### Article

## Asymptotic Behavior of the Solution to a Linear Stochastic Differential Equation and Almost Sure Optimality for a Controlled Stochastic Process

The asymptotic behavior of a stochastic process satisfying a linear stochastic differential equation is analyzed. More specifically, the problem is solved of finding a normalizing function such that the normalized process tends to zero with probabilty 1. The explicit expression found for the function involves the parameters of the perturbing process, and the function itself has a simple interpretation. The solution of the indicated probles makes it possible to considerably improve almost sure optimality results for a stochastic linear regulator on an infinite time interval.