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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 + 0 t a(t-τ)AX(τ)dτ+ 0 t ψ(τ)dW(τ),t[0,T](1)

is studied.

Where X 0 H, aL loc 1 (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 ψ is L 2 0 -predictable process such that

ψ T =E 0 T |ψ(τ)| L 2 0 2 dτ 1 2 <,

L 2 0 is set of all Hilbert-Schmidt operators from Q 1 2 (U) into H.

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

|a(t-τ)AX(τ)| H dτ<P-a.s.

and for any t[0,T] equation (1) holds P-a.s.

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

W ψ (t)= 0 t S(t-τ)ψ(τ)dW(τ),

where S(t) is a resolvent determined by operator A, is a strong solution to (1) with X 0 =0.

MSC:
60H20Stochastic integral equations
45D05Volterra integral equations
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