Solution and reflected solutions of forward SDEs with generalized Wiener functional approach (Q2722262)

This paper deals with the forward stochastic differential equation NEWLINE\[NEWLINE Y(t)=\alpha(t)+\int_{0}^{t}b(s,Y(s)) ds+\int_{0}^{t}\sigma(s,Y(s)) d W(s), NEWLINE\]NEWLINE where \(\alpha(t)\) is an adapted generalized Wiener functional and \(W(t)\) is a standard Wiener process. Existence and uniqueness of the solution to the equation are investigated. The objective is to look for an adapted generalized Wiener process \(Y(t)\) satisfying the above-mentioned equation. The second objective of the paper is to show the existence and uniqueness of a reflected solution for this equation with \(\alpha\in D_{2,-k}\) (Sobolev space) and \({\mathcal F}_{0}\)-measurable. The existence of the reflected solution is obtained by introducing a maximal monotone operator on the Sobolev space \(D_{2,-k}.\) Under Lipschitz condition the existence and uniqueness of reflected solutions for the equation are shown by using a penalization argument.