On the φ-asymptotic behaviour of solutions of stochastic differential equations

Abstract

In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic dfferential equation dX(t) = g(X(t))dt +σ(X(t))dW(t), X(0) = b > 0, where g and σ are positive continuous functions and W is a Wiener process. Making use of the theory of pseudo-regularly varying (PRV) functions, we find conditions on g, σ and φ, under which φ(X(•)) can be approximated a.s. by φ(μ(•), where μ is the solution of the ordinary differential equation dμ(t) = g(μ(t))dt, μ(0) = b. As an application of these results we discuss the problem of φ-asymptotic equivalence for solutions of stochastic differential equations.