線性正倒向隨機微分方程與里卡蒂方程

Linear Forward-Backward Stochastic Differential Equations and a Riccati Type Equation

林柏佐(Po-Tso Lin)

指導教授 ： 姜祖恕

共同指導教授 ： 謝南瑞

國立臺灣大學/理學院/數學研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.01178

摘要

本篇論文中我們探討線性正倒向隨機微分方程的可解性。我們在本篇論文中我們探討特殊情形時( b A = O)，可解性的充分必要條件。我們是針對Ma & Yong (2000)工作的延伸。最後我們提出了一個正向微分方程與倒向微分方程的關係，藉由解一個矩陣的常微分方程(里卡蒂方程)提出了類似的充分必要條件。

關鍵字

正倒向隨機微分方程 ； 里卡蒂方程

並列摘要

In this paper we investigate the solvability of linear forward-backward stochastic differential equations (FBSDEs, for short). We give sufficient and necessary conditions of the solvability in linear forward-backward stochastic differential equations and prove it in a special case ($widehat A=O$). These results are extensional work of Ma & Yong (2000). Then we introduce the relationship between forward equation and backward equation, we also can get similar sufficient and necessary conditions to solve linear forward-backward stochastic differential equations by solving a matrix ordinary differential equation (a Riccati type equation).

並列關鍵字

Forward-Backward Stochastic Differential Equation ； Riccati Type Equation

參考文獻

Karoui, N. E., Peng, S., & Quenez, M. C. (1997). Backward stochastic differential equations in finance. Mathematical Finance, Vol. 7., No. 1, 1-71.

Lax, P. D., (2002). Functional Analysis. Wiley.

Ma, J., & Yong, J. (2000). Forward-backward stochastic equations and their applications. Springer.

Oksendal, B. (2003). Stochastic differential equations (6th ed.) Springer.

Zhou, X. Y., & Li, D. (2000). Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Applied Mathematics Optimization, 19-33.