines
10844:Itô's lemma
10619:Predictable
10294:Percolation
10277:Potts model
10272:Ising model
10236:White noise
10194:Differences
10056:Itô process
9996:Cox process
9892:Loop-erased
9887:Random walk
9729:Otto, Felix
9422:. Then the
8656:centred on
8514:defined by
8484:, then the
7435:Itô's lemma
7303:defined by
6835:-dynamics.
6516:functional
6514:free energy
5980:: i.e., if
4774:containing
3505:denote the
3150:, known as
2223:is denoted
1204:(i.e. each
1167: Borel
1022:denote the
811:expectation
562:Itô process
533:coefficient
102:mathematics
70:introducing
11044:Statistics
10824:Filtration
10725:Kolmogorov
10709:Blumenthal
10634:Stationary
10574:Continuous
10562:Properties
10447:Hull–White
10189:Martingale
10076:Local time
9964:Fractional
9942:pure birth
9703:References
8982:, denoted
8672:under all
7439:chain rule
7277:martingale
7237:with mean
6403:where the
6048:, and so ρ
5583:operator.
4951:functions
4295:Definition
2710:An example
2054:Definition
1027:filtration
805:, and let
651:almost all
626:Continuity
373:condition
140:Kiyosi Itô
10956:Classical
9969:Geometric
9959:Excursion
9741:CiteSeerX
9627:∫
9623:−
9522:∫
9518:−
9498:τ
9320:∫
9224:τ
9214:∫
9148:∫
9070:χ
9058:τ
9048:∫
8958:stays in
8874:μ
8855:φ
8847:∂
8843:∫
8827:φ
8766:τ
8722:φ
8674:rotations
8670:invariant
8654:open ball
8586:∈
8575:τ
8526:μ
8366:≥
8348:ω
8336:τ
8231:τ
8130:τ
8121:∫
8074:τ
7851:Σ
7696:Σ
7582:σ
7577:⊤
7553:∇
7539:∫
7490:∫
7350:∫
7346:−
7275:is not a
7180:−
7169:κ
7166:β
7160:−
7152:
7128:π
7120:κ
7117:β
7092:∞
7088:ρ
7076:given by
7031:−
7020:κ
6993:Ψ
6933:−
6929:β
6902:−
6886:κ
6883:−
6829:-function
6751:ρ
6748:
6733:ρ
6715:∫
6705:ρ
6655:ρ
6643:Ψ
6625:∫
6615:ρ
6577:ρ
6566:β
6552:ρ
6537:ρ
6520:given by
6461:Ψ
6458:β
6455:−
6449:
6428:∫
6373:Ψ
6370:β
6367:−
6361:
6350:−
6328:∞
6324:ρ
6239:−
6235:β
6198:Ψ
6195:∇
6192:−
6105:∈
6078:∞
6074:ρ
6068:∗
5928:α
5914:−
5906:α
5824:−
5816:α
5808:α
5684:α
5681:−
5671:∞
5662:∫
5624:α
5581:resolvent
5563:is not a
5507:∂
5503:∂
5468:∑
5430:∂
5426:∂
5404:∑
5365:Δ
5345:, where Δ
5170:∂
5156:∂
5142:∂
5116:⊤
5102:σ
5090:σ
5069:∑
5028:∂
5020:∂
4989:∑
4919:∈
4879:and that
4852:⊆
4778:, define
4664:≥
4643:τ
4589:∞
4574:⋂
4553:⊆
4514:open sets
4478:τ
4445:−
4426:τ
4386:↓
4243:in time,
4190:∈
4127:∈
4074:−
4025:∂
4017:∂
3900:∫
3896:−
3888:
3744:∈
3718:ρ
3696:ρ
3674:∈
3636:ρ
3631:∗
3598:∂
3593:ρ
3590:∂
3459:ρ
3450:∫
3435:∈
3317:∈
3254:∈
3185:∂
3177:∂
2910:∂
2896:∂
2883:∑
2842:∂
2828:∂
2814:∂
2798:δ
2782:∑
2673:∇
2663:∇
2649:⊤
2635:σ
2623:σ
2582:∇
2578:⋅
2532:Frobenius
2482:∂
2468:∂
2454:∂
2428:⊤
2414:σ
2402:σ
2381:∑
2340:∂
2332:∂
2301:∑
2271:and that
2234:), while
2179:−
2131:↓
2064:generator
1987:τ
1960:τ
1956:Σ
1934:τ
1623:Σ
1459:In fact,
1411:ω
1382:ω
1363:Σ
1281:σ-algebra
1248:for each
1150:⊆
1138:≤
1132:≤
1117:Ω
1114:⊆
1097:−
1079:σ
1062:Σ
1049:Σ
998:Markovian
667:(ω) is a
620:diffusion
475:−
461:≤
444:σ
441:−
429:σ
401:−
287:σ
78:July 2013
11092:Category
11077:Category
10961:Abstract
10495:Bühlmann
10101:Compound
9779:(2003).
9763:13890235
9691:See also
8942:and let
8808:and all
8498:boundary
8421:interval
8387:∉
8194:, +
8188:interval
6831:for the
6002:for any
5882:lies in
4689:∉
4263:must be
3559:for all
3377:. Let ρ(
3138:for all
2524:gradient
2264:lies in
1850:at time
840:that is
683:so that
502:and all
146:Overview
134:Japanese
120:used in
10584:Ergodic
10472:Vašíček
10314:Poisson
9974:Meander
9799:2001996
9771:1617171
9722:0193671
9438:;
9401:
9389:
9379:,
8496:on the
8480:within
8453:
8441:
8437:
8425:
7441:) that
7291:;
6839:Example
6040:, so ρ(
5991:, then
5744:, that
5349:is the
5339:
5327:
5271:,
5257:
5245:
5233:
5221:
3804:;
3537:,
3360:adjoint
3115:,
3007:;
1198:adapted
1024:natural
809:denote
210:process
130:viscous
122:physics
66:improve
10924:Tanaka
10609:Mixing
10604:Markov
10477:Wilkie
10442:Ho–Lee
10437:Heston
10209:Super-
9954:Bridge
9902:Biased
9787:
9761:
9743:
9371:where
8680:about
8652:is an
7973:since
7640:, for
7410:where
6974:where
6824:is an
6598:where
6296:(i.e.
4766:}. If
4676:
4670:
3796:is in
3142:, and
2953:i.e.,
2528:scalar
1126:
1120:
1033:: for
786:, let
570:sample
539:; the
341:is an
337:where
10777:Tools
10553:M/M/c
10548:M/M/1
10543:M/G/1
10533:Fluid
10199:Local
9759:S2CID
9383:) is
8699:: if
8492:) of
8455:, so
8279:at a
7068:is a
6010:: if
5761:and α
5275:): a
3107:then
1018:Let Σ
942:: if
834:Borel
832:be a
645:is a
531:drift
208:is a
154:This
108:– an
51:, or
10729:Lévy
10528:Bulk
10412:Chen
10204:Sub-
10162:Both
9785:ISBN
8930:Let
8688:on ∂
8664:on ∂
8644:and
8617:and
8609:for
8488:(or
7433:and
6512:the
4714:for
4631:and
4314:The
4299:The
4115:>
3986:The
3662:>
3545:, ρ(
3533:, ρ(
3501:Let
3242:>
2973:and
2530:and
2526:and
1271:and
1255:Let
1235:has
1211:is Σ
1200:to Σ
820:Let
586:the
579:the
572:and
349:and
10309:Cox
9751:doi
9593:in
9589:is
9136:by
8998:by
8978:at
8970:of
8676:of
8668:is
8504:of
8476:is
8465:is
8357:inf
8033:is
7980:is
7149:exp
6850:on
6745:log
6446:exp
6358:exp
5860:if
5785:is
5773:if
5453:det
5388:det
4655:inf
4379:lim
4239:is
3885:exp
3836:by
3509:of
3392:on
3366:of
2124:lim
2094:by
1862:of
1196:is
860:by
547:of
535:of
170:in
162:A (
100:In
11094::
10727:,
10723:,
10719:,
10715:,
10711:,
9796:MR
9768:MR
9757:.
9749:.
9737:29
9735:.
9719:MR
9597:,
9407:.
8916:.
8816:,
8692:.
8625:.
8190:(−
7648:,
7263:.
7226:,
6312::
5547:.
5347:LB
5343:LB
4811:0.
4718:.
4311:.
4286:.
3567::
3404:,
3154::
3123:,
2961:.
2537:,
1866:.
1300::
817:.
775:.
510:∈
506:,
166:)
142:.
55:,
47:,
10731:)
10707:(
9828:e
9821:t
9814:v
9793:.
9765:.
9753::
9675:.
9672:)
9669:y
9665:d
9661:,
9658:x
9655:(
9652:G
9648:)
9645:y
9642:(
9639:f
9636:A
9631:D
9620:=
9617:)
9614:x
9611:(
9608:f
9595:D
9587:f
9570:.
9567:)
9564:y
9560:d
9556:,
9553:x
9550:(
9547:G
9543:)
9540:y
9537:(
9534:f
9531:A
9526:D
9514:]
9509:)
9502:D
9493:X
9489:(
9485:f
9481:[
9475:x
9470:E
9465:=
9462:)
9459:x
9456:(
9453:f
9440:R
9436:R
9434:(
9432:C
9428:f
9420:D
9416:x
9412:E
9405:D
9398:2
9395:/
9392:1
9381:y
9377:x
9375:(
9373:G
9356:,
9353:y
9349:d
9344:)
9341:y
9338:,
9335:x
9332:(
9329:G
9324:H
9316:=
9313:)
9310:H
9307:,
9304:x
9301:(
9298:G
9285:X
9268:.
9264:]
9260:s
9256:d
9251:)
9246:s
9242:X
9238:(
9235:f
9228:D
9218:0
9209:[
9203:x
9198:E
9193:=
9190:)
9187:y
9183:d
9179:,
9176:x
9173:(
9170:G
9166:)
9163:y
9160:(
9157:f
9152:D
9134:R
9130:D
9126:f
9109:,
9105:]
9101:s
9097:d
9092:)
9087:s
9083:X
9079:(
9074:H
9062:D
9052:0
9043:[
9037:x
9032:E
9027:=
9024:)
9021:H
9018:,
9015:x
9012:(
9009:G
8996:R
8992:H
8988:x
8986:(
8984:G
8980:x
8976:D
8972:X
8964:D
8960:H
8956:X
8952:H
8948:A
8944:X
8940:R
8936:D
8932:A
8897:.
8894:)
8891:y
8888:(
8883:x
8878:G
8869:d
8864:)
8861:y
8858:(
8850:G
8839:=
8836:)
8833:x
8830:(
8814:G
8810:x
8806:H
8802:G
8785:,
8781:]
8777:)
8770:H
8761:X
8757:(
8754:f
8750:[
8744:x
8739:E
8734:=
8731:)
8728:x
8725:(
8709:R
8705:R
8701:f
8690:D
8682:x
8678:D
8666:D
8662:B
8658:x
8650:R
8646:D
8642:R
8638:x
8634:R
8630:B
8623:G
8619:F
8615:G
8611:x
8593:]
8589:F
8579:G
8570:X
8565:[
8559:x
8554:P
8549:=
8546:)
8543:F
8540:(
8535:x
8530:G
8511:G
8506:G
8502:G
8500:∂
8494:X
8482:R
8474:G
8460:τ
8457:B
8450:2
8447:/
8444:1
8434:2
8431:/
8428:1
8417:B
8413:X
8396:.
8393:}
8390:H
8382:t
8378:X
8373:|
8369:0
8363:t
8360:{
8354:=
8351:)
8345:(
8340:H
8319:R
8315:H
8308:X
8277:X
8273:X
8256:.
8251:2
8247:R
8243:=
8240:]
8235:R
8227:[
8222:0
8217:E
8201:R
8196:R
8192:R
8171:.
8167:]
8163:s
8159:d
8154:)
8149:s
8145:X
8141:(
8138:f
8135:A
8125:0
8116:[
8110:x
8105:E
8100:+
8097:)
8094:x
8091:(
8088:f
8085:=
8082:]
8079:)
8070:X
8066:(
8063:f
8060:[
8055:x
8050:E
8035:C
8031:R
8027:R
8023:f
8019:E
8015:A
8011:X
7984:s
7982:F
7977:s
7975:M
7958:,
7953:s
7949:M
7945:=
7940:]
7933:s
7929:F
7923:|
7916:s
7912:M
7906:[
7899:x
7894:E
7889:=
7885:]
7879:s
7875:F
7869:|
7862:]
7855:s
7845:|
7838:t
7834:M
7828:[
7821:x
7816:E
7810:[
7804:x
7799:E
7794:=
7791:]
7786:s
7782:F
7777:|
7771:t
7767:M
7763:[
7758:x
7753:E
7725:.
7720:s
7716:M
7712:=
7707:]
7700:s
7690:|
7683:t
7679:M
7673:[
7666:x
7661:E
7646:s
7642:t
7638:B
7634:∗
7617:.
7612:s
7608:B
7603:d
7598:)
7593:s
7589:X
7585:(
7573:)
7567:s
7563:X
7559:(
7556:f
7548:t
7543:0
7535:+
7532:s
7528:d
7523:)
7518:s
7514:X
7510:(
7507:f
7504:A
7499:t
7494:0
7486:+
7483:)
7480:x
7477:(
7474:f
7471:=
7468:)
7463:t
7459:X
7455:(
7452:f
7431:f
7427:X
7423:∗
7420:F
7416:X
7412:A
7395:,
7392:s
7388:d
7383:)
7378:s
7374:X
7370:(
7367:f
7364:A
7359:t
7354:0
7343:)
7338:t
7334:X
7330:(
7327:f
7324:=
7319:t
7315:M
7301:R
7297:M
7293:R
7289:R
7287:(
7285:C
7281:f
7273:X
7261:m
7256:t
7254:X
7250:m
7245:t
7243:X
7239:m
7230:t
7228:X
7224:t
7218:.
7205:)
7199:2
7193:2
7188:|
7183:m
7177:x
7173:|
7156:(
7143:2
7140:n
7134:)
7125:2
7111:(
7106:=
7103:)
7100:x
7097:(
7074:∞
7066:X
7049:,
7044:2
7039:|
7034:m
7028:x
7024:|
7014:2
7011:1
7005:=
7002:)
6999:x
6996:(
6980:R
6976:m
6959:,
6954:t
6950:B
6945:d
6936:1
6925:2
6920:+
6917:t
6913:d
6908:)
6905:m
6897:t
6893:X
6889:(
6880:=
6875:t
6871:X
6866:d
6852:R
6848:X
6833:X
6827:H
6822:F
6818:t
6814:F
6806:F
6802:F
6798:t
6794:F
6790:∞
6786:Z
6769:x
6765:d
6760:)
6757:x
6754:(
6742:)
6739:x
6736:(
6726:n
6721:R
6711:=
6708:]
6702:[
6699:S
6673:x
6669:d
6664:)
6661:x
6658:(
6652:)
6649:x
6646:(
6636:n
6631:R
6621:=
6618:]
6612:[
6609:E
6583:,
6580:]
6574:[
6571:S
6563:1
6558:+
6555:]
6549:[
6546:E
6543:=
6540:]
6534:[
6531:F
6518:F
6510:R
6502:∞
6485:.
6482:x
6478:d
6473:)
6470:)
6467:x
6464:(
6452:(
6439:n
6434:R
6424:=
6421:Z
6408:Z
6388:,
6385:)
6382:)
6379:x
6376:(
6364:(
6353:1
6346:Z
6342:=
6339:)
6336:x
6333:(
6306:∞
6302:∞
6298:X
6294:∞
6290:R
6286:R
6265:,
6260:t
6256:B
6251:d
6242:1
6231:2
6226:+
6223:t
6219:d
6214:)
6209:t
6205:X
6201:(
6189:=
6184:t
6180:X
6175:d
6161:X
6149:X
6145:X
6141:A
6137:f
6120:.
6115:n
6110:R
6102:x
6098:,
6095:0
6092:=
6089:)
6086:x
6083:(
6064:A
6050:∞
6046:∞
6042:t
6038:t
6033:t
6031:X
6027:t
6023:∞
6019:∞
6015:0
6012:X
6008:∞
6004:t
6000:∞
5995:t
5993:X
5989:∞
5985:0
5982:X
5978:X
5974:R
5970:X
5942:.
5939:g
5936:=
5933:g
5924:R
5920:)
5917:A
5910:I
5903:(
5886:A
5884:D
5880:g
5877:α
5874:R
5870:R
5866:R
5862:g
5842:;
5839:f
5836:=
5833:f
5830:)
5827:A
5820:I
5813:(
5804:R
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