Backward stochastic dynamics on a filtered probability space

TL;DR

This paper reformulates backward stochastic differential equations as functional differential equations on path spaces, eliminating the need for Itô integrals and martingale representation, thus providing new tools especially for partial information scenarios.

Contribution

It introduces a novel reformulation of BSDEs as functional differential equations, broadening analytical tools and enabling study of complex, nonlocal PDEs without Itô calculus.

Findings

Unique solutions exist under certain conditions.

Applicable to BSDEs with partial information.

Handles nonlinear, nonlocal PDEs with integral operators.

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 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.