The Skorokhod embedding problem for inhomogeneous diffusions

2018/10/11 by Stefan Ankirchner, Ankirchner, Stefan, Stefan Engelhardt +5
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1810.05098

39 pages, 2 pictures, To appear in Annales de l'Institut Henri Poincare (B) Probability and Statistics

arxiv created 2019/06/18 · arxiv updated 2019/06/19

Abstract

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form d A_t =μ(t, A_t) d t + σ(t, A_t) d W_t. We provide sufficient conditions guaranteeing that for a given probability measure ν on ℝ there exists a bounded stopping time τ and a real a such that the solution (A_t) of the SDE with initial value a satisfies A_τ∼ ν. We hereby distinguish the cases where (A_t) is a solution of the SDE in a weak or strong sense. Our construction of embedding stopping times is based on a solution of a fully coupled forward-backward SDE. We use the so-called method of decoupling fields for verifying that the FBSDE has a unique solution. Finally, we sketch an algorithm for putting our theoretical construction into practice and illustrate it with a numerical experiment.

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