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Bismut–Elworthy–Li formula, singular SDEs, fractional Brownian motion, Malliavin calculus, stochastic flows, stochastic volatility
Communications in Mathematical Sciences ( IF 1 ) Pub Date : 2020-12-11 , DOI: 10.4310/cms.2020.v18.n7.a3
Oussama Amine 1 , Emmanuel Coffie 2 , Fabian Harang 1 , Frank Proske 1
Affiliation  

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 $H \lt 1/2$. “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 the $\delta$ price sensitivity of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDEs. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed “local time variational calculus”.

中文翻译:

Bismut–Elworthy–Li公式,奇异的SDE,分数布朗运动,Malliavin演算,随机流量,随机波动率

在本文中,我们针对具有加性噪声的奇异随机微分方程(SDE)的强解,导出了Bismut–Elworthy–Li型公式,该加性噪声是由多维分数布朗运动给出的,Hurst参数为$ H \ lt 1/2 $ 。这里的“单数”是指这样的方程式的漂移矢量场仅是有界和可积的。作为应用程序,我们使用此表示公式来研究基于具有随机波动性的股票价格模型的金融债权对价格的敏感性,该模型通过分数布朗运动驱动的SDE来描述其动态。我们获得这些结果的方法是基于Malliavin微积分和最近开发的“本地时间变分微积分”的论据。
更新日期:2020-12-12
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