Multidimensional backward stochastic differential equations with uniformly continuous coefficients
Multidimensional backward stochastic differential equations with uniformly continuous coefficients
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DOI：
10.3150/bj/1065444816
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发表时间：
2003-06
期刊：
影响因子：
1.5
通讯作者：
S. Hamadène
中科院分区：
文献类型：
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作者：
S. Hamadène
这种类型的方程，至少在非线性情况下，首先由Pardoux和Peng（1990 a）引入，他们在适当的假设下证明了解的存在性和唯一性，其中最重要的是f的Lipschitz连续性和？的平方可积性。他们的目的是给出一个二阶拟线性偏微分方程解的概率解释。此后，这些方程逐渐成为金融数学等诸多领域的重要数学工具（参见，例如，El-Karoui et al. 1997 a; 1997 b; Buckdahn and Hu 1998; Cvitanic and Karatzas 1996），随机博弈和最优控制（Hamadene和Lepeltier 1995 a; 1995 b; Hamadene等人1997; 1999; Cvitanic和Karatzas 1996; Dermoune等人1999），偏微分方程和均匀化（Pardoux and Peng 1990 b; 1992; Pardoux 1999; Peng 1991; Darling and Pardoux 1997; Buckdahn and Peng 1999）和F-鞅的构造（Darling 1995）。另一个被广泛讨论的问题是如何通过弱化f上的Lipschitz连续条件来改进Pardoux和Peng（1990 a）的存在唯一性结果。Hamadene（1996）、Kobylanski（2000）和Lepeltier and San Martin（1997; 1998）已经处理了Y是一维过程的情况。他们在不假设f为Lipschitz连续的情况下得到了一个存在性结果。然而，解决方案不一定是唯一的。Hamadene（1996）认为f只是局部Lipschitz，而Kobylanski
This type of equation, at least in the nonlinear case, was first introduced by Pardoux and Peng (1990a), who proved the existence and uniqueness of a solution under suitable assumptions on f and ?, the most important of which are the Lipschitz continuity of f and the square integrability of ?. Their aim was to give a probabilistic interpretation of a solution to a second-order quasilinear partial differential equation. Since then, these equations have gradually become an important mathematical tool in many fields such as financial mathematics (see, for example, El-Karoui et al. 1997a; 1997b; Buckdahn and Hu 1998; Cvitanic and Karatzas 1996), stochastic games and optimal control (Hamadene and Lepeltier 1995a; 1995b; Hamadene et al. 1997; 1999; Cvitanic and Karatzas 1996; Dermoune et al. 1999), partial differential equations and homogenization (Pardoux and Peng 1990b; 1992; Pardoux 1999; Peng 1991; Darling and Pardoux 1997; Buckdahn and Peng 1999) and construction of F-martingales (Darling 1995). A further problem under widespread discussion is how to improve the existence and uniqueness result of Pardoux and Peng (1990a) by weakening the Lipschitz continuity condition on f. Hamadene (1996), Kobylanski (2000) and Lepeltier and San Martin (1997; 1998) have dealt with the situation where Y is a unidimensional process. They obtained an existence result without assuming f Lipschitz continuous. However, the solution is not necessarily unique. Hamadene (1996) takes f to be just locally Lipschitz, while Kobylanski