隨機微分方程之數值研究

Abstract

對隨機微分方程而言，因為大多數的問題都很難找到精確解，所以數值解在這領域中非常的重要。而在隨機微分方程中，最簡單的情況是所有係數都符合全域Lipschitz條件且連續，而且有許多簡單且有名的數值方法都需要這一項假設，例如Euler-Maruyama法和Milstein法。不幸的是，許多重要的隨機微分方程模型都只符合部分區域Lipschitz條件且連續，而Euler-Maruyama法和Milstein法也已被證明在飄移項是超線性成長及只符合全域單邊Lipschitz條件且連續的情況下並不會強收斂至精確解。在近期，有人提出了調整的版本，名為drift-tamed Euler-Maruyama法及drift-tamed Milstein法。這兩種方法在係數只符合非全域的Lipschitz條件且連續的情況下也能夠強收斂到精確解。因為drift-tamed Milstein法會使用到擴散項的微分項，所以我們在這篇研究當係數只符合非全域Lipschitz條件且連續的情況下如何不使用微分項也能夠強收斂至精確解的數值方法。因此，我們在這篇論文中提出了一個新的數值方法: drift-tamed derivative-free Milstein法。我們證明出drift-tamed derivative-free Milstein法在隨機微分方程中的漂移項只符合超線性成長及全域單邊Lipschitz條件且連續的情況下將會一階強收斂至精確解。同時我們也對簡單的數值微分方程做數值模擬並比較drift-tamed derivative-free Milstein法及drift-tamed Milstein法的執行效率。

For stochastic differential equations (SDEs), the numerical solution is very important because most of SDEs are difficult to find the explicit solution. The simplest situation of SDEs is with globally Lipschitz continuous coefficients and many numerical methods such as Euler-Maruyama schemes and Milstein schemes require these assumptions to obtain the convergence. Unfortunately, many important SDE models are only with local Lipschitz continuous coefficients and the Euler-Maruyama and Milstein schemes were proven that thy fails to strongly converge to the exact solution of SDEs with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. Recently, in \cite{HJK12} and \cite{GW12}, the authors proposed tame versions, called drift-tamed Euler scheme and drift-tamed Milstein schemes with commutative noise. These two schemes are strongly convergent to the exact solution of SDEs with nonglobally Lipschitz continuous coefficients. In this thesis, we study how to simulate SDEs with nonglobally Lipschitz continuous coefficients without the derivatives of coefficients because the drift-tamed Milstein scheme should use the derivative of diffusion coefficients. Therefore, we propose the drift-tamed derivative-free Milstein scheme. We show that the proposed new scheme has strong convergence order one to the SDEs with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient and the numerical simulations of a simple SDE which are presented to confirm computational efficiency of the drift-tamed derivative-free Milstein scheme compared to the drift-tamed Milstein scheme.