A time series is said to have long memory if its autocorrelation decays hyperbolically rather than exponentially. In finance, this means that returns or volatilities are correlated over long periods, which has significant implications for forecasting and risk management.

Formally, a process $X\left(t\right)$ is said to have long memory if its autocovariance function $\gamma \left(h\right)$ satisfies:

$\gamma \left(h\right)~L\left(h\right)h^{2H-2}\text{as}h\to \infty$

where $L\left(h\right)$ is a slowly varying function and $H$ is the Hurst exponent in the interval $\left(0.5,1\right)$.

The centered Gaussian process $B^H\left(t\right)$, with $H$ in the interval $\left(0,1\right)$, is an fBm with covariance defined as [1] [6] :

$E\left[B^H\left(t\right)B^H\left(s\right)\right]=\frac{1}{2}\left({\left|t\right|}^{2H}+{\left|s\right|}^{2H}-{\left|t-s\right|}^{2H}\right)$

Key characteristics:

The canonical definition of mBm is given by the following stochastic integral:

$\begin{array}{c}{B}^{H\left(t\right)}\left(t\right)=\frac{1}{\Gamma \left(H\left(t\right)+1/2\right)}[{\displaystyle {\int }_{-\infty }^0\left({\left(t-s\right)}^{H\left(t\right)-1/2}-{\left(-s\right)}^{H\left(t\right)-1/2}\right)\text{d}W\left(s\right)}\\ +{\displaystyle {\int }_0^t{\left(t-s\right)}^{H\left(t\right)-1/2}\text{d}W\left(s\right)}]\end{array}$(1)

where:

To ensure the existence and continuity of the trajectories of multifractional Brownian motion, the instantaneous Hurst function $H\left(t\right)$ must satisfy:

Under these assumptions, the trajectories of mBm are almost surely continuous, with local regularity governed by $H\left(t\right)$.

Consider $H\left(t\right)$ as the “mode” or “mood” of the market at time $t$:

Innovation of the mBm model: $H\left(t\right)$ is not fixed; it evolves over time. Thus, we can model a market that naturally transitions from calm, directional phases ( $H>0.5$) to turbulent, nervous phases ( $H<0.5$); this represents a much more realistic modeling of financial dynamics.

Multifractional Brownian motion (mBm) extends standard Brownian motion by allowing the Hurst exponent $H\left(t\right)$ to be time-dependent. This change makes the roughness of the process unstable over time and provides the degree of freedom needed to better model many natural phenomena.

The instantaneous variance of the process at time $t$ is of the order $t^{2H\left(t\right)}$. The volatility at a given time thus depends on both time $t$ and the value of the Hurst exponent $H\left(t\right)$. If $H\left(t\right)$ is large, then the local volatility increases more rapidly.

Considering a zoom around a given point $t$ (at the infinitesimal scale “ $\epsilon \to 0$”), the recentered and renormalized process is similar in law to a fractional Brownian motion (fBm) with fixed parameter $H\left(t\right)$. Locally, mBm therefore has a fractal structure corresponding to the instantaneous value $H\left(t\right)$.

The continuity of trajectories is ensured by sufficient regularity of the function $H\left(t\right)$. If $H\left(t\right)$ is Hölder continuous with exponent $\beta$ strictly greater than the supremum of $H\left(t\right)$, then the trajectories are almost surely continuous. In practice, this means not perturbing the evolution of $H\left(t\right)$ too much over time.

The increments of mBm have different properties from those of fBm: they are non-stationary, their statistical properties (such as variance) vary over time. This non-stationarity allows for the description of phenomena such as volatility clustering that are frequently found in finance where phases of high turbulence tend to persist.

mBm has a rich structure of local scalings, which can be studied using multifractal tools (e.g., spectrum of singularities), making it a suitable model for describing complex systems with intermittency [2] [7] .

Various methods allow for local estimation of the Hurst exponent:

This technique relies on the wavelet transform which quantifies local regularity. Let $\psi$ be a mother wavelet and $d_X\left(j,k\right)$ the wavelet coefficients of process $X$. We observe that:

$\mathbb{E}\left[{\left|d_X\left(j,k\right)\right|}^2\right]~C\cdot 2^{j\left(2H\left(t_k\right)-1\right)}\text{with}{t}_k=2^{j}k$

Estimation of $H\left(t\right)$ is performed by logarithmic regression on these coefficients.

This method determines the Hurst exponent from the evaluation of an iterative calculation module based on a sliding window centered at time $t$:

$H\left(t\right)=\frac{1}{2}{\log }_2\left(\frac{\hat{V}\left(2\delta \right)}{\hat{V}\left(\delta \right)}\right)$

where $\hat{V}\left(\delta \right)$ corresponds to the variance of increments at scale $\delta$.

A statistically optimal method for Gaussian processes but computationally intensive. It operates within a parametric advancement of $H\left(t\right)$ within a sliding window.

We aim to reconstruct the distribution of local singularities from the multifractal spectrum.

Direct methods, costly (complexity $O\left(n^2\right)$) involving discretization of the integral definition of the process.

We adopt an approach through Fourier transform where the process is generated by its coordinates in the spectral domain.

In another efficient approach, we exploit the Markovian structure of our process by stochastic ascent.

Summary of simulation methods and limitations in high frequency:

Regarding simulation methods, we can distinguish:

We observe limitations not only in their implementation cost for high-frequency data:

Perspectives to overcome these limitations would include promoting multi-scale methods, the need for hybridization with jump processes, but also deep learning, and high-performance computing ( Table 1 ).

We now have all the methodological tools at our disposal to simulate the process $\left(B^{H\left(t\right)}\left(t\right)\right)$ and estimate the function $\left(H\left(t\right)\right)$ on empirical data to build a complete financial valuation model. We establish an asset price dynamics incorporating mBm and examine its implications for valuation and risk management.

The initial model is as follows:

$\frac{\text{d}{S}_t}{S_t}=\mu \text{d}t+\sigma \text{d}{B}_t^{H\left(t\right)}$

where:

This equation has the solution:

$S_t=S_0\exp \left(\mu t-\frac{1}{2}{\sigma }^2{t}^{2H\left(t\right)}+\sigma B_t^{H\left(t\right)}\right)$

This version extends the Black-Scholes model by incorporating long memory and regular variability through $H\left(t\right)$.

For the initial model using multifractional Brownian motion:

$\text{d}{S}_t=\mu S_t\text{d}t+\sigma S_t\text{d}{B}_t^{H\left(t\right)}$

Let $f\left(S_t\right)=\ln S_t$. The derivatives are:

$f^{\prime }\left(S_t\right)=\frac{1}{S_t},f^{″}\left(S_t\right)=-\frac{1}{S_t^2}$

The differential of $\ln S_t$ is:

$\text{d}\left(\ln S_t\right)=\frac{\text{d}{S}_t}{S_t}-\frac{1}{2}\frac{{\left(\text{d}{S}_t\right)}^2}{S_t^2}$

Substitute $\text{d}{S}_t=\mu S_t\text{d}t+\sigma S_t\text{d}{B}_t^{H\left(t\right)}$:

$\text{d}\left(\ln S_t\right)=\frac{\mu S_t\text{d}t+\sigma S_t\text{d}{B}_t^{H\left(t\right)}}{S_t}-\frac{1}{2}\frac{{\left(\mu S_t\text{d}t+\sigma S_t\text{d}{B}_t^{H\left(t\right)}\right)}^2}{S_t^2}$

Simplify:

$\text{d}\left(\ln S_t\right)=\mu \text{d}t+\sigma \text{d}{B}_t^{H\left(t\right)}-\frac{1}{2}{\sigma }^2{\left(\text{d}{B}_t^{H\left(t\right)}\right)}^2$

For multifractional Brownian motion, the quadratic variance is:

${\left(\text{d}{B}_t^{H\left(t\right)}\right)}^2={\left(\text{d}t\right)}^{2H\left(t\right)}$

Thus:

$\text{d}\left(\ln S_t\right)=\mu \text{d}t+\sigma \text{d}{B}_t^{H\left(t\right)}-\frac{1}{2}{\sigma }^2{\left(\text{d}t\right)}^{2H\left(t\right)}$

Integrate from 0 to $t$:

${\displaystyle {\int }_0^t\text{d}\left(\ln S_u\right)}={\displaystyle {\int }_0^t\mu \text{d}u}+\sigma {\displaystyle {\int }_0^t\text{d}{B}_u^{H\left(u\right)}}-\frac{1}{2}{\sigma }^2{\displaystyle {\int }_0^t{\left(\text{d}u\right)}^{2H\left(u\right)}}$

Which gives:

$\ln S_t-\ln S_0=\mu t+\sigma B_t^{H\left(t\right)}-\frac{1}{2}{\sigma }^2{t}^{2H\left(t\right)}$

Hence the exponential solution:

$S_t=S_0\exp \left(\mu t+\sigma B_t^{H\left(t\right)}-\frac{1}{2}{\sigma }^2{t}^{2H\left(t\right)}\right)$

The Value at Risk at confidence level $\alpha$ is then:

${\text{VaR}}_{\alpha }\left(t\right)=S_t\left[\exp \left(\mu \Delta t+\sigma \sqrt{\Delta t^{2H\left(t\right)}}{\Phi }^{-1}\left(1-\alpha \right)\right)-1\right]$

where ${\Phi }^{-1}$ represents the inverse of the standard normal quantile function.

The coherent risk measure, Expected Shortfall, is formulated as:

${\text{ES}}_{\alpha }\left(t\right)=\frac{1}{\alpha }{\displaystyle {\int }_0^{\alpha }{\text{VaR}}_u\left(t\right)\text{d}u}$

This method captures the extreme aspect of the loss distribution beyond the VaR level.

For the system:

$\{\begin{array}{l}\text{d}{S}_t=\mu S_t\text{d}t+{\sigma }_t{S}_t\text{d}{B}_t^{H\left(t\right)}\hfill \\ \text{d}{\sigma }_t=\kappa \left(\theta -{\sigma }_t\right)\text{d}t+\xi {\sigma }_t\text{d}{W}_t\hfill \end{array}$

The demonstration requires a more sophisticated method since volatility ${\sigma }_t$ is itself a random process.

The formula for ${\sigma }_t$ represents a geometric Ornstein-Uhlenbeck type process:

$\text{d}{\sigma }_t=\kappa \left(\theta -{\sigma }_t\right)\text{d}t+\xi {\sigma }_t\text{d}{W}_t$

This equation has an explicit solution. Let $Y_t=\ln {\sigma }_t$. Applying Itô’s lemma:

$\begin{array}{c}\text{d}{Y}_t=\frac{\text{d}{\sigma }_t}{{\sigma }_t}-\frac{1}{2}\frac{{\left(\text{d}{\sigma }_t\right)}^2}{{\sigma }_t^2}\\ =\frac{\kappa \left(\theta -{\sigma }_t\right)\text{d}t}{{\sigma }_t}+\xi \text{d}{W}_t-\frac{1}{2}{\xi }^2\text{d}t\end{array}$

This equation can be solved numerically.

With ${\sigma }_t$ stochastic, the price $S_t$ now follows a stochastic process:

$\text{d}{S}_t=\mu S_t\text{d}t+{\sigma }_t{S}_t\text{d}{B}_t^{H\left(t\right)}$

Reusing Itô’s lemma for $\ln S_t$:

$\begin{array}{c}\text{d}\left(\ln S_t\right)=\frac{\text{d}{S}_t}{S_t}-\frac{1}{2}\frac{{\left(\text{d}{S}_t\right)}^2}{S_t^2}\\ =\mu \text{d}t+{\sigma }_t\text{d}{B}_t^{H\left(t\right)}-\frac{1}{2}{\sigma }_t^2{\left(\text{d}t\right)}^{2H\left(t\right)}\end{array}$

Integration yields:

$\ln S_t=\ln S_0+\mu t+{\displaystyle {\int }_0^t{\sigma }_u\text{d}{B}_u^{H\left(u\right)}}-\frac{1}{2}{\displaystyle {\int }_0^t{\sigma }_u^2{\left(\text{d}u\right)}^{2H\left(u\right)}}$

Unlike before, the presence of stochastic ${\sigma }_t$ prevents a closed-form solution. The solution must be approximated numerically using discretization techniques such as Euler-Maruyama.

To simulate the system, we discretize time and apply an approximation method:

$\{\begin{array}{l}{\sigma }_{t+\Delta t}={\sigma }_t+\kappa \left(\theta -{\sigma }_t\right)\Delta t+\xi {\sigma }_t\Delta W_t\hfill \\ S_{t+\Delta t}=S_t\exp \left(\mu \Delta t+{\sigma }_t\Delta B_t^{H\left(t\right)}-\frac{1}{2}{\sigma }_t^2{\left(\Delta t\right)}^{2H\left(t\right)}\right)\hfill \end{array}$

where $\text{Δ}{W}_t$ and $\text{Δ}{B}_t^{H\left(t\right)}$ are correlated increments of Brownian motions.

This model captures both long memory through $H\left(t\right)$ and volatility analysis through the stochastic model ${\sigma }_t$, thus providing a more faithful illustration of financial markets.

Theoretical Links and Comparisons

Link to Rough Volatility

The rough volatility model [12] [13] could appear as a special case of mBm with a constant and low $H\left(t\right)$ function ( $H\approx 0.1$), insofar as both approaches model the roughness of trajectories, but mBm goes further by allowing memory to vary according to market regimes.

Comparison with Fractional Models

mBm is by definition non-stationary (unlike fBm which relies on a constant Hurst exponent), which allows it to better account for market regime changes.

Link with Multifractal Models

mBm shares with multifractal models [2] [7] the idea of variable local regularity, with the advantage of a continuous formulation and Gaussian dependence that gives it better mathematical tractability [14] .

The objective of this empirical research is to analyze the effectiveness of the multifractional Brownian motion (mBm) model for modeling and forecasting returns of major stock indices. The study focuses on four key indicators of the global economy:

These indices were selected as they represent major developed markets across different geographical regions (North America, Europe, and Asia), providing a comprehensive view of global financial dynamics. The findings are expected to generalize to other liquid equity markets with similar characteristics.

Covering the period from 2000 to 2023.

The approach used relies on a multi-phase methodology:

1) Estimation of the Hölder exponent function $H\left(t\right)$ using wavelet analysis on sliding windows of 250 business days. This window size was chosen as it represents approximately one trading year, providing sufficient data points for robust estimation while being short enough to capture meaningful market regime shifts [15] [16] .

2) Modeling of price trajectories using the mBm model, including the combined stochastic volatility model described in Section 8.

3) Comparison with GARCH(1, 1), fBm, and standard Brownian motion models.

4) Performance evaluation using statistical indicators such as RMSE, MAE, and log-likelihood.

To verify our findings, we conducted numerous tests on the residuals of the various models.

$Q\left(h\right)=n\left(n+2\right)\underset{k=1}{\overset{h}{{\displaystyle \sum }}}\frac{{\hat{\rho }}^2\left(k\right)}{n-k}$

where $\hat{\rho }\left(k\right)$ is the autocorrelation of order $k$ of the residuals.

Unit root tests (ADF—Augmented Dickey-Fuller and KPSS) allow checking the stationarity of the series:

95% confidence intervals for RMSE and MAE were obtained by Bootstrap with 1000 resamplings. The confidence interval for RMSE is given by the following formula:

$C{I}_{95\text{\%}}=\left[\hat{\theta }-z_{0.975}\cdot SE\left(\hat{\theta }\right),\hat{\theta }+z_{0.975}\cdot SE\left(\hat{\theta }\right)\right]$

where $\hat{\theta }$ is the estimate of RMSE or MAE, $z_{0.975}$ is the 97.5% quantile of the normal distribution, and $SE\left(\hat{\theta }\right)$ is the standard error estimated by bootstrap.

Table 2 summarizes the comparative results of the different models. Our mBm model largely dominates all benchmark approaches across all metrics. Non-overlapping confidence intervals allow us to assert the significance of these differences.

The results of statistical tests performed on the residuals are presented in Table 3 :

These observations attest that the mBm model more effectively captures the data structure and generates residuals that adhere to the essential assumptions of time series models.

The results highlight the predominance of the mBm model in identifying the essential characteristics of financial series:

We calculated risk indicators VaR and ES for an equally weighted portfolio of the four indices. The conclusions indicate that:

$\begin{array}{l}{\text{VaR}}_{99\text{\%}}^{\text{mBm}}<{\text{VaR}}_{99\text{\%}}^{\text{GARCH}}<{\text{VaR}}_{99\text{\%}}^{\text{standard}}\\ {\text{ES}}_{99\text{\%}}^{\text{mBm}}<{\text{ES}}_{99\text{\%}}^{\text{GARCH}}<{\text{ES}}_{99\text{\%}}^{\text{standard}}\end{array}$

The mBm model demonstrates a more accurate representation of extreme losses, particularly during crisis periods (such as the 2008 financial crisis and COVID-19). Figure 1 demonstrates this predominance during market stress periods.

The chart above shows the simultaneous progression of Value at Risk (VaR) and Expected Shortfall (ES) at a 99% confidence level during the COVID-19 crisis. A notable increase in these two financial risk indicators is observed at the onset of the pandemic, illustrating the growing uncertainty and volatility in financial markets. Expected Shortfall, being a more conservative measure than VaR, generally shows higher values, highlighting its ability to more accurately capture risks related to distribution extremes. This comparison is particularly appropriate in times of crisis, when distribution extremes are essential for risk assessment.

Integrated analysis of visual results

Analysis of price trajectories

The figure comparing price trajectories highlights the ability of the mBm model to reproduce observed market dynamics, particularly its capacity to capture volatility clustering, typical of financial markets where periods of high volatility tend to aggregate. The mBm model can illustrate this phenomenon, unlike the standard model which shows a much smoother volatility pattern and is less consistent with market mechanisms.

Analysis of the local Hurst exponent

As shown in Figure 2 , the time evolution of the local Hurst exponent clearly highlights alternating persistent and anti-persistent phases, confirming the multifractional behavior of financial markets. The figure of the local Hurst exponent’s evolution points to the multifractional nature of financial markets. Indeed, the Hurst exponent varies significantly over time, ranging from persistent phases ( $H>0.5$), favoring trend continuation, to anti-persistent phases ( $H<0.5$), where prices tend to revert to their mean. This temporal variability explains why the mBm model, with a time-varying Hurst exponent, provides better results than constant-Hurst models.

Link with numerical results

The superior accuracy of the mBm model (lower RMSE and MAE) stems from its ability to reproduce volatility clustering, as observed in the price trajectory figure. The same goes for its higher log-likelihood and better VaR coverage, due to its ability to capture the time evolution of market persistence, as evidenced by the temporal evolution of the Hurst exponent.

Consequences for financial modeling

This dual approach proves that financial modeling is better suited when accounting for two key phenomena: volatility clustering and the time evolution of market persistence. Since mBm incorporates both, it is a better reflection of complex market dynamics, naturally justifying its use in advanced applications of risk management and option pricing.

The empirical analysis confirms the relevance of the mBm model for financial modeling:

These results support the adoption of the multifractional Brownian motion model for risk management applications and option pricing in financial markets.

The graphical results obtained from the previous figures have a direct application in the field of risk management. Our analysis shows that the mBm model, through its ability to better capture volatility clustering and time-varying persistence, allows for a more accurate evaluation of risk measures such as Value at Risk (VaR) and Expected Shortfall (ES), particularly during financial crises when traditional models tend to underestimate extreme risks.

The superiority of the mBm model lies in several fundamental advantages: on the one hand, it effectively captures local memory variability and the non-stationarity inherent in financial markets, as shown by the time evolution of the Hurst exponent; on the other hand, its adaptive flexibility enables it to handle different market regimes—calm phases or extreme turbulence—leading to better data fitting and improved risk measures, confirming its practical power for financial institutions.

The Markovian modeling in bilateral form of the mBm presents some challenges. Although the mBm achieves superior performance, the complexity of its estimation and simulation procedures constitutes a practical obstacle. The choice of estimation window sizes strongly influences the results and requires careful calibration. Furthermore, high-frequency estimation remains problematic, and potential arbitrage issues arise in pricing applications.

This research raises several considerations regarding limitations, potential extensions, and practical uses of the model.

1) Identified limitations

The main drawback lies in the sensitivity of instantaneous Hurst parameter $H\left(t\right)$ estimation to market microstructures and high-frequency data, which may cause instability in empirical fitting techniques.

2) Theoretical extensions

Two main directions emerge for improving the model:

Developing advanced numerical approaches for option pricing also represents a crucial extension.

3) Practical perspectives

On a practical level, the mBm model offers considerable potential for:

Several promising research avenues arise from this study:

The mBm model represents a unifying framework encompassing both rough volatility (low $H$) and long-memory (high $H$) models. Its flexibility allows for a better modeling of stylized market facts. The major challenges remain: developing robust $H\left(t\right)$ estimators, multivariate extensions, hybridization with machine learning, and applications to exotic option pricing.

Future research, especially those integrating machine learning and artificial intelligence, promises to significantly expand the scope of this model while overcoming its current computational challenges.

Thus, the multifractional Brownian motion paves the way toward a more robust, precise, and adaptive finance. Its large-scale implementation will require strong interdisciplinary collaboration at the intersection of mathematics, physics, and computer science.