Asymptotic properties of processes that appear in optimal investment problems

Abstract

Asymptotic behavior of random process take one of the main part in many sections of insurance and financial mathematics. In this paper we study properties some processes which have applications in optimal investment problems. We study the limit behavior of a random process, which is modeled, for example, as $$ dX(t) = [\eta+m\varphi(t)]dt + \sigma_1 \theta_1(t)dW_1(t)+\sigma_2\theta_2(t) dW_2(t),$$ where $\eta, \mu, \sigma_{i}, i=1,2,$ are positive numbers, $\varphi$ is a continuous positive function, $\theta_i, i=1,2$ are continuous functions, Wiener processes $W_1$ and $W_2$ aren't necessary independent. We only consider those processes $X$ for which $\underset{t\to\infty} \lim X(t)=\infty$ and find conditions under which random process $(X(t), t\geq 0)$ can be approximated by the solution of some ordinary differential equation. Examples are given to prove the significance of new theorems.