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Adapted solution of a backward stochastic differential equation. (English) Zbl 0692.93064
Summary: Let $\{W\sb t$; $t\in [0,1]\}$ be a standard $k$-dimensional Wiener process defined on a probability space ($\Omega$,${\cal F},P)$, and let $\{$ ${\cal F}\sb t\}$ denote its natural filtration. Given a ${\cal F}\sb 1$ measurable d-dimensional random vector X, we look for an adapted pair of processes $\{$ x(t), y(t); $t\in [0,1]\}$ with values in ${\bbfR}\sp d$ and ${\bbfR}\sp{d\times k}$ respectively, which solves an equation of the form: $$x(t)+\int\sp{1}\sb{t}f(s,x(s),y(s))ds+\int\sp{1}\sb{t}[g(s,x(s))+y(s)]dW\sb s=X.$$ 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: $$x(t)+\int\sp{1}\sb{t}f(s,x(s),y(s))ds+\int\sp{1}\sb{t}g(s,x(s),y(s))dW\sb s=X$$ 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
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##### References:
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