Perturbed stochastic hereditary differential equations

• Published in: Stochastic analysis and applications Vol. 20; no. 3; pp. 567 - 589
• Main Authors: JANKOVIC, Svetlana, JOVANOVIC, Miljana
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 2002; Taylor & Francis
• Subjects: AMS Subject Classification: 60H10; Closeness in the (2m)th mean; Exact sciences and technology; Mathematics; Parametric perturbations; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic analysis; Stochastic differential equation; Burkholder Davis Gundy inequality; Itô equation; Differential equation; Gronwall Bellman inequality; Perturbation; Stochastic process; Parametric method; Hereditary system; Sufficient condition; Integrodifferential equation; Stochastic equation; Stochastic differential equation; Parametric perturbation
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1081/SAP-120004115

The aim of the present paper is to investigate the solution of the stochastic hereditary integrodifferential equation of Itô type with "small" perturbations which is at least a non-Markovian process, by comparing it with the solution of a simpler equation, i.e. of an unperturbed stochastic hereditary differential equation the solution of which is certainly a Markovian process. More precisely, we estimate the closeness of the (2m)th moments of these solutions. If the solutions are defined on unbounded intervals, we give sufficient conditions under which they are closed on intervals whose length goes to infinity.