Stochastic differential equations with path-independent solutions
Probability
2012-09-05 v1
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.
Cite
@article{arxiv.1209.0623,
title = {Stochastic differential equations with path-independent solutions},
author = {Imme van den Berg},
journal= {arXiv preprint arXiv:1209.0623},
year = {2012}
}