BACKWARD STOCHASTIC DYNAMICS ON A FILTERED PROBABILITY SPACE

• 成果类型: Article
• 署名作者: Liang, Gechun; Lyons, Terry; Qian, Zhongmin
• 署名单位: University of Oxford; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP588
• 发表日期: 2011
• 页码: 1422-1448
• 关键词: differential-equations; Utility maximization; MAXIMUM PRINCIPLE; WEAK SOLUTIONS; sdes

摘要

We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither Ito'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 - Sigma(d)(i=1) f(i)(j) (t, Y-t) d B-t(i) + d M-t(j) with Y-T = xi, on a general filtered probability space (Omega, F, 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.