Multiple stochastic integrals and their application to the solution of an Itô-Volterra stochastic integral equation. (English. Russian original) Zbl 0749.60062

Pikovskij, I. E.

Theory Probab. Math. Stat. 43, 131-137 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 117-123 (1990).

The author proves an existence and uniqueness theorem for solutions of a Volterra stochastic integral equation \[ X_ t=X_{0,t}+\int_ 0^ t K_ 1(t,s,X_ s)dw_ s+\int_ 0^ t K_ 2(t,s,X_ s)ds.\tag{*} \] The first part of the paper gives the known construction of the multiple Itô-integral. This is used to obtain a representation of the first integral in (*) as a double stochastic integral relative to the Wiener process \((w_ t)\): \[ \int_ 0^ t K_ 1(t,s,X_ s)dw_ s=\int_ 0^ t \int_ s^ t L(r,s,X_ s)dw_ r dw_ s. \] Under natural (mainly Lipschitz) assumptions on the functions \(L\) and \(K_ 2\) it is then proved that (*) has a unique weak solution, which is a continuous semimartingale.

Reviewer: E.Dettweiler (Reutlingen)

MSC:

60H20	Stochastic integral equations
60H05	Stochastic integrals
45D05	Volterra integral equations

Keywords:

existence and uniqueness theorem; Volterra stochastic integral equation; multiple Itô-integral; weak solution; semimartingale