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Adapted solution of a backward semilinear stochastic evolution equation. (English) Zbl 0736.60051

Consider the following semilinear stochastic evolution equation of the backward form:

$dx+Ax\left(t\right)dt=f\left(t,x\left(t\right),y\left(t\right)\right)dt+\left(g\left(t,x\left(t\right)\right)+y\left(t\right)\right)dW\left(t\right),\phantom{\rule{2.em}{0ex}}x\left(T\right)=X,$

where $A$ is the infinitesimal generator of a ${C}_{0}$- semigroup $\left\{{e}^{At}\right\}$ on $H$, a separable Hilbert space. The precise meaning of the equation is

$x\left(t\right)+{\int }_{t}^{T}{e}^{A\left(s-t\right)}f\left(s,x\left(s\right),y\left(s\right)\right)ds+{\int }_{t}^{T}{e}^{A\left(s-t\right)}\left(g\left(s,x\left(s\right)\right)+y\left(s\right)\right)dW\left(s\right)={e}^{A\left(T-t\right)}X·\phantom{\rule{2.em}{0ex}}\left(*\right)$

The authors establish existence and uniqueness results for an adapted pair $\left\{x\left(t\right),y\left(t\right);t\in \left[0,T\right]\right\}$ which solves the equation (*), under some suitable conditions on the functions involved in (*). Finally, a result on the backward stochastic partial differential equation is also given.

##### MSC:
 60H10 Stochastic ordinary differential equations