Backward stochastic dynamics on a filtered probability space

Details Backward stochastic dynamics on a filtered probability space Backward stochastic dynamics on a filtered probability space

2009-04-02

arxiv.org/abs/0904.0377v3

Mathematics

Probability

28 pages

Scientific paper

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}^{d}f_{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},\boldsymbol{P}), where B is a d-dimensional Brownian motion, L is a prescribed (non-linear) 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 non-linear, but also may be non-local and involve integral operators.

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