Stochastic Optimization Of Risk Measures Based On Relative Entropy

How to control the risk has been an important point in financial mathematic.And stochastic control theory is one of the powerful tool in financial research.This paper combines entropic risk measure and stochastic control in order to find the optimal strategy to avoid risk In addition,we use numerical simulation to verify the theoretical resultsThis paper is mainly divided into five chapters:In the first part,we introduce the relevant background about entropic risk measure and the development of stochastic differential games and relative entropy.Then the main results are given.What shown in the second is the basic knowledge related to the main study:entropic risk measures,stochastic processes,stochastic calculus and stochastic controlsChapter 3 is about our mainly conclusion.At the beginning,we introduce a compound investment process that combines the risky product like stock with fixed income instruments like bond.Based on the given process,we use convex entropic risk measure and coherent entropic risk measure to describe risk.For convex entropic risk measure,we study the stochastic control problem and ergodicity control problem and find the optimal control.Then,we illustrate some connections between them.For coherent entropic risk measure,we prove the optimal control of the stochastic control problem.We put the numerical simulations in the fourth part.Firstly,we draw pictures about different stochastic processes with optimal controls.Then,we present some numerical results about the given processes.Finally,comparing theoretical and simulation results,we come to the conclusion.In the final chapter,we summarize the paper and put forward further study.