In this paper, we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process by using Malliavin calculus. Similar to the celebrated Black–Scholes formula, we aim to express the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of stop-loss contracts, the building block is given by the distribution function of the terminal cumulated loss taken at the Value at Risk when computing the expected shortfall risk measure.

中文翻译：

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使用Malliavin微积分的保险衍生产品定价公式

在本文中，我们通过使用Malliavin演算为依赖于一般损失过程的不同类别的精算和金融合同提供了一个估值公式。与著名的Black-Scholes公式类似，我们的目标是以构件为基础来表达预期的现金流量。前者与损失过程有关，损失过程是由索赔的双重随机Poisson过程索引的累积和，允许其依赖于计数过程的强度和跳跃时间。例如，在止损合约的情况下，当计算预期的空头风险度量时，通过在风险价值处采用的终端累计损失的分布函数来给出构件。