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July 2011 Backward stochastic dynamics on a filtered probability space
Gechun Liang, Terry Lyons, Zhongmin Qian
Ann. Probab. 39(4): 1422-1448 (July 2011). DOI: 10.1214/10-AOP588

Abstract

We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither Itô’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 YT = ξ, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbf{P})$, 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.

Citation

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Gechun Liang. Terry Lyons. Zhongmin Qian. "Backward stochastic dynamics on a filtered probability space." Ann. Probab. 39 (4) 1422 - 1448, July 2011. https://doi.org/10.1214/10-AOP588

Information

Published: July 2011
First available in Project Euclid: 5 August 2011

zbMATH: 1238.60064
MathSciNet: MR2857245
Digital Object Identifier: 10.1214/10-AOP588

Subjects:
Primary: 60H10 , 60H30
Secondary: 60J45

Keywords: Brownian motion , BSDE , SDE , Semimartingale

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 4 • July 2011
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