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Stochastic optimal control theory: new applications to finance and insurance

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2017
Akdoğan, Emre
In this study, the literature, recent developments and new achievements in stochastic optimal control theory are studied. Stochastic optimal control theory is an important direction of mathematical optimization for deriving control policies subject to timedependent processes whose dynamics follow stochastic differential equations. In this study, this methodology is used to deal with those infinite-dimensional optimization programs for problems from finance and insurance that are indeed motivated by the real life. Stochastic optimal control problems can be further treated and solved along different avenues, two of the most important ones of being (i) Pontryagin’s maximum principle together with stochastic adjoint equations (within both necessary and sufficient optimality conditions), and (ii) Dynamic Programming principle together with Hamilton-Jacobi-Bellman (HJB) equations (within necessary and sufficient versions, e.g., a verification analysis). Here we introduce the needed instruments from economics and from Ito calculus, such as the theory of jump-diffusion and Ĺevy processes. We then present Dynamic Programing Principle, HJB Equations, Verification Theorem, Sufficient Maximum Principle for stochastic optimal control of diffusions and jump diffusions, and we state some connections and differences between Maximum Principle and the Dynamic Programing Principle. As financial applications, we investigate mean-variance portfolio selection problem and Merton optimal portfolio and consumption problem. From actuarial sciences, we study the optimal investment and liability ratio problem for an insurer and the problem of purchase of optimal life insurance, optimal investment and consumption of a wage-earner within a market of severallife-insuranceproviders,respectively. Inourexamples,weshallrefertovarious utility functions such as exponential, power and logarithmic ones, and to different parameters of risk averseness. We provide some graphical representations of the optimal solutions to illustrate the theoretical results. The thesis ends with a conclusion and an outlook to future studies, addressing elements of information, memory and stochastic robust optimal control problems.