[Paper Review] Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion

Alexandros Beskos, Joseph Dureau+1 more|arXiv (Cornell University)|Jun 30, 2013

Financial Risk and Volatility Modeling37 references3 citations

TL;DR

This paper develops a computationally efficient Bayesian inference method for partially observed stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm), using a reparameterization via the Davies and Harte algorithm and a mesh-free hybrid Monte Carlo sampler. The approach enables stable, high-dimensional posterior inference for the Hurst parameter H, with empirical results favoring H < 1/2 in S&P500/VIX data, indicating medium-range dependence.

ABSTRACT

We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a non-trivial likelihood given the latent path. Due to the non-Markovianity and high-dimensionality of the latent paths, estimating posterior expectations is a computationally challenging undertaking. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use this framework to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based on a version of hybrid Monte Carlo that delivers increased efficiency when applied on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model allowing for memory in the volatility increments through a fractional specification. The methodology is illustrated on simulated data and on the S&amp;P500/VIX time series and is shown to be effective. Contrary to a long range dependence attribute of such models often assumed in the literature, with Hurst parameter larger than 1/2, the posterior distribution favours values smaller than 1/2, pointing towards medium range dependence.

Motivation & Objective

• To address the computational challenge of Bayesian inference in partially observed SDEs driven by fractional Brownian motion, which are non-Markovian and involve high-dimensional latent paths.
• To develop a Markov chain Monte Carlo (MCMC) algorithm that maintains stable mixing time as the number of time points increases, overcoming issues from non-Markovian dependence and high-dimensional path inference.
• To enable full Bayesian inference over all parameters, including the Hurst index H, avoiding non-likelihood-based methods such as least squares.
• To construct a data augmentation scheme that treats the latent fBm path as a high-dimensional latent variable, with likelihoods depending non-trivially on the entire path.
• To apply the method to a stochastic volatility model with fractional dynamics, allowing memory in volatility increments through H

Proposed method

• Uses the Davies and Harte algorithm to reparameterize the fractional Brownian motion path, enabling efficient sampling of the high-dimensional Gaussian latent process.
• Applies a hybrid Monte Carlo (HMC) algorithm with a Hamiltonian formulation to update the latent path and parameters jointly, ensuring mesh-free mixing time.
• Employs a time-discretized approximation of the continuous path on a grid of size N, with the posterior computed via data augmentation: p(θ, X | Y) ∝ p(Y | X, θ) p(X | θ) p(θ).
• Implements a version of HMC that preserves the target measure Q_N(x,v) = exp{-H(x,v;M)} by ensuring volume preservation and using a probabilistic approach to handle infinite-dimensional Gaussian measures.
• Derives the density ratio dQ^(i)/dQ_0 using recursive application of transition maps, with explicit expressions for the Girsanov-type change of measure in the velocity updates.
• Uses the Cameron–Martin space theory to ensure that the Girsanov correction terms are well-defined and finite under the assumption that ∇_zΦ(z,θ) ∈ ℓ₂

Experimental results

Research questions

• RQ1 Can efficient Bayesian inference be performed for partially observed SDEs driven by fractional Brownian motion when the latent path is infinite-dimensional and non-Markovian?
• RQ2 How can the non-Markovian dependence structure of fBm be handled in MCMC sampling without deteriorating mixing time as the number of time points increases?
• RQ3 What reparameterization strategy enables stable and efficient sampling of the high-dimensional latent fBm path in the context of Bayesian inference?
• RQ4 Does the hybrid Monte Carlo algorithm maintain good mixing properties in infinite-dimensional state spaces when applied to fBm-driven SDEs?
• RQ5 What is the posterior inference for the Hurst parameter H in real financial data, and does it support medium-range or long-range dependence?

Key findings

• The proposed MCMC algorithm based on Davies and Harte reparameterization and hybrid Monte Carlo achieves mesh-free mixing time, ensuring stable performance as the number of time points N increases.
• The method enables full Bayesian inference over all parameters, including the Hurst index H, without relying on non-likelihood-based estimation techniques.
• In the stochastic volatility model, the posterior distribution for H favors values below 1/2, indicating medium-range dependence in the volatility process.
• Empirical analysis of S&P500/VIX data shows strong posterior support for H < 1/2, suggesting that volatility increments exhibit negative autocorrelation and slower decay than exponential.
• Theoretical analysis confirms that the HMC algorithm preserves the target measure Q_N through a recursive derivation of the density ratio dQ^(i)/dQ_0, with cancellations in the log-density leading to a well-defined Hamiltonian structure.
• The method is applicable to higher-dimensional systems with multiple Hurst parameters, though computational cost increases accordingly