Forward-backward stochastic differential equations and quasilinear parabolic PDEs
type: Item
instance of: scholarly article
title: Forward-backward stochastic differential equations and quasilinear parabolic PDEs
zbMATH Open document ID: 0943.60057
author: Etienne Pardoux
author: Shanjian Tang
publication date: 2000-08-30T00:00:00Z
review text: This paper considers a forward-backward stochastic differential equation (FBSDE) which is a system of the type \[ \begin{aligned} X(t) &= x+\int_0^t f(s,X(s),Y(s),Z(s)) ds +\int_0^t \sigma(s,X(s),Y(s),Z(s)) dB(s),\\ Y(t)&= h(X(T))+\int_t^T g(s,X(s),Y(s),Z(s)) ds -\int_0^t Z(s) dB(s) \end{aligned} \] for \(t\in[0,T]\). Under some technical conditions like Lipschitz, linear growth, or measurability and a simple and very natural monotonicity condition on \(f\) and \(g\), the authors prove results on existence and uniqueness of a solution \(\{(X(s),Y(s),Z(s)):s\in[0,T]\}\). Furthermore, they establish a priori estimates and continuous dependence upon a parameter. Finally, they connect FBSDEs to systems of quasilinear parabolic PDEs of second order. Using their purely probabilistic approach, they prove the existence of viscosity solutions of the PDE.
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 93E03
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1332833
zbMATH Keywords: forward-backward stochastic differential equations
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: monotonicity condition
zbMATH Keywords: viscosity solution
MaRDI profile type: Publication
full work available at URL: s004409970001
OpenAlex ID: W2081458347
DOI: 10.1007/S004409970001
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1332833