A converse comparison theorem for anticipated BSDEs and related non-linear expectations
type: Item
instance of: scholarly article
title: A converse comparison theorem for anticipated BSDEs and related non-linear expectations
zbMATH Open document ID: 1264.60043
author: Zhe Yang
author: Robert J. Elliott
published in: Stochastic Processes and their Applications
publication date: 2013-01-24T00:00:00Z
review text: The authors consider the anticipated backward stochastic differential equation \[ \begin{aligned} -dY_t& =f(t,Y_t,Z_t,Y_{t+\delta(t)})\,dt-Z_t\,dW_t,\quad t\in[0,T] \\ Y_t& =\xi_t t\in[T,T+K], \end{aligned} \] where \(\delta\) is a continuous \(\mathbb R_+\)-valued function satisfying some technical assumptions, \(Y\) is a real process, \(Z\) is \(\mathbb R^d\)-valued, \(W\) is a Brownian motion in \(\mathbb R^d\), \(\xi\) is a terminal condition process and \(f\) is, roughly speaking, a random variable with a Lipschitz-like dependence on the last three variables (the domain for the last variable being \(L^2(\mathcal F_r)\)). The underlying filtration is generated by \(W\). The paper contains two main results. The first one asserts the existence and uniqueness of solutions to the above equation, whereby \(T\) is admitted to be a finite stopping time rather than a positive constant, provided that \(f\) satisfies an eventually vanishing driver condition. For the second result, the notions of \((f,\delta)\)-expectation and conditional \((f,\delta)\)-expectation are introduced, and it is proved that such expectations exist under reasonable conditions. The \((f,\delta)\)-expectation assigns to an adapted process \(X\) the process \(\mathcal E^\delta_f[X](s)=\operatorname{E}[Y_s]\), where \(Y\) is the solution of the equation \[ \begin{aligned} Y_r &=X_t+\int_r^tf(s,Y_s,Z_s,Y_{s+\delta(s)})\,ds- \int_r^tZ_s\,dW_s,\quad r\in[0,t] \\ Y_r &=X_r\,\, r\in[t,t+K]. \end{aligned} \] The converse comparison theorem then asserts that, if two equations with drivers \(f_1\) and \(f_2\) and with the same anticipating function \(\delta\) are considered, then \(f_1(t,y,z,\theta_{t+\delta(t)})\geq f_2(t,y,z,\theta_{t+\delta(t)})\) holds for every \((t,y,z,\theta)\) if and only if \(\mathcal E^\delta_{f_1}[X](0)\geq\mathcal E^\delta_{f_2}[X](0)\) holds for every \(X\).
reviewed by: Martin Ondreját
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 6128613
zbMATH Keywords: anticipated BSDEs
zbMATH Keywords: stopping times
zbMATH Keywords: \((f, \delta )\)-expectations
zbMATH Keywords: converse comparison theorem
MaRDI profile type: Publication
full work available at URL: j.spa.2012.09.006
OpenAlex ID: W2049689185
DOI: 10.1016/J.SPA.2012.09.006
description: scientific article; zbMATH DE number 6128613