The Stochastic Volterra Equation

Abstract

We study the stochastic (Skorohod) integral equation of the Volterra type $$ {X_t}(\omega ) = {Y_t}(\omega ) + \int\limits_0^t {b(t,s){X_s}(\omega )ds} + \int\limits_0^t {\sigma (t,s){X_s}(\omega )\delta {B_s}(\omega )} $$ where Y, b and a are given functions; b and a are bounded, deterministic and Yt is stochastic, not necessarily adapted. The stochastic integral (δB) is taken in the Skorohod sense. In general there need not exist a classical stochastic process Xt(w) satisfying this equation. However, we show that a unique solution exists in the following extended senses: Moreover, in both cases we find explicit solution formulas. The formulas are similar to the formulas in the deterministic case (σ≡0), but with Wick products in stead of ordinary (pointwise) products.

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