The behavior of solutions of stochastic differential inequalities
type: Item
instance of: scholarly article
title: The behavior of solutions of stochastic differential inequalities
zbMATH Open document ID: 0844.60031
DOI: 10.1007/BF01246336
publication date: 1996-08-27T00:00:00Z
review text: Let \(X\) and \(Z\) be \(R^d\)-valued solutions of the stochastic differential inequalities \[ dX_t \leq a(t,X_t)dt + \sigma(t,X_t)dW_t \quad \text{and} \quad b(t,Z_t)dt + \sigma(t,Z_t)dW_t \leq dZ_t, \] respectively, with a fixed \(R^m\)-valued Wiener process \(W\). The authors give conditions on \(a\), \(b\) and \(\sigma\) under which the relation \(X_0 \leq Z_0\) of the initial values leads to the same relation between the solutions with probability one. A detailed discussion whether in general their conditions can be weakened or not is given. Inspired by the deterministic theory they introduce the notions of maximal/mimimal solution of a stochastic differential inequality. Using the comparison result a sufficient condition for existence of such solutions as well as some Gronwall-type estimates are derived.
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 822729
zbMATH Keywords: stochastic differential inequalities
zbMATH Keywords: comparison result
zbMATH Keywords: Gronwall-type estimates
author: Sigurd Assing
author: Ralf Manthey
reviewed by: B. G. Pachpatte
MaRDI profile type: Publication
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cites work: On the uniqueness of solutions of stochastic differential equations
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 822729