Multifractional Process with Random Exponent

• MPRE is a class of stochastic processes with a random scaling exponent that allows local variations in regularity and multifractality.
• They extend classical models like fractional Brownian motion by dynamically randomizing the Hurst or stability index via methods such as Itô integrals and wavelet expansions.
• MPRE models provide robust frameworks for simulation, estimation, and applications in fields including biophysics, climatology, and financial time series analysis.

A Multifractional Process with Random Exponent (MPRE) is a class of stochastic processes that extend classical models such as fractional Brownian motion by allowing their scaling exponent—often the Hurst or stability index—to fluctuate randomly in time or space. This represents a flexible modeling paradigm for phenomena exhibiting non-stationary regularity, multi-scale behavior, and dynamic forms of predictability. MPRE models integrate the ideas of multifractionality (local changes in fractal properties) and stochasticity in the exponent itself, enabling analysis of systems where local irregularity, jump intensity, or long-memory structure varies stochastically.

1. Mathematical Construction and Main Types

MPRE models are built by modifying fractional or stable stochastic process representations so that the scaling (self-similarity, “roughness”) parameter is a stochastic function of time, integration variable, or location.

• Fractional Brownian Motion Generalizations. Standard fractional Brownian motion (fBm) is given by

$$B_H(t) = \int_{-\infty}^{+\infty} [(t-x)_+^{H-1/2} - (-x)_+^{H-1/2}] dW(x),$$

with constant Hurst $H$. In the multifractional setting, $H$ is replaced by $H(t)$,

$$B_{H(t)}(t) = \int_{-\infty}^{+\infty} [(t-x)_+^{H(t)-1/2} - (-x)_+^{H(t)-1/2}] dW(x).$$

In MPRE, $H(t)$ is a stochastic process (possibly adapted to the Brownian filtration), or more generally a random field depending on time or the integration variable (Ayache et al., 2018, Ayache, 2011, Loboda et al., 2020).

• Riemann-Liouville Process Extensions. Another construction replaces the constant exponent in the Riemann-Liouville integral by a stochastic process $A(s)$,

$X(t) = \int_{0}^{1} (t-s)_+^{A(s)-1/2}\,dB(s),$

where $A(s)$ is random and adapted. This “integration-variable randomization” yields an MPRE with Itô integral representation, facilitating simulation and analysis (Ayache et al., 2018). The series expansion of the kernel in Haar basis and corresponding simulation algorithms are specifically described.

• Telegraphic and Incremental Models. Certain models use stochastic processes with telegraphic dynamics for the exponent, e.g. a smoothed two-state jump process with relaxation, resulting in a stationary beta distribution for $H(t)$ (Balcerek et al., 20 Apr 2025). Incremental multifractional Brownian motion (IMFBM) extends FBM so new increments reflect updated local $H(t)$ and $D(t)$ values, but history “remembers” previous regimes (Slezak et al., 2023).
• Multistable and Stable Lévy Extensions. MPRE concepts generalize to the stable context (linear multifractional stable motion, multistable subordinators (Molchanov et al., 2014, Dang, 2017)). Here, the stability index $\alpha$ or localisability index $h(t)$ become time-dependent (possibly random), allowing modeling of jumps and heavy-tails with non-constant intensities.

MPRE Construction	Exponent Randomization	Integration Variable
Wavelet Series	$H(t)$ random	Time parameter
Itô Integral	$A(s)$ random	Integration variable $s$
Telegraph Process	$H(t)$ telegraph	Time parameter

2. Regularity, Scaling, and Local Structure

The scaling laws and sample path regularity in MPRE models are fundamentally governed by the random exponent, leading to pronounced non-stationarity and multi-scale behavior.

• Incremental Moment Scaling. For a broad class of multifractional and multistable processes, the moments of increments scale locally as

$$\mathbb{E}[|Y(t+\epsilon)-Y(t)|^{\eta}] \sim C(t,\eta)\,\epsilon^{\eta h(t)},$$

where $h(t)$ is the local regularity index (“localisability index”), and the constant $C(t,\eta)$ may depend on the law of the tangent process (Guével et al., 2010).

• Local Hölder Exponents. The pointwise Hölder exponent $H_t$ of the process is almost surely bounded above by the local exponent. For certain explicit Gaussian MPRE counterexamples, the pointwise exponent is genuinely random:

$$\alpha_X(s,\omega) = \begin{cases} H(s), & \text{if } \partial_H B(s,H(s),\omega)\ne 0, \ \min\{H(s),2\alpha_H(s)\}, & \text{if } \partial_H B(s,H(s),\omega)=0. \end{cases}$$

This randomness can be quantified probabilistically; the zero-level sets of the derivative can have positive Hausdorff dimension (Ayache, 2011).

• Local Structure and Tangency. The tangent process at time $t$ is typically an fBm or stable process with exponent $h(t)$ (or random $H(t)$), confirming local asymptotic self-similarity (Loosveldt, 2023, Guével et al., 2010). In the Hermite process setting, the tangent process is the standard Hermite process with parameter $H(t)$ (Loosveldt, 2023).

3. Statistical Estimation and Simulation Methodologies

MPRE estimation leverages discrete variations, wavelet and Haar expansions, and scaling relations of increments.

• Empirical Estimators. Ratio-type estimators on dyadic intervals and negative power variations allow strong uniform and moment-consistent estimation of the pathwise $H(t)$, even for stable or heavy-tailed MPREs (Ayache et al., 2014, Dang, 2017). Estimation is feasible with log-ratios of empirical interval variation, robust even under intricate dependence.
• Simulation Algorithms. Haar series expansion and dyadic interval averaging facilitate efficient simulation—MPRE Itô-type integrals can be discretized using partial sums and Brownian increments (Ayache et al., 2018). Tensor network schemes, such as MERA (multi-scale entanglement renormalization ansatz), allow for fast $O(N\log N)$ simulation, significant for high-resolution paths (Descamps, 2016).
• Model Identification. Model selection between FBM, FBM with random constant exponent (FBMRE), and true MPRE can be performed by segmentwise TAMSD (time-averaged mean squared displacement) and autocovariance analysis of the estimated exponents. For telegraphic MPREs, a decaying ACVF of the local exponent time series discriminates temporally-varying from ensemble-heterogeneous processes (Balcerek et al., 20 Apr 2025).

4. Applications in Physical, Biological, and Financial Systems

MPRE models have been invoked in diverse contexts where local regularity and memory are time or space dependent.

• Biophysics and Single-Particle Tracking. MPRE frameworks with beta-distributed Hurst exponents (telegraphic models) accurately capture the heterogeneity observed in biological diffusion, e.g., motion of cellular organelles, where transitions in dynamics are gradual and statistically bounded (Balcerek et al., 20 Apr 2025, Slezak et al., 2023).
• Climate and Environmental Data. Spherical multifractional models (e.g., GMBM) enable analysis of variable regularity on the sphere for cosmological datasets—e.g., cosmic microwave background (CMB) maps—facilitating anomaly detection through spatial clustering of estimated local Hölder exponents (Broadbridge et al., 2021).
• Financial Time Series and Volatility. MPRE is used to reconceptualize volatility in finance not as mere dispersion, but as a function of local predictability. “Fair volatility” is defined by the volatility implied under Hurst–Hölder exponent $H(t)=1/2$ in efficient markets:

$$\mathrm{sd}\big(X(t+h)-X(t)\big|\mathcal{F}_t^{H,\nu}\big) \sim |h|^{H(t)}\,\nu(t)\,\sqrt{A(H(t))}, \qquad A(H) = \frac{\Gamma(H+\frac{1}{2})^2}{2H\sin(\pi H)\Gamma(2H)}$$

(Bianchi et al., 23 Sep 2025). Deviations from $H=1/2$ indicate inefficiency, with $H>1/2$ signaling momentum and suppressed volatility, $H<1/2$ indicating mean reversion and excess volatility.

Application Domain	MPRE Feature	Modeling Benefit
Biophysics, SPT	Random $H(t)$, beta	Heterogeneous anomalous diffusion
Cosmology, CMB	GMBM on spheres	Detection of spatial anomalies
Finance	Volatility–regularity	Absolute risk/interpretable inefficiency

5. Advanced Theoretical Developments and Extensions

• Multistable Poisson Processes. MPRE concepts are extended to jump processes, with time-varying stability indices modeled via multistable subordinators, supporting non-stationary jump intensity and heavy-tailed behavior (Molchanov et al., 2014).
• Hermite and Higher-Order MPREs. Multifractional Hermite processes use a time-dependent Hurst function in multiple Wiener–Itô integrals. Their sample path Hölder regularity is governed by $H(t)$, and fractal dimensions of the graph can be exactly characterized in some cases by Malliavin calculus and small-ball probability techniques (Loosveldt, 2023).
• New Continuity Criteria. A polynomial moment condition generalizing Kolmogorov–Chentsov yields regularity results for MPREs with random local Hölder exponents, decoupling pathwise regularity from the roughness of the functional exponent (Loboda et al., 2020).

6. Implications, Limitations, and Future Directions

MPRE models offer a high degree of flexibility for modeling systems with non-stationary or random regularity, unifying multifractal analysis with temporal or spatial heterogeneity.

• Implications: They enable separation of trajectory regularity from long-memory/volatility phenomena, provide absolute measures of risk or anomaly (e.g., “fair volatility”), and furnish frameworks for identifying regime shifts, clustering, or transitions in data with underlying contextual variation.
• Limitations and Open Problems: Challenges remain in precise small-ball probabilities for higher-order Wiener chaos MPREs, robust estimation under sparse or noisy data, and extending analytic tractability to non-stationary, non-ergodic or ageing systems. Identifying MPREs in practice often requires advanced statistical methods, such as likelihood-based or machine learning approaches, capable of distinguishing between temporal versus ensemble heterogeneity.
• Future Directions: There is scope for MPRE extensions to systems with stochastic diffusivity, full spatio-temporal kernels, nonlinear driving processes, and for coupling MPRE models to stochastic PDEs. Applications in risk modeling, material science, geophysics, and high-frequency finance are poised to benefit from further theoretical and computational advances.

• (Guével et al., 2010) Incremental moments and Hölder exponents of multifractional multistable processes
• (Ayache, 2011) Continuous Gaussian multifractional processes with random pointwise Hölder regularity
• (Lim et al., 2014) Some Fractional and Multifractional Gaussian Processes: A Brief Introduction
• (Molchanov et al., 2014) Multifractional Poisson process, multistable subordinator and related limit theorems
• (Ayache et al., 2014) Uniformly and strongly consistent estimation for the Hurst function of a Linear Multifractional Stable Motion
• (Descamps, 2016) (Quantum) Fractional Brownian Motion and Multifractal Processes under the Loop of a Tensor Networks
• (Ayache et al., 2018) A new Multifractional Process with Random Exponent
• (Ayache et al., 2020) Wavelet series representation for multifractional multistable Riemann-Liouville process
• (Loboda et al., 2020) Regularity of multifractional moving average processes with random Hurst exponent
• (Broadbridge et al., 2021) On Multifractionality of Spherical Random Fields with Cosmological Applications
• (Loosveldt, 2023) Multifractional Hermite processes: definition and first properties
• (Slezak et al., 2023) Minimal model of diffusion with time changing Hurst exponent
• (Balcerek et al., 20 Apr 2025) Multifractional Brownian motion with telegraphic, stochastically varying exponent
• (Bianchi et al., 23 Sep 2025) Fair Volatility: A Framework for Reconceptualizing Financial Risk

This synthesis defines the MPRE class, its mathematical principles, estimation and simulation methodologies, applications, and major theoretical advances, focusing exclusively on rigorously established results and analytic constructs as found in the cited research.