Indefinite Risk-Sensitive Control for a Class of Nonlinear Systems

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.