Applications of Discrete and Continuous Time Stochastic Control to the Models of Economic Dynamics and Finance

本研究涉及离散和连续时间随机控制在经济增长、数学金融和数学保险等数学模型中的新应用，以及为此目的开发新的非传统控制技术。 在离散时间控制领域，我们将研究和分析凸值动力系统，类似于用于分析经济增长的数学模型，我们将使用这些结果分析复杂的离散时间金融市场模型。 在我们研究的第二部分中，我们将研究扩散过程的最优控制在金融和保险优化模型中的新应用。 我们将研究适合于这些问题的混合正则奇异随机和脉冲随机控制。 我们还打算研究一类非线性偏微分（或积分微分）方程的梯度约束，形成一个适当的分析框架，这些models.The这些活动的重要性是在发展的新方法的应用随机控制在数学金融和保险。 研究目标的成功完成将增加我们对金融和保险优化模型的政策结构的理解，并将使人们更好地了解上市金融公司应该坚持的最优股息分配政策的性质。 更重要的是，它将使人们了解这样一个机构应该遵循的风险控制政策。

This research deals with new applications of discrete and continuous time stochastic control to the mathematical models of economic growth, mathematical finance and mathematical insurance, as well as developing novel non-traditional control techniques useful for this purpose. In the area of discrete time control, we will study and analyze convex-valued dynamical systems, similar to the ones used for the analysis of the mathematical models of economic growth and we will use these results in analysis of complex discrete time financial market models. In the second part of our research, we will study new applications of the optimal control of diffusion processes to optimization models in finance and insurance. We will investigate mixed regular-singular stochastic and impulse stochastic control appropriate for those problems. We also intend to study the class of nonlinear partial differential (or integral-differential) equations with gradient constraints that form an appropriate analytical framework for these models.The importance of these activities is in the development of new methodology of applications of stochastic control in mathematical finance and insurance. Successful completion of the research objectives will increase our understanding of the structure of the policy the financial and insurance optimization models and will allow one to get a better insight into the nature of the optimal dividend distribution policy to which a publicly traded financial corporation should adhere. More importantly, it will provide understanding of the risk control policy such an institution should follow.