Forward-backward stochastic differential equations and their applications
type: Item
instance of: scholarly article
title: Forward-backward stochastic differential equations and their applications
zbMATH Open document ID: 0927.60004
DOI: 10.1007/978-3-540-48831-6
author: Jin Ma
author: Jiongmin Yong
published in: Lecture Notes in Mathematics
publication date: 1999-08-12T00:00:00Z
review text: The authors being the intellectual fathers of the forward-backward stochastic differential equations (FBSDE) in their general form, give with the present book a nice introduction to this theory. The eaerliest version of an FBSDE was introduced by \textit{J.-M. Bismut} [J. Math. Anal. Appl. 44, 383-404 (1973; Zbl 0276.93060)], in a decoupled, linear form of a usual (forward) stochastic differential equation (SDE) and a linear backward stochastic differential equation (BSDE) in order to describe the adjoint equation in the Pontryagin-type maximum principle in the stochastic control theory. The real breakthrough in the theory of BSDEs was reached only more than fifteen years later, by \textit{E. Pardoux} and \textit{S. G. Peng} with the proof of existence and uniqueness of a solution of the general, nonlinear BSDE [Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)] and by Peng (1992) who showed that the BSDE generalizes the Feynman-Kac formula and describes the solution of semilinear parabolic partial differential equations (PDE). Their discovery initiated an extensive study of BSDEs, motivated by a manifold of applications in stochastic control, theory of PDEs and in finance. These applications have led finally to FBSDEs, a coupled system of SDEs and BSDEs. After a first work by \textit{F. Antonelli} [Ann. Appl. Probab. 3, No. 3, 777-793 (1993; Zbl 0780.60058)] who studied FBSDEs over a sufficiently small time interval which allows the application of the the point argument, the authors of the present book have started a systematic study which has led to two methods, the four-step-scheme [\textit{J. Ma}, \textit{Ph. Protter} and \textit{J. Yong}, Probab. Theory Relat. Fields 98, No. 3, 339-359 (1994; Zbl 0794.60056)] and a method using a monotonicity condition [\textit{Y. Hu} and \textit{S. Peng}, ibid. 103, No. 2, 273-283 (1995; Zbl 0831.60065)], and it has also given rise to other subjects that are interesting in their own rights. The book is organised as f…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1324009
zbMATH Keywords: forward-backward stochastic differential equation
zbMATH Keywords: comparison theorem
zbMATH Keywords: four-step-scheme
zbMATH Keywords: continuation method
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Black-Scholes formula
zbMATH Keywords: American options
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-48831-6
OpenAlex ID: W659339155
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description: scientific article; zbMATH DE number 1324009