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 Mat. Sb. (N.S.), 1976, Volume 100(142), Number 2(6), Pages 266–284 (Mi msb2874)

On explicit formulas for solutions of stochastic equations

A. Yu. Veretennikov, N. V. Krylov

Abstract: The article is devoted to the proof of some criteria for the existence of a strong solution of a stochastic integral equation of the form $dx_t=\sigma(t,x_t) dw_t+b(t,x_t) dt$. One of the criteria appears as a Fredholm alternative; others are formulated in terms of the theory of differential equations of parabolic type. The proof of these criteria is based on finding formulas expressing $\mathsf M\{\varphi(x_t)|\mathscr F^w_t\}$ via multiple stochastic integrals, formulas which in the case $\varphi(x)\equiv x$ give an expression for $x_t$, if $x_t$ is a strong solution of the stochastic equation.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1976, 29:2, 239–256

Bibliographic databases:

UDC: 519.2
MSC: 60H20

Citation: A. Yu. Veretennikov, N. V. Krylov, “On explicit formulas for solutions of stochastic equations”, Mat. Sb. (N.S.), 100(142):2(6) (1976), 266–284; Math. USSR-Sb., 29:2 (1976), 239–256

Citation in format AMSBIB
\Bibitem{VerKry76} \by A.~Yu.~Veretennikov, N.~V.~Krylov \paper On explicit formulas for solutions of stochastic equations \jour Mat. Sb. (N.S.) \yr 1976 \vol 100(142) \issue 2(6) \pages 266--284 \mathnet{http://mi.mathnet.ru/msb2874} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=410921} \zmath{https://zbmath.org/?q=an:0353.60059} \transl \jour Math. USSR-Sb. \yr 1976 \vol 29 \issue 2 \pages 239--256 \crossref{https://doi.org/10.1070/SM1976v029n02ABEH003666} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1976EZ91500009} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Yu. Veretennikov, “On strong solutions and explicit formulas for solutions of stochastic integral equations”, Math. USSR-Sb., 39:3 (1981), 387–403
2. Hiroshi Kunita, “Cauchy problem for stochastic partial differential equations arizing in nonlinear filtering theory”, Systems & Control Letters, 1:1 (1981), 37
3. A. I. Kirillov, “Infinite-dimensional analysis and quantum theory as semimartingale calculus”, Russian Math. Surveys, 49:3 (1994), 43–95
4. Sergey Lototsky, Remigijus Mikulevicius, Boris L. Rozovskii, “Nonlinear Filtering Revisited: A Spectral Approach”, SIAM J Control Optim, 35:2 (1997), 435
5. R. Mikulevicius, B. Rozovskii, “Linear Parabolic Stochastic PDE and Wiener Chaos”, SIAM J Math Anal, 29:2 (1998), 452
6. R. Mikulevicius, B. Rozovskii, “Fourier–Hermite Expansions for Nonlinear Filtering”, Theory Probab Appl, 44:3 (2000), 606
7. Theory Probab. Appl., 54:2 (2010), 189–202
8. Hu Yaozhong, “A random transport-diffusion equation”, Acta Mathematica Scientia, 30:6 (2010), 2033
9. S.V. Lototsky, B.L. Rozovskii, D. Seleši, “On generalized Malliavin calculus”, Stochastic Processes and their Applications, 2011
10. Glinyanaya E. V., “Discrete analogue of the Krylov–Veretennikov expansion”, Theory Stoch. Process., 17(33):1 (2011), 39–49
11. Alexey Rudenko, “Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times”, Stochastic Processes and their Applications, 2012
12. G. V. Riabov, “Itô-Wiener expansion for functionals of the Arratia's flow n-point motion”, Theory Stoch. Process., 19(35):2 (2014), 64–89
13. E. V. Glinyanaya, “Krylov–Veretennikov representation for the $m$-point motion of a discrete-time flow”, Theory Stoch. Process., 20(36):1 (2015), 63–77
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