Stochastic differential equations with path-independent solutions

TL;DR

This paper establishes a specific condition linking drift and diffusion coefficients in stochastic differential equations to ensure the existence of path-independent solutions, and extends this to more general solutions involving Ito-processes.

Contribution

It provides a new criterion for path-independent solutions of SDEs and generalizes to solutions involving Ito-processes with coefficients depending only on time.

Findings

Condition relating {} and {\u0016} for unique solutions

Solutions derived from solving two PDEs reduced to ODEs

Extension to solutions involving Ito-processes with time-dependent coefficients

Abstract

We present a condition for a stochastic differential equation dX_{t}={\mu}(t,X_{t})dt+{\sigma}(t,X_{t})dB_{t} to have a unique functional solution of the form Z(t,B_{t}). The condition expresses a relation between {\mu} and {\sigma}. A generalization concerns solutions of the form Z(t,Y_{t}), where Y_{t} is an Ito-process satisfying a stochastic differential equation with coefficients only depending on time, to be determined from {\mu} and {\sigma}. The solutions in question are obtained by solving a system of two partial differential equations, which may be reduced to two ordinary differential equations.