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Strong solutions to stochastic Volterra equations. (English) Zbl 1158.60028

In the present paper the stochastic Volterra equation in a separable Hilbert space $H$

$X\left(t\right)={X}_{0}+{\int }_{0}^{t}a\left(t-\tau \right)AX\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}d\tau +{\int }_{0}^{t}\psi \left(\tau \right)\phantom{\rule{0.166667em}{0ex}}dW\left(\tau \right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right]\phantom{\rule{2.em}{0ex}}\left(1\right)$

is studied.

Where ${X}_{0}\in H$, $a\in {L}_{\text{loc}}^{1}\left({R}_{+}\right)$, $A$ is a closed unbounded linear operator in $H$ with a dense domain $D\left(A\right)$, $W$ is a cylindrical Wiener process with covariance operator $Q$, $Q$ is a linear bounded symmetric nonnegative operator in separable Hilbert space $U$, ${X}_{0}$ is an $H$-valued, ${F}_{0}$-measurable random variable and $\psi$ is ${L}_{2}^{0}$-predictable process such that

${\parallel \psi \parallel }_{T}={\left\{E\left({\int }_{0}^{T}{|\psi \left(\tau \right)|}_{{L}_{2}^{0}}^{2}\phantom{\rule{0.166667em}{0ex}}d\tau \right)\right\}}^{\frac{1}{2}}<\infty ,$

${L}_{2}^{0}$ is set of all Hilbert-Schmidt operators from ${Q}^{\frac{1}{2}}\left(U\right)$ into $H$.

An $H$-valued predictable process $X\left(t\right)$, $t\in \left[0,T\right]$, is said to be a strong solution to (1), if $X$ has a version such that $P\left(X\left(t\right)\in D\left(A\right)\right)=1$ for almost all $t\in \left[0,T\right]$; for any $t\in \left[0,T\right]$

${\int |a\left(t-\tau \right)AX\left(\tau \right)|}_{H}\phantom{\rule{0.166667em}{0ex}}d\tau <\infty \phantom{\rule{4pt}{0ex}}P\text{-a.s.}$

and for any $t\in \left[0,T\right]$ equation (1) holds $P\text{-a.s}$.

Under certain assumptions the authors show that (1) has a strong solution. Precisely, the stochastic convolution

${W}^{\psi }\left(t\right)={\int }_{0}^{t}S\left(t-\tau \right)\psi \left(\tau \right)\phantom{\rule{0.166667em}{0ex}}dW\left(\tau \right),$

where $S\left(t\right)$ is a resolvent determined by operator $A$, is a strong solution to (1) with ${X}_{0}=0$.

##### MSC:
 60H20 Stochastic integral equations 45D05 Volterra integral equations
##### References:
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