Stochastic differential equations with path-independent solutions
Abstract
We present a condition for a stochastic differential equation dXt=μ(t,Xt)dt+σ(t,Xt)dBt to have a unique functional solution of the form Z(t,Bt). The condition expresses a relation between μ and σ. A generalization concerns solutions of the form Z(t,Yt), where Yt is an Ito-process satisfying a stochastic differential equation with coefficients only depending on time, to be determined from μ and σ. The solutions in question are obtained by solving a system of two partial differential equations, which may be reduced to two ordinary differential equations.
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