Risk-sensitive control for a class of non-linear systems and its financial applications

This thesis studies the risk-sensitive control problem for a class of non-linear stochastic systems and its financial applications. The nonlinearity is of the squareroot type, and is inspired by applications. The problems of optimal investment and consumption are also considered under several different assumptions on the stochastic interest rate and stochastic volatility. At the beginning, we systematically investigate the nonlinearity of risk-sensitive control problem. It consists of quadratic and square-root terms in the state. Such an optimal control problem can be solved in an explicit closed form by the completion of squares method. As an application of the risk-sensitive control in financial mathematics, the optimal investment problem will be described in the Chapter 4. A new interest rate, which follows the stochastic process with mixed CoxIngersoll-Ross (CIR) model and quadratic affine term structure model (QATSM) is introduced. Such an interest rate model admits an explicit price for the zerocoupon bond. In Chapter 5, we consider a portfolio optimization problem on an infinite time horizon. The stochastic interest rate consists not only of the quadratic terms, but also of the square-root terms. On the other hand, the double square root process is also introduced to establish the interest rate model. Under some sufficient conditions, the unique solution of the optimal investment problem is found in an explicit closed form. Furthermore, the optimal consumption problem is considered in Chapter 6 and 7. It can be solved in an explicit closed form via the methods of completion of squares and the change of measure. We provide a detailed discussion on the existence of the optimal trading strategies. Such trading strategies can be deduced for both finite and infinite time horizon cases.