A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
The first part of this paper is devoted to establish an extended version of the Itô-Ventzell formula for anticipating processes that can be written as the sum of an absolutely continuous term plus an indefinite Skorokhod integral. The proof of this result is based on an Itô formula for nonadapted Hilbert-valued processes which generalizes the formula obtained by the reviewer and the second author [Probab. Theory Relat. Fields 78, 535-581 (1988; Zbl 0629.60061)] in the finite-dimensional case. A similar Itô-Ventzell formula for Stratonovich stochastic integrals is also presented. In the second part, these results are applied to show the existence and uniqueness of a solution for a Stratonovich stochastic differential equation of the type \[ X_ t=X_ 0+\int^{t}_{0}b(s,X_ s)ds+\int^{t}_{0}\sum^{k}_{i=1}\sigma_ i(s,X_ s)\circ dW^ i_ s, \] where \(X_ 0\) and \(\{\) b(t,x); \(t\geq 0\), \(x\in R^ d\}\) are random and may depend on the whole \(\sigma\)-field generated by the Brownian motion. The solution of the above equation is of the form \(X_ t=\phi_ t(Y_ t)\), where \(\phi_ t(x)\) denotes the flow associated to the same equation with \(b=0\) and \(Y_ t\) solves a certain ordinary differential equation with random coefficients.
Itô-Ventzell formula for anticipating processes
ordinary differential equation with random coefficients
scientific article; zbMATH DE number 4104121