Forward-backward Stochastic Equations Research Articles

Forward-backward stochastic equations: a functional fixed point approach

We introduce forward-backward stochastic equations (FBSEs) that incorporate fully-coupled forward-backward structure into backward stochastic equations (BSEs) introduced in Cheridito and Nam (Ann. Probab. 45(6A):3795–3828, 2017). Such a system generalizes the classical backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs) in previous literature. We transform an FBSE into a fixed point equation on the space of random variables and then apply general fixed point theorems to derive the existence and/or uniqueness of a solution. As a result, we obtain novel existence and/or uniqueness results for fully-coupled FBSDEs with functional drivers, which are either Lipschitz or non-Lipschitz.

Forward-Backward Stochastic Equations Associated with Systems of Quasilinear Parabolic Equations and Comparison Theorems

1,x ∈ R d . Our approach is based on the possibility to reduce the original quasilinear parabolic system to a quasilinear parabolic equation in an alternative phase space and derive forward-backward stochastic differential equations associated with it. This reduction allows us to prove some comparison theorems for BSDEs, and, as a result, to construct a probabilistic representation of a viscosity solution of the original Cauchy problem. Bibliography: 16 titles.