Malliavin calculus of Bismut type without probability

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

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A Malliavin Calculus method to study densities of additive functionals of SDE’s with irregular drifts

Annales de l Institut Henri Poincaré Probabilités et Statistiques ◽

10.1214/11-aihp418 ◽

2012 ◽

Vol 48 (3) ◽

pp. 871-883 ◽

Cited By ~ 8

Author(s):

Arturo Kohatsu-Higa ◽

Akihiro Tanaka

Keyword(s):

Malliavin Calculus ◽

Additive Functionals

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Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application

Random Operators and Stochastic Equations ◽

10.1515/rose-2020-2047 ◽

2020 ◽

Vol 28 (4) ◽

pp. 291-306

Author(s):

Tayeb Bouaziz ◽

Adel Chala

Keyword(s):

Brownian Motion ◽

Maximum Principle ◽

Control Problem ◽

Fractional Brownian Motion ◽

Stochastic Control ◽

Malliavin Calculus ◽

Stochastic Maximum Principle ◽

Hurst Parameter ◽

Principle Of Optimality ◽

Initial Cost

AbstractWe consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter {H\in(\frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.

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