Backward stochastic dynamics on a filtered probability space
We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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2009
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| _version_ | 1826274502178242560 |
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| author | Liang, G Lyons, T Qian, Z |
| author_facet | Liang, G Lyons, T Qian, Z |
| author_sort | Liang, G |
| collection | OXFORD |
| description | We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: \[dY_t^j=-f_0^j(t,Y_t,L(M)_t) dt-\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j\] with $Y_T=\xi$, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$, where $B$ is a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution $(Y,M)$. In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators. |
| first_indexed | 2024-03-06T22:44:24Z |
| format | Journal article |
| id | oxford-uuid:5ca9e176-3c5e-4003-bb8c-fa631107847f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:44:24Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:5ca9e176-3c5e-4003-bb8c-fa631107847f2022-03-26T17:29:33ZBackward stochastic dynamics on a filtered probability spaceJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5ca9e176-3c5e-4003-bb8c-fa631107847fEnglishSymplectic Elements at Oxford2009Liang, GLyons, TQian, ZWe demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: \[dY_t^j=-f_0^j(t,Y_t,L(M)_t) dt-\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j\] with $Y_T=\xi$, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$, where $B$ is a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution $(Y,M)$. In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators. |
| spellingShingle | Liang, G Lyons, T Qian, Z Backward stochastic dynamics on a filtered probability space |
| title | Backward stochastic dynamics on a filtered probability space |
| title_full | Backward stochastic dynamics on a filtered probability space |
| title_fullStr | Backward stochastic dynamics on a filtered probability space |
| title_full_unstemmed | Backward stochastic dynamics on a filtered probability space |
| title_short | Backward stochastic dynamics on a filtered probability space |
| title_sort | backward stochastic dynamics on a filtered probability space |
| work_keys_str_mv | AT liangg backwardstochasticdynamicsonafilteredprobabilityspace AT lyonst backwardstochasticdynamicsonafilteredprobabilityspace AT qianz backwardstochasticdynamicsonafilteredprobabilityspace |