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Stochastic Volterra equations with anticipating coefficients. (English) Zbl 0717.60073

The authors study stochastic Volterra equations on \({\mathbb{R}}^ d\) of the following type \[ X_ t=X_ 0+ \int^{t}_{0}F(t,s,X_ s)ds+ \sum^{K}_{i=1} \int^{t}_{0}G_ i(H_ t;t,s,X_ s)dW^ i_ s, \] where \(\{H_ t\}\) is a p-dimensional adapted process, and the coefficients F(t,s,x) and \(G_ i(h;t,s,x)\), \(0\leq s\leq t\), are adapted to \({\mathcal F}_ t\) and to \({\mathcal F}_ s\), respectively. The composition \(G_ i(H_ t;t,s,X_ s)\) will be \({\mathcal F}_ t\)-measurable and, therefore, it anticipates the increments of \(\{W_ t\}\) between s and t. The stochastic integral is then interpreted as a non-adapted Skorokhod integral. The main result of the paper is the existence and uniqueness of a solution assuming Lipschitz conditions on the coefficients. The solution is an adapted process and is shown to be a semimartingale if the coefficients are smooth enough.
Reviewer: D.Nualart

MSC:

60H20 Stochastic integral equations
60H05 Stochastic integrals
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