Multidimensional SDE with anticipating initial process and reflection

TL;DR

This paper establishes strong solutions for multidimensional stochastic differential equations with reflection and anticipating initial conditions, using substitution formulas for Stratonovich integrals and continuity of functionals.

Contribution

It introduces a novel approach to handle anticipating initial variables in reflected SDEs through substitution formulas and continuity arguments.

Findings

Proved existence of strong solutions with reflecting boundaries and anticipating initial data.

Developed substitution formulas for Stratonovich integrals in this context.

Demonstrated continuity of solution functionals under the given conditions.

Abstract

In this paper, the strong solutions $ (X, L)$ of multidimensional stochastic differential equations with reflecting boundary and possible anticipating initial random variables is established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of $ (X, L)$.