The paper is concerned with optimal control of a stochastic differential system reflected in a domain. The cost functional is implicitly defined via a generalized backward stochastic differential equation developed by Pardoux and Zhang [Probab. Theory Relat. Fields 110 (1998) 535–558]. The value function is shown to be the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The proof requires new estimates for the reflected stochastic differential system.

DOI: 10.1051/cocv/2014062

Keywords: Hamilton–Jacobi–Bellman equation, nonlinear Neumann boundary, value function, backward stochastic differential equations, dynamic programming principle, viscosity solution

^{1}; Tang, Shanjian

^{2}

@article{COCV_2015__21_4_1150_0, author = {Li, Juan and Tang, Shanjian}, title = {Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1150--1177}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2014062}, mrnumber = {3395759}, zbl = {1341.49020}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014062/} }

TY - JOUR AU - Li, Juan AU - Tang, Shanjian TI - Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1150 EP - 1177 VL - 21 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014062/ DO - 10.1051/cocv/2014062 LA - en ID - COCV_2015__21_4_1150_0 ER -

%0 Journal Article %A Li, Juan %A Tang, Shanjian %T Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1150-1177 %V 21 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014062/ %R 10.1051/cocv/2014062 %G en %F COCV_2015__21_4_1150_0

Li, Juan; Tang, Shanjian. Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1150-1177. doi : 10.1051/cocv/2014062. http://www.numdam.org/articles/10.1051/cocv/2014062/

Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differ. Equ. 106 (1993) 90–106. | DOI | MR | Zbl

,Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973) 384–404. | DOI | MR | Zbl

,J. Bismut, Contrôl des systèmes linéares quadratiques, in Applications de L’intégrale Stochastique, Séminaire de Probabilité XII, Vol. 649 of Lect. Notes Math. Springer, Berlin, Heidelberg, New York (1978) 180–264. | Numdam | MR | Zbl

An introductory approach to duality in optimal stochastic control. SIAM Rev. 20 (1978) 62–78. | DOI | MR | Zbl

,An approximation result for a nonlinear Neumann boundary value problem via BSDEs. Stoch. Proc. Appl. 114 (2004) 331–350. | DOI | MR | Zbl

and ,M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with ${L}^{1}$-time dependence and Neumann boundary conditions. Available on http://www.phys.univ-tours.fr/˜barles/artL1-1.pdf. | MR | Zbl

Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J. Control. Optim. 47 (2008) 444–475. | DOI | MR | Zbl

and ,User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. | DOI | MR | Zbl

, and ,Backwards SDE with random terminal time, and applications to semilinear elliptic PDE. Ann. Probab. 25 (1997) 1135–1159. | MR | Zbl

and ,Neumann-Type Boundary Conditions for Hamilton–Jacobi Equations in Smooth Domains. Appl. Math. Optim. 53 (2006) 359–381. | DOI | MR | Zbl

,Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146 (2010) 291–336. | DOI | MR | Zbl

and ,Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. | DOI | MR | Zbl

, and ,Probabilistic interpretation for a system of quasilinear elliptic partial differential equations with Neumann boundary conditions. Stochastic. Process. Appl. 48 (1993) 107–121. | DOI | MR | Zbl

,Neumann type boundary conditions for Hamilton–Jacobi equations. Duke Math. J. 52 (1985) 793–820. | DOI | MR | Zbl

,Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511–537. | DOI | MR | Zbl

and ,Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983) 733–744. | DOI | MR | Zbl

,Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55–61. | DOI | MR | Zbl

and ,E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications. Vol. 176 of Proc. IFIP Int. Conf., Charlotte/NC (USA) (1991), Lect. Notes Control Inf. Sci. Springer (1992) 200–217. | MR | Zbl

Symmetric reflected diffusions. Ann. Inst. Henri Poincaré 30 (1994) 13–62. | Numdam | MR | Zbl

and ,Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields 110 (1998) 535–558. | DOI | MR | Zbl

and ,S. Peng, BSDE and stochastic optimizations (in Chinese), in: Chap. 2 of Topics in stochastic analysis, edited by J. Yan, S. Peng, S. Fang and L. Wu. Science Press, Beijing (1997).

A generalized dynamic programming principle and Hamilton–Jacobi–Bellman equation. Stoch. Stoch. Rep. 38 (1992) 119–134. | DOI | MR | Zbl

,Stochastic differential equations for multidimensional domains with refecting boundary. Probab. Theory Relat. Fields 74 (1987) 455-477. | DOI | MR | Zbl

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