PRV property and the asymptotic behaviour of solutions of stochastic differential equations

• Дата: 2005
• Автори: Buldygin, V.V., Klesov, O.I., Steinebach, J.G.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2005
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/4424
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: PRV property and the asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 42–57. — Бібліогр.: 17 назв.— англ.

We consider the a.s. asymptotic behaviour of a solution of the stochastic differential equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW(t), with X(0) ≡ b > 0, where g(.) and σ(.) are positive continuous functions and W(.) is the standard Wiener process. By applying the theory of PRV and PMPV functions, we find the conditions on g(.) and σ(.), under which X(.) resp. f(X(.)) may be approximated a.s. on {X(t)→∞} by μ(.) resp. f(μ(.)), where μ( ) is a solution of the deterministic differential equation dμ(t) = g(μ(t))dt with μ(0) = b, and f(.) is a strictly increasing function. Moreover, we consider the asymptotic behaviour of generalized renewal processes connected with this SDE.