倒向随机微分方程和Malliavin导数

发布时间：2018-03-28 01:21

  本文选题：倒向随机微分方程　切入点：比较定理　出处：《中国科学技术大学》2017年硕士论文

【摘要】：本文以 Shi[1]的"Backward Stochastic Differential Equations in Finance" 为基础,简述倒向随机微分方程(BSDE)相关基础知识及应用。Liang等人[2]在"Backward Stochastic Dynamics on a filtered probability Space" 中介绍了 BSDE可以重新表示为某一轨道空间上的一般微分方程,Shi将其思路应用到如下类型的倒向随机微分方程dYtj=-fj(t,Yt,L(M)t)dt + dMtj,YTj= ξj,其中L是将M映射到适应过程L(M)的给定非线性映射,修正项M是需要确定的鞅。Liang等人给出了某些条件下L2解的存在唯一性,Shi将其推广到Lp解并证明了这些条件下Lp解的存在唯一性。更进一步,Shi给出了这类倒向随机微分方程关于L2解的比较定理。最后,基于Liang等人的文章,Shi研究了经典BSDE dYt =-f(t,Yt,Zt)dt + Zt*dBt,YT = ξj,L2解的Malliavin导数。基于本文已有的结论,重新回顾并证明了一些其他文献中重要的定理。最后,简要的介绍了本文结论在金融市场中的应用,例如,收益为ξ≥0,到期日为T的欧式期权定价。并且应用Malliavin导数研究了某些条件下的期权定价。
[Abstract]:This paper is based on "Backward Stochastic Differential Equations in Finance" by Shi [1]. In "Backward Stochastic Dynamics on a filtered probability Space", the author introduces that BSDE can be rerepresented as a general differential equation in an orbital space. A type of backward stochastic differential equation, dYtjn -fjnt, Ytn, YTJ = 尉 _ j, where L is a given nonlinear mapping of M to the adaptive process LJ _ (M), The modified term M is a martingale. Liang et al. We give the existence and uniqueness of L2 solution under some conditions. Shi generalizes it to LP solution and proves the existence and uniqueness of LP solution under these conditions. Furthermore, Shi gives this kind of backward random solution. Comparison Theorems for L2 Solutions of differential equations. Based on Liang et al.'s paper, we have studied the Malliavin derivative of the solution of the classical BSDE dYt (BSDE dYt) -ftn (Ytn) Ztnt t (ZtBtT) = 尉 JN L2. Based on the conclusions in this paper, we have reviewed and proved some important theorems in other literatures. Finally, some important theorems in this paper are reviewed and proved. This paper briefly introduces the application of this conclusion in the financial market, for example, the European option pricing with a return of 尉 鈮，

本文编号：1674128