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

Karczewska, Anna; Lizama, Carlos

J. Math. Anal. Appl. 349, No. 2, 301-310 (2009).

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 \(X_0=0\).

Reviewer: Maria Stolarczyk (Katowice)

MSC:

60H20	Stochastic integral equations
45D05	Volterra integral equations

Keywords:

stochastic linear Volterra equation; resolvent; strong solution; stochastic convolution

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