在非Lipschitz係數條件及L′evy過程下隨機微分方程之特性

Some properties of stochastic differential equation on L′evy processes with non-Lipschitz coefficients

林敬堯(Ching-Yao Lin)

指導教授 ： 姜祖恕

國立臺灣大學/理學院/數學研究所/碩士(2007年)

https://doi.org/10.6342/NTU.2007.01618

摘要

我們在這篇論文主要探討的是Lévy 過程隨機微分方程解的三個性質。一個是在non-Lipschitz conditions條件下，強解的存在性，而Shizan Fang and Tusheng Zhang [1]已經討論過diffusion case。其二，我們關心在non-Lipschitz conditions條件下，解相對於初始值的相依性，而Shizan Fang and Tusheng Zhang [1]也已經討論過diffusion case。其三，我們比較兩條Lévy 過程隨機微分方程解的差異，僅在drift這一項不一樣，而Ikeda and Watanabe [3]也已經討論過diffusion case。

關鍵字

L′evy 過程 ； L′evy 型隨機微分方程 ； Comparison theorem

並列摘要

We shall discauss three properties of stochastic differential equation on Lèvy processes. One is the existence of the strong solutions of stochastic differential equation on Lèvy processes with non-Lipschitz conditions, the diffusion case with non-Lipschitz conditions have been studied by Shizan Fang and Tusheng Zhang [1]. Second subject is that we would generalize to the dependence of the solutions with respect to the initial values with Lévy processes.(The results are discussed by Shizan Fang and Tusheng Zhang [1] with diffusion cases) The third is the comparison theorem that says that there are two stochastic differential equation on Lèvy processes with different drift terms and we can compare the solutions by the two drift terms. The comparison theorem with diffusion case have studied by Ikeda and Watanabe [3].

並列關鍵字

L′evy processes ； stochastic differential equation on L′evy processes ； Comparison theorem

參考文獻

[1] Shizan Fang and Tusheng Zhang. (2005) A study of a class of stochastic differential equations with non-Lipschitzian coefficients.Probability Theory and Related Fields, vol.132, pp.356-390.

[2] Applebaum, David(2003) , L′evy Process and Stochastic Calculus , Cambridge University Press.

[3] Ikeda I.,Watanabe S.(1989) , Stochastic Differential equation and Diffusion Processes, North-Holland

Mathematical Library.

[4] J.M.Steele(2001) , Stochastic Calculus and Financial Applications , Springer.