Optimal investment in a market with borrowing, unbounded random coefficients, and a combined interest rate model

We consider the problem of optimal investment in a market with a higher interest rate for borrowing than for lending, and the power utility form terminal wealth. The market coefficients are random and unbounded in general. A certain combined interest rate model is introduced that contains known previous models, such as the Hull-White model and the quadratic-affine model, as special cases. The solution method is based on linear backward stochastic differential equation with possibly unbounded coefficients, and the minimization of a piece-wise quadratic function. An explicit closed-form solution is obtained as a linear state-feedback control with random coefficients