## 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 $$\mathcal F_t$$-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.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

### Keywords:

stochastic Volterra equations; anticipating calculus
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### References:

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