Forward-backward stochastic differential equations and quasilinear parabolic PDEs

Abstract

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward-backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.

Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients

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Abstract.We consider a system of fully coupled forward-backward stochastic differential equations.First we generalize the results of Pardoux-Tang [7] concerning the regularity of the solutions with respect to initial conditions. Then, we prove that in some particular cases this system leads to a probabilistic representation of solutions of a second-order PDE whose second order coefficients depend on the gradient of the solution.

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“…This has led to a huge amount of work. We refer to Pardoux-Tang [7], Ma-Yong [5] and the references therein for more precise results on the subject.…”

“…This case is partially studied in [7] concerning existence and uniqueness of the solution of the FBSDE but nothing else is done for that case. In the next section, we thus study a general FBSDE with σ that depends also on z.…”

“…Some conditions for existence and uniqueness of solutions of the FBSDE (2) have been given in [7]. We will study here a Picard iteration that leads to other conditions for existence and uniqueness of the solutions as a by-product but the purpose of this subsection is the final Theorem 2.7.…”

On the homotopy analysis method for backward/forward-backward stochastic differential equations

“…In 1991, Peng [2,3] further found the relationship between FBSDEs and a kind of second-order quasi-linear parabolic partial differential equation. Since then, extensive researches of the BSDEs and FBSDEs have been done [4][5][6][7][8][9]. Besides, many interesting properties and applications of BSDEs were presented [10].…”