On Backward Doubly Stochastic Differential Evolutionary System

In this paper, we are concerned with backward doubly stochastic differential evolutionary systems (BDSDESs for short). By using a variational approach based on the monotone operator theory, we prove the existence and uniqueness of the solutions for BDSDESs. We

[1]  Wei Liu,et al.  SPDE in Hilbert space with locally monotone coefficients , 2010, 1005.0632.

[2]  Zhen Wu,et al.  Maximum Principle for Backward Doubly Stochastic Control Systems with Applications , 2010, SIAM J. Control. Optim..

[3]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[4]  J. Bismut An Introductory Approach to Duality in Optimal Stochastic Control , 1978 .

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

[6]  Brahim Boufoussi,et al.  Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions , 2007, 0708.4138.

[7]  Alain Bensoussan,et al.  Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions , 1983 .

[8]  Shanjian Tang,et al.  Maximum Principle for Quasi-linear Backward Stochastic Partial Differential Equations , 2011 .

[9]  E. Zeidler Nonlinear functional analysis and its applications , 1988 .

[10]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[11]  Feng-Yu Wang,et al.  Stochastic generalized porous media and fast diffusion equations , 2006, math/0602369.

[12]  G. Minty Monotone (nonlinear) operators in Hilbert space , 1962 .

[13]  Freddy Delbaen,et al.  Harmonic analysis of stochastic equations and backward stochastic differential equations , 2008, 0801.3505.

[14]  Shanjian Tang,et al.  The Maximum Principle for Partially Observed Optimal Control of Stochastic Differential Equations , 1998 .

[15]  E. Pardouxt,et al.  Stochastic partial differential equations and filtering of diffusion processes , 1980 .

[16]  Shige Peng,et al.  Backward doubly stochastic differential equations and systems of quasilinear SPDEs , 1994 .

[17]  Xicheng Zhang ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS , 2007 .

[18]  L. Denis,et al.  A General Analytical Result for Non-linear SPDE's and Applications , 2004 .

[19]  Rainer Buckdahn,et al.  Stochastic viscositysolutions for nonlinear stochastic partial di#erential equations. Part II , 2001 .

[20]  Shige Peng,et al.  Stochastic Hamilton-Jacobi-Bellman equations , 1992 .

[21]  Xun Yu Zhou,et al.  A duality analysis on stochastic partial differential equations , 1992 .

[22]  Huaizhong Zhao,et al.  Stationary Solutions of SPDEs and Infinite Horizon BDSDEs , 2007 .

[23]  Bernard Delyon,et al.  Lp solutions of backward stochastic differential equations , 2003 .

[24]  Rainer Buckdahn,et al.  Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs , 2002 .

[25]  B. Øksendal,et al.  On Backward Stochastic Partial Differential Equations. , 2001 .

[26]  F. Browder Nonlinear elliptic boundary value problems , 1963 .

[27]  Bernard Delyon,et al.  L p solutions of Backward Stochastic Dierential Equations , 2003 .

[28]  Shanjian Tang,et al.  Lp Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space , 2010, 1006.1171.

[29]  J. Weidmann Linear Operators in Hilbert Spaces , 1980 .

[30]  Naoyuki Ichihara Homogenization problem for stochastic partial differential equations of Zakai type , 2004 .

[31]  B. Rozovskii,et al.  Stochastic evolution equations , 1981 .

[32]  J. Yong,et al.  On semi-linear degenerate backward stochastic partial differential equations , 1999 .

[33]  D. Laurent Solutions of stochastic partial differential equations considered as Dirichlet processes , 2004 .

[34]  David Nualart,et al.  Stochastic calculus with anticipating integrands , 1988 .

[35]  Zhi-Ming Ma,et al.  Introduction to the theory of (non-symmetric) Dirichlet forms , 1992 .

[36]  J. Real,et al.  Some Results on Nonlinear Backward Stochastic Evolution Equations , 2004 .

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

[38]  J. Bismut,et al.  Controle des systemes lineaires quadratiques : Applications de l’integrale stochastique , 1978 .

[39]  Nikolaos Englezos,et al.  Utility Maximization with Habit Formation: Dynamic Programming and Stochastic PDEs , 2009, SIAM J. Control. Optim..

[40]  Hai-ping Shi Backward stochastic differential equations in finance , 2010 .