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We consider a class of nonlinear stochastic systems with  square-root  nonlinearities appearing in the diffusion terms. The optimal control problems with  indefinite  quadratic criteria in both finite and infinite horizon are formulated and solved in an explicit closed form. It turns out that  all  optimal controls are of an  affine  state-feedback form, despite the fact that the system is nonlinear. We use the method of completion of squares and  new  types of Riccati differential and algebraic equations to find the solutions. An application to the problem of optimal investment in a market with a stochastic interest rate is given.

Optimal regulators for a class of nonlinear stochastic systems