Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

In this paper we introduce a new kind of backward stochastic differential equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness, and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to the optimal ergodic control of stochastic partial differential equations.

[1]  N. Karoui,et al.  Backward Stochastic Differential Equations , 1997 .

[2]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[3]  G. Samorodnitsky.  Random series and stochastic integrals:single and multiple , 1994 .

[4]  E. J. McShane,et al.  On Filippov’s implicit functions lemma , 1967 .

[5]  M. Fuhrman.  A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, st , 2003 .

[6]  B. Maslowski,et al.  Ergodic Control of Semilinear Stochastic Equations and the Hamilton–Jacobi Equation☆☆☆ , 1999 .

[7]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[8]  Ying Hu,et al.  BSDE on an infinite horizon and elliptic PDEs in infinite dimension , 2007 .

[9]  Ying Hu,et al.  Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs , 1998 .

[10]  Jan Seidler,et al.  Ergodic behaviour of stochastic parabolic equations , 1997 .

[11]  Federica Masiero,et al.  Stochastic Optimal Control Problems and Parabolic Equations in Banach Spaces , 2008, SIAM J. Control. Optim..

[12]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[13]  P. Lions,et al.  ON ERGODIC STOCHASTIC CONTROL , 1998 .

[14]  S. Cerrai.  Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach , 2001 .

[15]  S. Peng,et al.  Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control , 1999 .

[16]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

[17]  M. Royer,et al.  BSDEs with a random terminal time driven by a monotone generator and their links with PDEs , 2004 .

[18]  Marco Fuhrman,et al.  Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces , 2004 .

[19]  A. Bensoussan,et al.  On Bellman equations of ergodic control in Rn , 1992 .

[20]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[21]  F. Masiero.  Regularizing Properties for Transition Semigroups and Semilinear Parabolic Equations in Banach Spaces , 2007 .

[22]  G. Tessitore,et al.  Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control , 2002 .

[23]  S. Peng,et al.  Ergodic Backward SDE and Associated PDE , 1999 .

[24]  G. Tessitore,et al.  The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces , 2002 .

[25]  Tyrone E. Duncan,et al.  Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems , 1994 .