Backward stochastic differential equations with time delayed generators - results and counterexamples
Backward stochastic differential equations with time delayed generators - results and counterexamples
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DOI：
10.1214/09-aap663
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发表时间：
2010-05
影响因子：
1.8
通讯作者：
L. Delong;P. Imkeller
中科院分区：
文献类型：
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作者：
L. Delong;P. Imkeller
研究了具有时滞生成元的倒向随机微分方程。在这种新类型的方程中，在时间t的生成器可以依赖于过去的解的值，用时间延迟函数加权，例如移动平均类型。我们证明了一个充分小的时域或充分小的Lipschitz常数的生成元的解的存在性和唯一性。我们给出的例子，具有时间延迟的发生器，有多个解决方案或没有解决方案的bsde。我们证明了一些特殊类的发电机的存在性和唯一性可能仍然保持为任意的时间范围和任意Lip schitz常数。这个类包括我们更详细研究的线性时间延迟发生器。研究了一类具有时滞生成元的边值问题解的性质，包括有界性对终端条件的继承性、比较原理、测度解的存在性和BMO鞅性质.我们给出了他们可能失败的例子。
We deal with backward stochastic differential equations with time de layed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a suffi ciently small Lipschitz constant of a generator. We give examples of bsde with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lip schitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a bsde with time delayed generator, including the inheritance of bound edness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.