Abstract
We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.
Highlights
The Skorokhod embedding problem (SEP) stimulates research in probability theory for over 50 years
The spirit of our approach is related to an approach to the original Skorokhod embedding problem by Bass [Bas83] that was later extended to the Brownian motion with linear drift in [AHI08]
As an immediate consequence of the previous lemma we observe the following fact: If we have a solution (Y, Z) ∈ S2(R) × H2(R) of equation (2.3), we obtain a weak solution to the Skorokhod embedding problem, i.e. a Gaussian process of the form (2.1), a starting point c, and an integrable random time such that our process stopped at this time possesses a given distribution
Summary
The Skorokhod embedding problem (SEP) stimulates research in probability theory for over 50 years. The spirit of our approach is related to an approach to the original Skorokhod embedding problem by Bass [Bas83] that was later extended to the Brownian motion with linear drift in [AHI08]. The procedure of both papers can be briefly summarized and divided into the following four steps. As an immediate consequence of the previous lemma we observe the following fact: If we have a solution (Y, Z) ∈ S2(R) × H2(R) of equation (2.3), we obtain a weak solution to the Skorokhod embedding problem, i.e. a Gaussian process of the form (2.1), a starting point c, and an integrable random time such that our process stopped at this time possesses a given distribution. We have to deal with a fully coupled FBSDE which in addition possesses a not globally Lipschitz continuous coefficient in the forward component
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