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A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

Communications in Mathematical Sciences(2020)

University of Oslo | Mathematics And Statistics

Cited 6|Views15
Abstract
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter H<1/2. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".
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Volatility Modeling
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要点】:本文推导了针对由具有H<1/2的Hurst参数的多维分数布朗运动驱动的奇异随机微分方程(SDEs)的Bismut-Elworthy-Li公式,并将其应用于基于分数布朗运动驱动的SDEs描述的随机波动率股票价格模型中的金融索赔价格敏感性的研究,创新点在于将此公式应用于奇异SDEs以及结合Malliavin微积分和局部时间变分微积分的新方法。

方法】:研究采用Malliavin微积分和局部时间变分微积分方法。

实验】:文中未明确描述具体实验,但通过理论推导和应用分析,得出了针对奇异SDEs的Bismut-Elworthy-Li公式的结论,未提及使用的数据集名称。