Stochastic differential equations in mathematical finance and game theory

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Building a bridge from mathematical finance to probability theory, stochastic differential equations (SDEs) are of great importance in mathematics and finance. SDEs are differential equations involving stochastic processes and can be used to model processes such as prices of stocks. The aim of the proposed research is to address some financial and game-theoretical problems and to develop mathematical tools for the related SDEs.****The problems I will consider go from utility maximization over option valuation to equilibrium questions in game theory, all with an eye to the application of techniques for the related SDEs. In utility maximization, where one seeks at maximizing utility of an agent who can invest on some financial market, I will study a situation where the instantaneous risk on the market is unbounded. A problem related to utility maximization is the valuation of options by using the so-called indifference valuation, whose basic idea is to find a value which makes the agent indifferent between buying and not buying the option. Here I will examine convergence and approximation questions in a model driven by Brownian motions. The game-theoretical part of the project concerns games in continuous time when a player can only imperfectly monitor the other players' actions, in the sense that observations are distorted by noise of Brownian motions. I will use stochastic methods to tackle equilibrium questions in such models.************