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 & 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, 921 (1982), 237.

[2]

G. Ben Arous, Flots et séries de Taylor stochastiques,, Probab. Theory Related Fields, 81 (1989), 29. 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. 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. 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. 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), 28 (2011), 27.

[7]

T. Hida and N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral,, in, (1967), 117.

[8]

A. Jentzen and P. E. Kloeden, Pathwise Taylor schemes for random ordinary differential equations,, BIT, 49 (2009), 113. 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 (1992).

[10]

H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (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. 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. 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, (1999), 1998.

[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. 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.

[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, 1908 (2004), 6.

[17]

D. Nualart, "The Malliavin Calculus and Related Topics,", Second edition, (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, (1991).

show all references

References:
[1]

R. Azencott, Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman,, in, 921 (1982), 237.

[2]

G. Ben Arous, Flots et séries de Taylor stochastiques,, Probab. Theory Related Fields, 81 (1989), 29. 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. 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. 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. 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), 28 (2011), 27.

[7]

T. Hida and N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral,, in, (1967), 117.

[8]

A. Jentzen and P. E. Kloeden, Pathwise Taylor schemes for random ordinary differential equations,, BIT, 49 (2009), 113. 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 (1992).

[10]

H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (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. 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. 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, (1999), 1998.

[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. 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.

[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, 1908 (2004), 6.

[17]

D. Nualart, "The Malliavin Calculus and Related Topics,", Second edition, (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, (1991).

[1]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[2]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[3]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078

[4]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093

[5]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[6]

John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667

[7]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[8]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[9]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618

[10]

Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163

[11]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965

[12]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640

[13]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735

[14]

MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777

[15]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[16]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521

[17]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[18]

B. M. Haines, Igor S. Aranson, Leonid Berlyand, Dmitry A. Karpeev. Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque. Communications on Pure & Applied Analysis, 2012, 11 (1) : 19-46. doi: 10.3934/cpaa.2012.11.19

[19]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[20]

Liang Zhao. New developments in using stochastic recipe for multi-compartment model: Inter-compartment traveling route, residence time, and exponential convolution expansion. Mathematical Biosciences & Engineering, 2009, 6 (3) : 663-682. doi: 10.3934/mbe.2009.6.663

2017 Impact Factor: 0.542

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]