General Stochastic Integral and It� Formula with Application to Stochastic Differential Equations and Mathematical Finance

Abstract

A general stochastic integration theory for adapted and instantly independent stochastic processes arises when we consider anticipative stochastic differential equations. In Part I of this thesis, we conduct a deeper research on the general stochastic integral introduced by W. Ayed and H.-H. Kuo in 2008. We provide a rigorous mathematical framework for the integral in Chapter 2, and prove that the integral is well-defined. Then a general Itô formula is given. In Chapter 3, we present an intrinsic property, near-martingale property, of the general stochastic integral, and Doob-Meyer's decomposition for near-submartigales. We apply the new stochastic integration theory to several kinds of anticipative stochastic differential equations and discuss their properties in Chapter 4. In Chapter 5, we apply our results to a general Black-Scholes model with anticipative initial conditions to obtain the properties of the market with inside information and the pricing strategy, which can help us better understand inside trading in mathematical finance. In dynamical systems, small random noises can make transitions between equilibriums happen. This kind of transitions happens rarely but can have crucial impact on the system, which leads to the numerical study of it. In Part II of this thesis, we emphasize on the temporal minimum action method introduced by X. Wan, using optimal linear time scaling and finite element approximation to find the most possible transition paths, also known as the minimal action paths, numerically. We show the method and its numerical analysis in Chapter 7. In Chapter 8, we apply this method to a stochastic dynamic system with time delay using penalty method.