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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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2010
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| Online Access: | https://hdl.handle.net/2134/15258 |
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| _version_ | 1818173434407944192 |
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| author | Qi Zhang Huaizhong Zhao |
| author_facet | Qi Zhang Huaizhong Zhao |
| author_sort | Qi Zhang (28502) |
| collection | Figshare |
| description | 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. |
| format | Default Article |
| id | rr-article-9388460 |
| institution | Loughborough University |
| publishDate | 2010 |
| record_format | Figshare |
| spelling | rr-article-93884602010-01-01T00:00:00Z Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients Qi Zhang (28502) Huaizhong Zhao (1247379) Other mathematical sciences not elsewhere classified Weak solutions Malliavin derivative Wiener-Sobolev compactness Stochastic differential equations Mathematical Sciences not elsewhere classified 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. 2010-01-01T00:00:00Z Text Journal contribution 2134/15258 https://figshare.com/articles/journal_contribution/Stationary_solutions_of_SPDEs_and_infinite_horizon_BDSDEs_with_non-Lipschitz_coefficients/9388460 CC BY-NC-ND 4.0 |
| spellingShingle | Other mathematical sciences not elsewhere classified Weak solutions Malliavin derivative Wiener-Sobolev compactness Stochastic differential equations Mathematical Sciences not elsewhere classified Qi Zhang Huaizhong Zhao Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title | Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title_full | Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title_fullStr | Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title_full_unstemmed | Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title_short | Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients |
| title_sort | stationary solutions of spdes and infinite horizon bdsdes with non-lipschitz coefficients |
| topic | Other mathematical sciences not elsewhere classified Weak solutions Malliavin derivative Wiener-Sobolev compactness Stochastic differential equations Mathematical Sciences not elsewhere classified |
| url | https://hdl.handle.net/2134/15258 |