Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the L (R ; R) ⊗ L (R ; R)-valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the L (R...

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Bibliographic Details
Main Authors: Qi Zhang, Huaizhong Zhao
Format: Default Article
Published: 2010
Subjects:
Other mathematical sciences not elsewhere classified
Weak solutions
Malliavin derivative
Wiener-Sobolev compactness
Stochastic differential equations
Mathematical Sciences not elsewhere classified
Online Access:https://hdl.handle.net/2134/15258
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Summary:We prove a general theorem that the L (R ; R) ⊗ L (R ; R)-valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the L (R ; R) ⊗ L (R ; R)-valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.

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