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Anticipating stochastic Volterra equations. (English) Zbl 0942.60045
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 $\cal F_t$-measurable. The integrands $G_i(t,s,X_s)$ are not necessarily $\cal F_s$-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.

60H10Stochastic ordinary differential equations
Full Text: DOI
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