An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

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 Equations (FBS-DEs), 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.MSC 2010: Primary: 60G40, 60H30; Secondary: 93E20.

Solvability of coupled FBSDEs with diagonally quadratic generators

“…These systems naturally appear in numerous areas of applied mathematics including stochastic control and mathematical finance. Moreover, they provide solutions or viscosity solutions to various types of parabolic partial differential equations, and as shown recently by Fromm et al [10], they can be used in the study of the Skorokhod embedding problem. In the Markovian setting, coupled FBSDEs are linked to parabolic PDEs, the solutions of which provide existence for the FBSDE, see Ma et al [19].…”

Strong solutions of forward-backward stochastic differential equations with measurable coefficients

“…These equations also provide a probabilistic approach to deal with quasilinear parabolic partial differential questions via the nonlinear Feynman-Kac formula initiated by Pardoux and Peng [30] and further developed notably in [3,8,15,31]. As a result, FBSDEs have received a lot of attention in the applied probability community and appear in various applications, we refer for instance to [6,[10][11][12]27] and the references therein. When the coefficients of the equations, i.e.…”

Multidimensional Markovian FBSDEs with super-quadratic growth

“…As shown for instance in Ma et al [23] and Cheridito and Nam [5], in the Markovian case, FBSDEs can be linked to parabolic PDEs. More recently Fromm et al [14] proved that FBSDEs can be used in the study of the Skorokhod embedding problem. BSDEs and FBSDEs with Lipschitz continuous generators are well understood, we refer to El Karoui et al [11] and Delarue [7].…”