An analytic approach to stochastic Volterra equations with completely monotone kernels

Abstract

We apply the semigroup setting of Desch and Miller to a class of stochastic integral equations of Volterra type with completely monotone kernels with a multiplicative noise term; the corresponding equation is an infinite dimensional stochastic equation with unbounded diffusion operator that we solve with the semigroup approach of Da Prato and Zabczyk. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.

Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

“…This will be the object of Section 4.4; notice that, in order to get this result, we must first prove a general result concerning existence and uniqueness of the solution for a stochastic evolution equation with unbounded operator terms. This is an extension of the results in [8], compare also [4]. We proceed with the study of the optimal control problem associated to the stochastic Volterra equation (1.1)(Section 5).…”

“…The result is basically known, see e.g. [4], but we include the proof for completeness and because it will be useful in the following. The argument is as follows: we define a mapping K from L p F (Ω; C([0, T ]; X η )) to itself by the formula…”

“…In order to handle the control problem, we first restate equation (1.1) in an evolution setting, by the state space setting first introduced in [19,9] and recently revised, for the stochastic case, in [17,3,4]. Thus we obtain a stochastic evolution equation in a (different) Hilbert space.…”

“…The proof follows the line of Theorem 4.4 in [4]; however, under the assumptions in Hypothesis 1.1, it requires to pass through the approximation sequence {v n }.…”

“…We notice that, since ψ n (t, v, 0) ≥ ψ(t, v, 0) ≥ 0 and by an application of the comparison theorem (see [4]), it holds 0 ≤ Y …”