Abstract: Under the assumption of exponent utility, this paper treats optimal investment policies of a firm with a random risk process. The firm’s choice on investment can be savings, loan and trading of risky asset. The principal achievements are as follows: According to the fact that the interest rate on loan is higher than the one on savings, we acquire the optimal policies that maximize the expectation of terminal utility. Furthermore, we present an example of application in the end.

Key words: stochastic control, portfolio theory, controlled diffusion process, rate difference on savings and loan, Hamiltion Jocobi Bellman equation