Viability of moving sets for stochastic differential equation.
type: Item
instance of: scholarly article
title: Viability of moving sets for stochastic differential equation.
zbMATH Open document ID: 1037.60055
author: Marc Quincampoix
author: Catherine Rainer
author: Aurel Răşcanu
author: Rainer Buckdahn
published in: Advances in Differential Equations
publication date: 2004-03-02T00:00:00Z
review text: The paper provides a necessary and sufficient characterization that guarantees the solution \( X_s^{t,x} , s \in [0,T]\), of a given forward stochastic differential equation stays (a.s. and \( \forall s \in [t,T] \)) in a prescribed set of closed, time dependent constraint \( K_0 (t), t \in [0,T]\) for all \( t \in [0,T] \) and \( x \in K_0 (t)\). This characterization is given in terms of viscosity super- and subsolution of some suitable partial differentiable equations (PDE). The above property, called viability, is stated for both forward and backward stochastic differential equations (BSDE) and is equivalent to the fact that the square of the distance function is a viscosity supersolution (subsolution, respectively) of the PDE for the forward SDE (BSDE, respectively) [see, e.g., \textit{R. Buckdahn}, \textit{M. Quincampoix} and \textit{A. Răşcanu}, Probab. Theory Relat. Fields 116, 485--504 (2000; Zbl 0969.60061)].
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 1977261
zbMATH Keywords: forward stochastic differential equations
zbMATH Keywords: semilinear parabolic partial differential equations
zbMATH Keywords: viability
zbMATH Keywords: viscosity subsolution
zbMATH Keywords: closed and time depending constraint
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: viscosity supersolution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1977261