Anticipating stochastic Volterra equations
Stochastic integral equations of the form \[ X_t=x_0+\int _0^tF(t,s,X_s)ds +\sum _{i=1}^k \int _0^t G_i(t,s,X_s)dW^{i}_s, \qquad t\in [0,T],\tag{1} \] are studied where \(W\) is a \(k\)-dimensional Brownian motion, and the coefficients \(F(t,s,x)\) and \(G_i(t,s,x)\) are \(\mathcal F_t\)-measurable. The integrands \(G_i(t,s,X_s)\) are not necessarily \(\mathcal F_s\)-measurable and the stochastic integrals in (1) are interpreted in the Skorokhod sense. Under suitable conditions on smoothness of the diffusion term, existence, uniqueness, time-continuity and semimartingale property of solutions are proved.
scientific article; zbMATH DE number 1409151
scientific article