[Paper Review] A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs

Frédéri Viens, Jianfeng Zhang|arXiv (Cornell University)|Dec 11, 2017

Stochastic processes and financial applications19 references3 citations

TL;DR

This paper develops a functional Itô calculus for Volterra-type processes, such as fractional Brownian motion, by introducing a martingale auxiliary process Θ to overcome time inconsistency and non-semimartingale properties. The key contribution is a path-dependent PDE framework that extends Dupire’s functional Itô calculus to non-Markovian, non-semimartingale settings, enabling conditional expectation computations and option pricing in rough volatility models.

ABSTRACT

In this paper we study dynamic backward problems, with the computation of conditional expectations as a main 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 \cite{Dupire} to our more general framework. In particular, unlike in \cite{Dupire} 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 in a financial market with rough volatility is presented.

Motivation & Objective

• To address the challenge of computing conditional expectations for non-Markovian, non-semimartingale processes such as fractional Brownian motion.
• To overcome the time inconsistency and lack of flow properties in Volterra-type SDEs that prevent standard PDE and Markovian methods from applying.
• To extend Dupire’s functional Itô calculus to non-semimartingale processes by introducing a martingale auxiliary process Θ.
• To derive and solve path-dependent PDEs for backward problems in a framework where the state process depends on its entire history.
• To enable application to option pricing and hedging in financial models with rough volatility, such as the rough Bergomi model.

Proposed method

• Introduce an auxiliary process Θ defined as Θₛᵗ = ∫₀ᵗ K(s,r) dWᵣ, which is a martingale in t for fixed s, to recover semimartingale properties.
• Use the orthogonal decomposition Xₛ = Θₛᵗ + (Xₛ − Θₛᵗ) to separate the past-dependent component from the future-conditional component.
• Reformulate the conditional expectation as Yₜ = u(t, X[₀,ₜ) ⊗ Θᵗ[ₜ,ₜ]), where u depends on both the observed path and a smooth future path observable derived from the distribution of future paths.
• Derive a new functional Itô formula that accounts for the path dependence on both the current path and the future path observable, extending Dupire’s framework.
• Establish a path-dependent PDE (PPDE) for the function u by leveraging the martingale property of Θ and the time-inconsistent structure of the process.
• Apply the framework to the rough Bergomi model, showing how the extended calculus enables conditional expectation and option pricing in models with rough volatility.

Experimental results

Research questions

• RQ1 How can functional Itô calculus be extended to non-semimartingale, non-Markov processes such as fractional Brownian motion?
• RQ2 What auxiliary structure is needed to restore a form of Markovian or flow property in time-inconsistent processes?
• RQ3 How can conditional expectations be represented in a path-dependent framework when the state process lacks semimartingale or Markov properties?
• RQ4 What is the correct generalization of Dupire’s functional Itô formula for Volterra-type processes with memory?
• RQ5 Can the proposed framework be applied to option pricing in financial models with rough volatility, such as the rough Bergomi model?

Key findings

• The paper introduces a novel functional Itô formula that extends Dupire’s framework to non-semimartingale processes by incorporating future path observables through the auxiliary martingale process Θ.
• The solution to backward problems is represented as a function of both the current path X[₀,ₜ) and the future path observable Θᵗ[ₜ,ₜ], enabling a generalized form of path-dependent PDEs.
• The method successfully recovers a form of flow property via the joint process (X, Θ), allowing PDE-based analysis despite the time inconsistency of the original process.
• The framework is applied to the rough Bergomi model, where it is shown that the underlying process may lead to strict local martingales due to exponential growth in the volatility drift.
• For the rough Bergomi model with ρ = 0 and H = 1/2, the paper proves that E[Sₜᵖ] = ∞ for p > 1, indicating that moments of order greater than 1 do not exist.
• The paper demonstrates that when the payoff function g has linear growth, the transformed function ̂g(x) = g(eˣ) grows exponentially, which violates integrability conditions required for standard PDE methods, highlighting the necessity of the proposed framework.