Backward stochastic differential equations with subdifferential operator and related variational inequalities
type: Item
instance of: scholarly article
title: Backward stochastic differential equations with subdifferential operator and related variational inequalities
zbMATH Open document ID: 0932.60070
DOI: 10.1016/S0304-4149(98)00030-1
author: Etienne Pardoux
author: Aurel Răşcanu
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: The main goal of the paper is to provide probabilistic representations for the viscosity solutions to parabolic variational inequalities on the whole space and also to the Dirichlet problem for an elliptic variational inequality, by means of the backward stochastic differential equations (BSDE's). To this end, the authors at first study a BSDE of the form \[ dY_t +F(t,Y_t,Z_t)dt\in \partial \varphi (Y_t)dt +Z_tdB_t,\quad 0 \leq t\leq \tau , \] with a terminal condition \(Y_{\tau}=\xi \) where \(B_t\) is a \(d\)-dimensional Brownian motion defined on a probability space \((\Omega , \mathcal F, P)\), \(\tau \) is an a.s. finite stopping time, the functions \(F:\Omega \times [0,\infty)\times R^k\times R^{r\times d} \to R^k\) and \(\varphi :R^k\to (-\infty , \infty ]\) satisfy suitable conditions, and \(\partial \) denotes the subdifferential. Existence, uniqueness and some other useful properties (including a priori estimates for a penalized equation) of solutions to the BSDE are proved in the first part of the paper. In the second part, the results are applied to obtain the probabilistic formulae of viscosity solutions to the variational inequalities.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35K85
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1364493
zbMATH Keywords: backward stochastic equations
zbMATH Keywords: subdifferential operators
zbMATH Keywords: variational inequalities
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00030-1
OpenAlex ID: W2093778414
description: scientific article; zbMATH DE number 1364493