Optimal investment in a market with borrowing, quadratic-affine interest rates, and Heston stochastic volatility

We consider the problem of optimal investment in an incomplete market with borrowing, random and possibly unbounded coefficients, and the power utility from terminal wealth. We use the Heston model for stochastic volatility, and the quadratic-affine model for interest rates. The resulting problem is an example of optimal stochastic control problem with a nonlinear system dynamics which is due to borrowing, the square-root non-linearity of Heston model, and the quadratic non-linearity of the interest rates. Explicit closed-form solution is obtained by a certain piece-wise completion of squares method. The resulting optimal control law is of linear state-feedback form the gain of which can be in up to three different regimes.