Optimal investment in a market with borrowing and unbounded random coefficients

We consider the problem of optimal investment in a market with borrowing and random coefficients. We assume that the bond interest rate, the borrowing interest rate, the appreciation rate and the volatility of stock, are random and possibly unbounded. Due to the possibility of borrowing, the formulated optimal investment problem is an optimal stochastic control problem with nonlinear system dynamics and possibly unbounded coefficients. Explicit closed-form solutions in terms of a linear backward stochastic differential equation are obtained for the power and logarithmic utility from terminal wealth. The optimal controls turn out to be of a linear state-feedback form.