arXiv 1701.061552017-01-22math.PR

随机泛函微分方程及其对初始路径的敏感性

Stochastic functional differential equations and sensitivity to their initial path

David R. Baños, Giulia Di Nunno, Hannes Haferkorn, Frank Proske

摘要：

我们考虑由随机泛函微分方程表示的具有记忆的系统。基本上，这些是随机微分方程，其系数取决于过程本身的过去历史。这样的系数因此被定义在函数空间上。具有记忆的模型出现在从生物学到金融学的许多应用中。在这里，我们考虑基于这些模型的一些评估结果（例如，一些金融产品的价格）和与这些模型的选择相关的风险。特别是，我们专注于初始条件对评估的影响。这个问题被称为对初始条件的敏感性分析，在金融术语中，它被称为Delta。在这项工作中的初始条件表示的随机泛函微分方程的有关过去的历史。这自然导致了Delta定义的重新设计。我们建议将它定义为一个泛函方向导数，这是一个自然的选择。为此，我们研究了一个表示公式，它允许其计算，而不需要评估功能是可微的。此功能与应用程序特别相关。通过研究Malliavin导数与泛函方向导数之间的一种恰当的关系，得到了该公式。为此，我们介绍了{ \it随机化的初始条件}的技术。

英文摘要：

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of {\it randomisation of the initial condition}.

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