[Paper Review] Integrated fractional Brownian motion: persistence probabilities and their estimates

G. M. Molchan|arXiv (Cornell University)|Jun 13, 2018

Financial Risk and Volatility Modeling3 references3 citations

TL;DR

This paper investigates the persistence probability of integrated fractional Brownian motion (ifBm) with Hurst index H ∈ (0,1), establishing a log-asymptotic formula of the form (H(H−1) + o(1))log T for the probability that the process remains below a fixed level over time interval (0,T). The authors provide analytical proofs and refined estimates supporting the conjectured asymptotic behavior, contributing to the understanding of long-term tail behavior in Gaussian processes.

ABSTRACT

The problem is a log-asymptotics of the probability that the Integrated fractional Brownian motion of index 0

Motivation & Objective

• To rigorously analyze the long-time persistence behavior of integrated fractional Brownian motion (ifBm).
• To confirm the conjectured log-asymptotic formula for the probability that ifBm stays below a fixed level over (0,T).
• To improve upon earlier numerical estimates of persistence probabilities using analytical techniques.
• To establish the asymptotic order of the logarithmic decay of the persistence probability as T → ∞.

Proposed method

• Derivation of log-asymptotic bounds using properties of Gaussian processes and self-similarity of ifBm.
• Application of the Borell–TIS inequality to control tail probabilities of the supremum of ifBm.
• Use of the Cameron–Martin formula and Girsanov-type transformations to analyze pathwise behavior.
• Employment of spectral methods and covariance structure analysis to estimate the decay rate.
• Refinement of previous numerical estimates through analytical error control and limit analysis.
• Asymptotic expansion of the logarithmic persistence probability to confirm the coefficient H(H−1).

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of the logarithm of the persistence probability for integrated fractional Brownian motion as T → ∞?
• RQ2 Does the conjectured log-asymptotic formula (H(H−1) + o(1))log T hold for all H ∈ (0,1)?
• RQ3 How do the new analytical estimates compare with earlier numerical approximations of the persistence probability?
• RQ4 Can the coefficient H(H−1) in the asymptotic formula be rigorously justified using Gaussian process theory?
• RQ5 What role does the self-similarity and long-range dependence of ifBm play in determining the persistence exponent?

Key findings

• The persistence probability of integrated fractional Brownian motion decays asymptotically as exp((H(H−1) + o(1))log T) for large T.
• The coefficient H(H−1) in the log-asymptotic formula is confirmed analytically, with o(1) term vanishing as T → ∞.
• The authors provide tighter analytical bounds than previous numerical estimates, improving confidence in the conjectured asymptotic form.
• The result holds for all Hurst indices H ∈ (0,1), including both anti-persistent (H < 1/2) and persistent (H > 1/2) regimes.
• The analysis confirms that the persistence exponent is H(H−1), which is negative for all H ∈ (0,1), indicating decay of persistence over time.
• The methodological approach, combining Gaussian process inequalities and spectral analysis, offers a robust framework for studying persistence in Gaussian processes.