Annals of Applied Probability

A martingale approach for fractional Brownian motions and related path dependent PDEs

Frederi Viens and Jianfeng Zhang

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 6 (2019), 3489-3540.

Received: December 2017
Revised: October 2018
First available in Project Euclid: 7 January 2020

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60H20: Stochastic integral equations 60H30: Applications of stochastic analysis (to PDE, etc.) 35K10: Second-order parabolic equations 91G20: Derivative securities

Fractional Brownian motion Volterra SDE Monte Carlo methods path dependent PDEs functional Itô formula rough volatility time inconsistency


Viens, Frederi; Zhang, Jianfeng. A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab. 29 (2019), no. 6, 3489--3540. doi:10.1214/19-AAP1486.

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