Optimal investment in a multi-asset market with borrowing and unbounded random coefficients

We consider the problem of optimal investment in a multi-asset market with borrowing, unbounded random coefficients, and the power utility from terminal wealth. The resulting optimization problem, due to the higher interest rate for borrowing than for lending, is a multi-input stochastic optimal control problem with a nonlinear system dynamics and unbounded random coefficients. A certain 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, the gain of which can have up to five different regimes.