Stochastic Volterra equations with singular kernels
Let us consider the following Volterra type equation \[ X(t) = Y(t) + \int_0^t\;K(t,s) X(s) dB(s),\quad t \geq 0, \] where \(Y\) is some possibly anticipating process, \(K\) is a deterministic kernel which can be singular at \(s = t\) and \(B(t)\) is the standard Brownian motion. Assume that \(|K(s,t) |\leq A (t - s)^{- 1/2}\) with a constant \(A\) and \(Y\) is a weakly measurable generalized process valued in Hida distribution \(({\mathcal S})^*\) and satisfies a boundedness condition. Then the above equation has a unique solution of generalized process in \(({\mathcal S})^*\). Assume that \(\int_0^{t \wedge t_n} |K(t,s) - K(t_n,s) |ds \to 0\), \(t_n \to t\), and the continuity of \(Y\). Then the solution \(X(t)\) is continuous in the strong topology of \(({\mathcal S})^*\).
scientific article; zbMATH DE number 757535