查看“︁倒向随机微分方程”︁的源代码

'''倒向随机微分方程'''（'''BSDE'''）是带有终点条件的[[随机微分方程]]，其解要根据底层滤波进行调整。BSDE自然地出现在各种应用中，如[[随机控制]]、[[金融数学]]与非线性[[费曼-卡茨公式]]。<ref>{{Cite book|last1=Ma|first1=Jin|last2=Yong|first2=Jiongmin|date=2007|title=Forward-Backward Stochastic Differential Equations and their Applications|series=Lecture Notes in Mathematics|volume=1702|url=https://link.springer.com/book/10.1007/978-3-540-48831-6|publisher=Springer Berlin, Heidelberg|doi=10.1007/978-3-540-48831-6|isbn=978-3-540-65960-0|access-date=2023-11-10|archive-date=2023-08-09|archive-url=https://web.archive.org/web/20230809201413/https://link.springer.com/book/10.1007/978-3-540-48831-6|dead-url=no}}</ref>

==背景==
1973年[[让-米歇尔·比斯姆]]提出了BSDE线性情形<ref>{{cite journal|last=Bismut|first=Jean-Michel|year=1973|title=Conjugate convex functions in optimal stochastic control|journal=Journal of Mathematical Analysis and Applications|volume=44|issue=2 |pages=384–404|doi=10.1016/0022-247X(73)90066-8}}</ref>，1990年法国学者{{link-en|Etienne Pardoux|Etienne Pardoux|Etienne Pardoux}}和中国学者[[彭实戈]]合作发表的论文中提出BSDE非线性情形，线性是广泛的非线性中的一特殊形式<ref>{{cite journal|last1=Pardoux|first1=Etienne|last2=Peng|first2=Shi Ge|year=1990|title=Adapted solution of a backward stochastic differential equation|url=https://archive.org/details/sim_systems-control-letters_1990-01_14_1/page/55|journal=Systems & Control Letters|volume=14|pages=55–61|doi=10.1016/0167-6911(90)90082-6}}</ref><ref>{{Cite web |author=陈欢欢 |date=2008-06-29 |title=彭实戈院士：倒向随机微分方程理论在金融决策中的应用 |url=https://news.sciencenet.cn/htmlnews/2008/6/2008630104511433208511.html |publisher=[[科学网]] |access-date=2024-01-07 |website=news.sciencenet.cn |publication-date=2008-06-29 |archive-date=2024-01-07 |archive-url=https://web.archive.org/web/20240107215006/https://news.sciencenet.cn/htmlnews/2008/6/2008630104511433208511.html |dead-url=no }}</ref>。

==数学框架==
固定终点时刻<math>T>0</math>与[[概率空间]]<math>(\Omega,\mathcal{F},\mathbb{P})</math>。令<math>(B_t)_{t\in [0,T]}</math>为[[布朗运动]]，其自然滤波<math>(\mathcal{F}_t)_{t\in [0,T]}</math>。BSDE是积分方程，其类型为

{{NumBlk|:|<math>Y_t = \xi - \int_t^T f(s,Y_s,Z_s) \mathrm{d}s - \int_t^T Z_s \mathrm{d}B_s,\quad t\in[0,T],</math>|{{EquationRef|1}}}}

其中<math>f:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}</math>称作BSDE的生成器，终点条件<math>\xi</math>是<math>\mathcal{F}_T</math>-可测随机变量，解<math>(Y_t,Z_t)_{t\in[0,T]}</math>包含随机过程<math>(Y_t)_{t\in[0,T]}</math>、<math>(Z_t)_{t\in[0,T]}</math>，其适应于过滤<math>(\mathcal{F}_t)_{t\in [0,T]}</math>。

===例子===
在<math>f\equiv 0</math>情形下，BSDE ({{EquationNote|1}})简化为

{{NumBlk|:|<math>Y_t = \xi - \int_t^T Z_s \mathrm{d}B_s,\quad t\in[0,T].</math>|{{EquationRef|2}}}}

若<math>\xi\in L^2(\Omega,\mathbb{P})</math>，则根据[[鞅表示定理]]，存在唯一的随机过程<math>(Z_t)_{t\in [0,T]}</math>使<math>Y_t = \mathbb{E} [ \xi | \mathcal{F}_t ]</math>、<math>Z_t</math>满足BSDE ({{EquationNote|2}})。

==另见==
* [[鞅表示定理]]
* [[随机控制]]
* [[随机微分方程]]

==参考文献==
{{Reflist}}

==阅读更多==
* {{cite book
| last1 = Pardoux
| first1 = Etienne
| last2 = Rӑşcanu
| first2 = Aurel
| title = Stochastic Differential Equations, Backward SDEs, Partial Differential Equations 
| series = Stochastic modeling and applied probability
| publisher = Springer International Publishing Switzerland
| year = 2014
}}
* {{cite book
| last = Zhang
| first = Jianfeng
| title = Backward stochastic differential equations
| series = Probability theory and stochastic modeling
| publisher = Springer New York, NY
| year = 2017
}}

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[[Category:随机微分方程]]