具有间断杀灭率的Lévy过程的费曼-卡茨公式
Feynman-Kac formula for Lévy processes with discontinuous killing rate
摘要：
将金融数学和数值分析的最新技术有效地结合起来的挑战激发了研究人员开发快速确定性期权定价方法。因此，通过求解偏积分微分方程来计算Lévy模型中的期权价格的高效算法已经被开发出来。为了给这些方法提供一个坚实的数学基础，我们推导出了描述非齐次Lévy过程泛函条件期望的偏积分微分方程变分解的Feynman-Kac表示.我们允许一个广泛的潜在的随机过程，包括布朗部分的过程，和广泛的一类纯跳跃过程，如广义双曲，多元正态逆高斯，回火稳定，和$α$ -半稳定的Lévy过程。由于我们对杀死率和偏微分方程的初始条件的温和的正则性假设，我们的结果不仅在金融数学中，而且在概率论和相对论量子力学中的许多应用提供了严格的基础。
英文摘要：
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman-Kac representation of variational solutions to partial integro differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow for a wide range of underlying stochastic processes, comprising processes with Brownian part, and a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and $α$-semi stable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial differential equation, our results provide a rigorous basis for numerous applications, not only in financial mathematics but also in probability theory and relativistic quantum mechanics.
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