Optimal growth in continuous-time with credit risk

We consider asset allocation strategies for the case where an investor can allocate his wealth dynamically between a risky stock, whose price evolves according to a geometric Brownian motion, and a risky bond, whose price is subject to negative jumps due to its credit risk and therefore has discontinuous sample paths. We derive optimal policies for a number of objectives related to growth. In particular, we obtain the policy that minimizes the expected time to reach a given target value of wealth in an exact explicit form. We also show that this policy is exactly equivalent to the policy that is optimal for maximizing logarithmic utility of wealth and, hence, the expected average rate at which wealth grows, as well as to the policy that maximizes the actual asymptotic rate at which wealth grows. Our results generalize and unify results obtained previously for cases where the bond was risk-free in both continuous- and discrete-time.