Electronic Communications in Probability

Representation theorems for SPDEs via backward doubly

Auguste Aman, Abouo Elouaflin, and Mamadou Diop

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In this paper we establish a probabilistic representation for the spatial gradient ofthe viscosity solution to a quasilinear parabolic stochastic partial differential equations(SPDE, for short) in the spirit of the Feynman-Kac formula, without using thederivatives of the coefficients of the corresponding backward doubly stochastic differentialequations (FBDSDE, for short).

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 64, 15 pp.

Accepted: 25 July 2013
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H20: Stochastic integral equations 60H30: Applications of stochastic analysis (to PDE, etc.)

Backward doubly SDEs stochastic partial differential equation stochastic viscosity

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Aman, Auguste; Elouaflin, Abouo; Diop, Mamadou. Representation theorems for SPDEs via backward doubly. Electron. Commun. Probab. 18 (2013), paper no. 64, 15 pp. doi:10.1214/ECP.v18-2223. https://projecteuclid.org/euclid.ecp/1465315603

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  • Bismut, Jean-Michel. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973), 384–404.
  • Bismut, Jean-Michel. An introductory approach to duality in optimal stochastic control. SIAM Rev. 20 (1978), no. 1, 62–78.
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl. 93 (2001), no. 2, 181–204.
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stochastic Process. Appl. 93 (2001), no. 2, 205–228.
  • Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67.
  • El Karoui, N.; Peng, S.; Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1–71.
  • Dellacherie, C. and Meyer, P., Probabilities and Potential. North Holland, (1978).
  • Duffie, Darrell; Epstein, Larry G. Stochastic differential utility. With an appendix by the authors and C. Skiadas. Econometrica 60 (1992), no. 2, 353–394.
  • GÄ«hman, Ĭ. Ī.; Skorohod, A. V. Stochastic differential equations. Translated from the Russian by Kenneth Wickwire. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. Springer-Verlag, New York-Heidelberg, 1972. viii+354 pp.
  • Hamadene, S.; Lepeltier, J.-P. Zero-sum stochastic differential games and backward equations. Systems Control Lett. 24 (1995), no. 4, 259–263.
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1
  • Lions, Pierre-Louis; Souganidis, Panagiotis E. Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 8, 735–741.
  • Ma, Jin; Zhang, Jianfeng. Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002), no. 4, 1390–1418.
  • Nualart, David. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, New York, 1995. xii+266 pp. ISBN: 0-387-94432-X
  • Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55–61.
  • Pardoux, É.; Peng, S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200–217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.
  • Pardoux, Étienne; Peng, Shi Ge. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 (1994), no. 2, 209–227.
  • Peng, Shi Ge. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61–74.
  • Protter, Philip. Stochastic integration and differential equations. A new approach. Applications of Mathematics (New York), 21. Springer-Verlag, Berlin, 1990. x+302 pp. ISBN: 3-540-50996-8