Article

# Backward stochastic dynamics on a filtered probability space

04/2009; DOI:10.1214/10-AOP588
Source: arXiv

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.

0 0
·
0 Bookmarks
·
75 Views

1 Download
Available from

### Keywords

$d$-dimensional Brownian motion

adapted process $L(M)$

associated partial differential equations

backward stochastic differential equations

BSDE

correction term

following type

integral operators

nonlinear

ordinary functional differential equations

square-integrable $M$

square-integrable martingale

unique solution