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Adapted solution of a backward stochastic differential equation. (English) Zbl 0692.93064

Summary: Let $\left\{{W}_{t}$; $t\in \left[0,1\right]\right\}$ be a standard $k$-dimensional Wiener process defined on a probability space (${\Omega }$,$ℱ,P\right)$, and let $\left\{$ ${ℱ}_{t}\right\}$ denote its natural filtration. Given a ${ℱ}_{1}$ measurable d-dimensional random vector X, we look for an adapted pair of processes $\left\{$ x(t), y(t); $t\in \left[0,1\right]\right\}$ with values in ${ℝ}^{d}$ and ${ℝ}^{d×k}$ respectively, which solves an equation of the form:

$x\left(t\right)+{\int }_{t}^{1}f\left(s,x\left(s\right),y\left(s\right)\right)ds+{\int }_{t}^{1}\left[g\left(s,x\left(s\right)\right)+y\left(s\right)\right]d{W}_{s}=X·$

A linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process. We also generalize our results to the following equation:

$x\left(t\right)+{\int }_{t}^{1}f\left(s,x\left(s\right),y\left(s\right)\right)ds+{\int }_{t}^{1}g\left(s,x\left(s\right),y\left(s\right)\right)d{W}_{s}=X$

under rather restrictive assumptions on g.

##### MSC:
 93E03 General theory of stochastic systems 93E20 Optimal stochastic control (systems) 34F05 ODE with randomness 49K45 Optimal stochastic control (optimality conditions) 60H10 Stochastic ordinary differential equations