基于Ito演算的Clark-Ocone型公式及其在金融中的应用
A Clark-Ocone type formula via Ito calculus and its application to finance
摘要：
将给出随机变量的一个显式鞅表示，它被描述为Levy过程的一个泛函。Clark-Ocone定理证明了鞅表示中的被积函数是由Malliavin导数的条件期望给出的。我们的目标是将其扩展到随机变量是不Malliavin微。为此，我们利用伊藤公式，而不是Malliavin演算。作为对金融数学的一个应用，我们将给出指数Levy模型数字期权局部风险最小化策略的一个显式表示。由于数字期权的收益是由一个指示函数描述的，我们还讨论了指示函数关于Levy过程的Malliavin可微性。
英文摘要：
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.
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