We consider the optimal control problem with an indefinite generalised risk-sensitive criterion, and a class of stochastic control systems with multiplicative noise that have a quadratic type nonlinearity in the state and control variables. All solutions to such an optimal control problem are obtained in an explicit closed-form as affine state-feedback controls with possibly random gains. An application to the optimal investment problem is also given.

We consider the problem of optimally tracking a financial benchmark in a market with random and possibly unbounded coefficients. This is an example of a stochastic linear-quadratic control problem with unbounded system and cost functional coefficients. We obtain the unique solution in an explicit closed-form as an affine state-feedback control. The coefficients of the control law are given in terms of two linear backward stochastic differential equations with unbounded coefficients, the solvability of which is also proved.