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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 $H$ $$X(t)=X_0+\int^t_0a(t-\tau)AX(\tau)\,d\tau+\int^t_0 \psi(\tau)\,dW (\tau),\quad t\in[0,T]\tag 1$$ is studied. Where $X_0\in H$, $a\in L^1_{\text{loc}}(R_+)$, $A$ is a closed unbounded linear operator in $H$ with a dense domain $D(A)$, $W$ 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 $L^0_2$-predictable process such that $$\|\psi\|_T=\left\{E\left(\int^T_0| \psi (\tau)|^2_{L^0_2} \,d\tau\right)\right\}^{\frac 12}<\infty,$$ $L^0_2$ is set of all Hilbert-Schmidt operators from $Q^{\frac 12}(U)$ into $H$. An $H$-valued predictable process $X(t)$, $t\in[0,T]$, is said to be a strong solution to (1), if $X$ has a version such that $P(X(t)\in D(A))=1$ for almost all $t\in [0,T]$; for any $t\in[0,T]$ $$\int|a(t-\tau)AX(\tau)|_H \,d\tau<\infty\ P\text{-a.s.}$$ and for any $t\in[0,T]$ equation (1) holds $P\text{-a.s}$. Under certain assumptions the authors show that (1) has a strong solution. Precisely, the stochastic convolution $$W^\psi(t)= \int^t_0S(t-\tau)\psi(\tau)\,dW(\tau),$$ where $S(t)$ is a resolvent determined by operator $A$, is a strong solution to (1) with $X_0=0$.

60H20Stochastic integral equations
45D05Volterra integral equations
Full Text: DOI arXiv
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