Stochastic differential equations in mathematical finance and game theory

随机微分方程架起了从数学金融学到概率论的桥梁，在数学和金融学中具有重要意义。SDE是涉及随机过程的微分方程，可用于对股票价格等过程进行建模。拟议研究的目的是解决一些金融和博弈论问题，并为相关的SDE开发数学工具。*我将考虑的问题从期权估值的效用最大化到博弈论中的均衡问题，所有这些都着眼于相关SDE的技术应用。在效用最大化中，一个人寻求最大化一个可以投资于某个金融市场的代理人的效用，我将研究市场上的瞬时风险是无限的情况。与效用最大化有关的一个问题是使用所谓的无差别估值对期权进行估值，其基本思想是找到一个价值，使代理人在购买和不购买期权之间没有差别。在这里，我将研究由布朗运动驱动的模型中的收敛和逼近问题。该项目的博弈论部分涉及连续时间的游戏，当一个玩家只能不完全地监控其他玩家的行为时，从某种意义上说，观测被布朗运动的噪声扭曲。我将使用随机方法来解决这些模型中的均衡问题。

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.************