American Institute of Mathematical Sciences

December  2011, 1(4): 437-468. doi: 10.3934/mcrf.2011.1.437

Pathwise Taylor expansions for Itô random fields

 1 Département de Mathématiques, Université de Bretagne-Occidentale, CS 93837, F-29238 Brest cedex 3, France 2 Universität Greifswald Institut für Mathematik und Informatik, Walther-Rathenau-Straβe 47, 17487 Greifswald, Germany 3 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States

Received  August 2010 Revised  June 2011 Published  November 2011

In this paper we study the pathwise stochastic Taylor expansion, in the sense of our previous work [3], for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work [3], our new expansion can be defined around any random time-space point (τ,ξ), where the temporal component τ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point (τ,ξ). As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the stochastic viscosity solution for such SPDEs, which is new in the literature, and potentially of essential importance in stochastic control theory.
Citation: Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control and Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437
References:
 [1] R. Azencott, Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman, in "Seminar on Probability," XVI, Supplement, Lecture Notes in Math., 921, Springer, Berlin-New York, (1982), 237-285. [2] G. Ben Arous, Flots et séries de Taylor stochastiques, Probab. Theory Related Fields, 81 (1989), 29-77. doi: 10.1007/BF00343737. [3] R. Buckdahn and J. Ma, Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab., 30 (2002), 1131-1171. doi: 10.1214/aop/1029867123. [4] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stochastic Process. Appl., 93 (2001), 181-204. doi: 10.1016/S0304-4149(00)00093-4. [5] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stochastic Process. Appl., 93 (2001), 205-228. doi: 10.1016/S0304-4149(00)00092-2. [6] M. Caruana, P. Friz, and H. Oberhauser, A (rough) pathwise approach to fully non-linear stochastic partial differential equations, Annals IHP (C), Nonlinear Analysis, 28 (2011), 27-46. [7] T. Hida and N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, in "Proc. Fifth Berkeley Symp. Math. Stat. Probab." (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. Calif. Press, Berkeley, Calif., (1967), 117-143. [8] A. Jentzen and P. E. Kloeden, Pathwise Taylor schemes for random ordinary differential equations, BIT, 49 (2009), 113-140. doi: 10.1007/s10543-009-0211-6. [9] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [10] H. Kunita, "Stochastic Flows and Stochastic Differential Equations," Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [11] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1085-1092. doi: 10.1016/S0764-4442(98)80067-0. [12] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 735-741. doi: 10.1016/S0764-4442(98)80161-4. [13] P.-L. Lions and P. E. Souganidis, Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, in "Seminaire: Équations aux Dérivées Partielles," 1998-1999, Sémin. Équ. Dériv. Partielles, Exp. No. I, 15 pp., École Polytech, Palaiseau, 1999. [14] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 617-624. doi: 10.1016/S0764-4442(00)00583-8. [15] P.-L. Lions and P. E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations. Viscosity solutions of differential equations and related topics, (Japanese) (Kyoto, 2001), RIMS Kokyuroku, 1287 (2002), 58-65. [16] T. Lyons, M. Caruana and T. Lévy, "Differential Equations Driven by Rough Paths," Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004, With an introduction concerning the Summer School by Jean Picard, Lecture Notes in Mathematics, 1908, Springer, Berlin, 2007. [17] D. Nualart, "The Malliavin Calculus and Related Topics," Second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. [18] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1991.

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References:
 [1] R. Azencott, Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman, in "Seminar on Probability," XVI, Supplement, Lecture Notes in Math., 921, Springer, Berlin-New York, (1982), 237-285. [2] G. Ben Arous, Flots et séries de Taylor stochastiques, Probab. Theory Related Fields, 81 (1989), 29-77. doi: 10.1007/BF00343737. [3] R. Buckdahn and J. Ma, Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab., 30 (2002), 1131-1171. doi: 10.1214/aop/1029867123. [4] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stochastic Process. Appl., 93 (2001), 181-204. doi: 10.1016/S0304-4149(00)00093-4. [5] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stochastic Process. Appl., 93 (2001), 205-228. doi: 10.1016/S0304-4149(00)00092-2. [6] M. Caruana, P. Friz, and H. Oberhauser, A (rough) pathwise approach to fully non-linear stochastic partial differential equations, Annals IHP (C), Nonlinear Analysis, 28 (2011), 27-46. [7] T. Hida and N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, in "Proc. Fifth Berkeley Symp. Math. Stat. Probab." (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. Calif. Press, Berkeley, Calif., (1967), 117-143. [8] A. Jentzen and P. E. Kloeden, Pathwise Taylor schemes for random ordinary differential equations, BIT, 49 (2009), 113-140. doi: 10.1007/s10543-009-0211-6. [9] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [10] H. Kunita, "Stochastic Flows and Stochastic Differential Equations," Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [11] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1085-1092. doi: 10.1016/S0764-4442(98)80067-0. [12] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 735-741. doi: 10.1016/S0764-4442(98)80161-4. [13] P.-L. Lions and P. E. Souganidis, Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, in "Seminaire: Équations aux Dérivées Partielles," 1998-1999, Sémin. Équ. Dériv. Partielles, Exp. No. I, 15 pp., École Polytech, Palaiseau, 1999. [14] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 617-624. doi: 10.1016/S0764-4442(00)00583-8. [15] P.-L. Lions and P. E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations. Viscosity solutions of differential equations and related topics, (Japanese) (Kyoto, 2001), RIMS Kokyuroku, 1287 (2002), 58-65. [16] T. Lyons, M. Caruana and T. Lévy, "Differential Equations Driven by Rough Paths," Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004, With an introduction concerning the Summer School by Jean Picard, Lecture Notes in Mathematics, 1908, Springer, Berlin, 2007. [17] D. Nualart, "The Malliavin Calculus and Related Topics," Second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. [18] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1991.
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