Optimal Investment in a Market with Borrowing, Unbounded Random Coefficients, and a Random Time-Horizon

We consider the problem of optimal investment in a market with borrowing, unbounded random coefficients, and the power utility from terminal wealth with a random time-horizon. The resulting optimization problem, due to the higher interest rate for borrowing than for lending, is an optimal stochastic control problem with a nonlinear system dynamics and unbounded random coefficients. A piece-wise completion of squares method and a linear backward stochastic differential equation are used to find an explicit closed-form solution as a linear state-feedback control.