Poisson-modulated fractional Brownian motion: Definition, preliminary properties, aspects of long-range dependence and an application
Date
2025
Authors
Abbott, William
Editors
Advisors
Roughan, Matthew
Peralta-Gutierrez, Oscar
Peralta-Gutierrez, Oscar
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Thesis
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Abstract
Stochastic processes are a mathematical model of random phenomena. Mod-ulated stochastic processes take this a step further and assist in modelling complex behaviour by changing parameters based on a separate external process. Fractional Brownian motion (fBm) is a stochastic process used extensively to model behaviour that demonstrates slower than normal decay in correlations. This property is termed long-range dependence (LRD) and whether or not fBm displays this property is dependent on a parameter H called the Hurst parameter, usually considered constant. Allowing this value to change in intervals of varying lengths would provide more accurate modelling of time-series such as the S&P 500 during times of high volatil-ity. However, in the existing literature there is no version of fBm that uses piece-wise constant Hurst parameters. We define a new process Poisson-modulated fractional Brownian motion (PmfBm) with piece-wise Hurst parameters, constant over intervals generated by Poisson arrival times. In Theorem 3.3.8 we show the covariance of PmfBm, and in the independent and identically distributed (iid) case, show that our process has stationary increments (Theorem 3.1.15), which are of a similar form to fBm. In Theorem 3.3.5, we show that the LRD properties of the process exist for deterministic sequences of Hurst parameters. Importantly, given an iid sequence of Hurst parameters, the regime switching times are irrelevant to both the variance and covariance, which suggests a link to multi-fractional Brownian motion (multi-fBm). In addition, using deterministic sequences of Hurst parameters suggests definitions for locally-LRD processes. Periods of global turmoil present visible behavioural changes in the move-ment of stock indices such as the S&P 500. Events such as the Global Financial Crisis (GFC) and COVID-19 Pandemic give evidence for Hurst parameter regime changes during their occurrence. This demonstrates real-world time-series that matches the modelling procedure of PmfBm. We expect this thesis to motivate research into PmfBm; specifically how modulating the Hurst parameter of fBm at Poisson intervals will benefit definitions of LRD and its applicability to modelling properties of asset prices.
School/Discipline
Mathematical Sciences
Dissertation Note
Thesis (M.Phil) -- University of Adelaide, School of Mathematical Sciences, 2025
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