Mathematical Sciences: Stochastic Integration and Differential Equations

首席研究员将与许多美国和外国科学家合作开展两个项目。这两个项目都与随机积分和随机微分方程有关。目前存在的许多类型的随机积分都是在特别的基础上发展起来的，以适应情况的特殊需要。主要研究人员将发展一个基于半鞅积分器的一般统一理论。希望在这个过程中，能够摒弃传统的渐近拟左连续假设。这将使这一理论更加适用。利息在金融理论中的一个特殊应用是布莱克-斯科尔斯模型的扩展。该项目的第二部分将处理定义具有预期被积的随机积分。在传统的随机积分定义中，被积函数(非预期的)被假定仅依赖于积分所针对的过程的过去。这使得积分的收敛性质易于证明。然而，在许多应用中，例如在物理中的Volterra方程中，被积函数不能被假定为非预期的。这项工作将使用首席调查员在过去开发的许多技术成果。

The principal investigator will work on two projects in collaboration with many US and foreign scientist. Both projects relate to stochastic integrals and stochastic differential equations. The many types of stochastic integrals that exist at the moment have developed on an ad hoc basis, to fit the particular needs of the situation. The principal investigator will develop a general unifying theory based on semi-martingale integrators. It is hoped that in the process, the traditional assumption of asymptotic quasi-left continuity can be dropped. This will make the theory much more applicable. A particular application of interest is in finance theory for the extension of the Black-Scholes model. The second part of the project will deal with defining stochastic integrals with anticipating integrands. In traditional definition of stochastic integrals, the integrand (non-anticipating) is assumed to be dependent only on the past of the process with respect to which the integral is carried out. This makes the convergence properties of the integrals easy to prove. However, in many applications, such as in Volterra equations in Physics, the integrand cannot be assumed to be non- anticipating. The work will use many technical results that the principal investigator has developed in the past.