Topics in risk-sensitive stochastic control

This thesis consists of three topics whose over-arching theme is based on risk sensitive stochastic control. In the �first topic (chapter 2), we study a problem on benchmark out-performance. We model this as a zero-sum risk-sensitive stochastic game between an investor who as a player wants to maximize the risk-sensitive criterion while the other player ( a stochastic benchmark) tries to minimize this maximum risk-sensitive criterion. We obtain an explicit expression for the strategies for both these two players. In the second topic (chapter 3), we consider a finite horizon risk-sensitive asset management problem. We study it in the context of a zero-sum stochastic game between an investor and the second player called the "market world" which provides a probability measure. Via this game, we connect two (somewhat) disparate areas in stochastics; namely, stochastic stability and risk-sensitive stochastic control in mathematical finance. The connection is through the Follmer-Schweizer minimal martingale measure. We discuss the impact of this measure on the investor's optimal strategy. In the third topic (chapter 4), we study the sufficient stochastic maximum principle of semi-Markov modulated jump diffusion. We study its application in the context of a quadratic loss minimization problem. We also study the finite-horizon risk-sensitive optimization in relation to the underlying sufficient stochastic maximum principle of a semi-markov modulated diffusion.