Open Access
December 2019 A martingale approach for fractional Brownian motions and related path dependent PDEs
Frederi Viens, Jianfeng Zhang
Ann. Appl. Probab. 29(6): 3489-3540 (December 2019). DOI: 10.1214/19-AAP1486


In this paper, we study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire (Quant. Finance 19 (2019) 721–729) to our more general framework. In particular, unlike in (Quant. Finance 19 (2019) 721–729) where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.


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Frederi Viens. Jianfeng Zhang. "A martingale approach for fractional Brownian motions and related path dependent PDEs." Ann. Appl. Probab. 29 (6) 3489 - 3540, December 2019.


Received: 1 December 2017; Revised: 1 October 2018; Published: December 2019
First available in Project Euclid: 7 January 2020

zbMATH: 07172340
MathSciNet: MR4047986
Digital Object Identifier: 10.1214/19-AAP1486

Primary: 60G22
Secondary: 35K10 , 60H20 , 60H30 , 91G20

Keywords: fractional Brownian motion , functional Itô formula , Monte Carlo methods , path dependent PDEs , Rough volatility , Time inconsistency , Volterra SDE

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.29 • No. 6 • December 2019
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