Mixed Fractional Brownian Motion

• Mixed Fractional Brownian Motion is a unified stochastic model that blends standard Brownian dynamics with fractional components to capture both short-term randomness and long-range dependence.
• It employs independent Brownian and fractional processes with detailed covariance and spectral analyses to elucidate path regularity, small-ball asymptotics, and non-semimartingale behavior.
• The model is extensively applied in financial mathematics and stochastic control, offering analytical tractability, simulation flexibility, and robust statistical inference tools.

A mixed fractional Brownian motion (mfBm) is a Gaussian process formed by superimposing a standard Brownian motion and an independent fractional Brownian motion, generalizing to finite or infinite mixtures of independent fBms with various Hurst parameters and weights. This construction provides a unified stochastic model that interpolates between short-memory (Brownian) and long-memory (fractional) regimes and is crucial for areas such as long-range dependence modeling, statistical inference for stochastic processes, filtering, and arbitrage-free financial mathematics.

1. Definitions and Canonical Construction

The classical mixed fractional Brownian motion, introduced by Cheridito (2001), is defined as

$M_t = a B_t + b B^H_t,$

where $B_t$ is a standard Brownian motion, $B^H_t$ is an independent fractional Brownian motion with Hurst index $H \in (0,1)$, and $a, b \in \mathbb{R}$. The process $B^H$ has covariance

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

For finite mixtures, one considers

$X_t = \sum_{i=1}^n \sigma_i B^{H_i}_t,$

with $B^{H_i}$ independent fBms and $\sigma_i > 0$. The infinite superposition, called multi-mixed fractional Brownian motion (mmfBm), is defined as the $L^2$-limit

$M_t = \sum_{k=1}^\infty \sigma_k B^{H_k}_t,$

under the condition $\sum_{k=1}^\infty \sigma_k^2 < \infty$ and $H_k \in (0,1)$. This formalism extends to infinite-dimensional settings by mapping the construction onto a real separable Hilbert space with a nuclear covariance operator (Almani et al., 2021, Rao, 2021).

Generalizations such as the mixed generalized fractional Brownian motion (MGfBM) allow for the inclusion of time-reversed fBm components:

$M_t^{H}(a, b, c) = a B_t + b B^H_t + c B^H_{-t},$

leading to a richer family encompassing sub-fBm, Zili's generalized fBm, and other classical Gaussian processes (Mliki et al., 2021).

2. Covariance Structure and Dependence Properties

The covariance function for mfBm (with independent Brownian and fBm components) is

$$\mathrm{Cov}(M_s, M_t) = a^2 \, (s \wedge t) + b^2 \frac{1}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right).$$

Its increments are, in general, neither stationary nor self-similar unless in Brownian or pure fBm limits. For the MGfBM,

$$C(t,s) = a^2 (t \wedge s) + \frac{b^2 + c^2 - (2^{2H}-2)bc}{2} (t^{2H} + s^{2H} - |t-s|^{2H}) - \frac{bc}{2} ((t+s)^{2H} - |t-s|^{2H}),$$

which demonstrates how the addition of $c B^H_{-t}$ modifies both autocovariance and increment stationarity (Mliki et al., 2021).

For multi-mixed fBm (mmfBm), the covariance is an infinite sum of the elementary fBm covariances:

$$r(t,s) = \frac{1}{2} \sum_{k=1}^\infty \sigma_k^2 \left( |t|^{2H_k} + |s|^{2H_k} - |t-s|^{2H_k} \right).$$

Analysis of the increment covariance shows that the largest Hurst index ($H_\text{sup}$) among the mixture governs the long-range dependence, while the smallest ($H_\text{inf}$) determines path roughness (Almani et al., 2021). In the ccmfBm (completely correlated mixed fBm), the covariance includes cross-terms from the integral representation of $B^H$ with respect to the same $W$:

$$R(t,s) = a^2 (t \wedge s) + ab \int_0^{t \wedge s} (K_H(t,u) + K_H(s,u)) du + b^2 \tfrac{1}{2}(t^{2H} + s^{2H} - |t-s|^{2H}),$$

where $K_H$ is the Molchan–Golosov kernel (Dufitinema et al., 2021).

3. Path Regularity and Semimartingale Properties

The pathwise regularity of mfBm reflects the most singular component in the mixture. For any finite or infinite superposition,

• The sample paths are almost surely Hölder continuous of any order less than $H_\text{inf}$, but not of order greater or equal (Almani et al., 2021, Dufitinema et al., 2021).
• The $p$-variation is finite, infinite, or zero according to whether $p H_\text{inf} > 1, =1, <1$ respectively; the $p$-variation index is $1/H_\text{inf}$ (Almani et al., 2021).
• When $H>1/2$, and $b \neq 0$, mfBm is not, in general, a semimartingale (except for $H>3/4$), nor is it Markov (Mliki et al., 2021, Li et al., 2023). This property impacts stochastic calculus and control, necessitating careful treatment, e.g., using pathwise (Young or Russo–Vallois) integrals or Malliavin calculus.

For ccmfBm, the quadratic variation is governed uniquely by the Brownian component: $[X, X]_t = a^2 t$, and the local Hölder index is $1/2$ regardless of the fBm component, but long-range dependence arises from the tail of the fractional kernel in $K_H$ (Dufitinema et al., 2021).

4. Spectral Theory and Small-Ball Asymptotics

The spectral properties of the covariance operator for mfBm show sharp stratification:

• For $H > 1/2$, the Brownian eigenvalues decay as $n^{-2}$ and dominate those of the fBm, which are order $n^{-2H-1}$; the converse holds for $H < 1/2$ (Chigansky et al., 2018).
• This separation leads directly to small-deviation (small-ball) probability asymptotics: for the $L^2$-norm,
  • If $H > 1/2$,

  $$\log \mathbb{P} \left( \| X \|_{L^2} < \varepsilon \right) = -\frac{1}{8} \varepsilon^{-2} - B(H) \varepsilon^{-1/H} + o(\varepsilon^{-1/H})$$ - If $H < 1/2$,

  $$\log \mathbb{P} \left( \| X \|_{L^2} < \varepsilon \right) = - B(H) \varepsilon^{-1/H} - \frac{1}{8} \varepsilon^{-2} + o(\varepsilon^{-2})$$

$B(H)$ is an explicit function of $H$. At $H = 1/2$, a logarithmic correction term appears (Chigansky et al., 2018, MacKay et al., 2018).

In the presence of deterministic trends, Girsanov-type factorization reduces small-ball problems to zero-trend cases; the optimal splitting of the trend is characterized by a Fredholm integral equation (MacKay et al., 2018).

5. Stochastic Differential Equations and Modelling

Solutions to mixed SDEs driven by mfBm are well-posed under mild conditions:

$$X_t = X_0 + \int_0^t a(s, X_s) ds + \int_0^t b(s, X_s) dW_s + \int_0^t c(s, X_s) dB^H_s$$

• Existence and uniqueness are shown for both independent and dependent noise sources, provided the coefficients satisfy regularity, Lipschitz, and linear growth conditions; the fBm integral is constructed via Young or fractional calculus (Mishura et al., 2011).
• Euler-type numerical schemes yield rates of convergence governed by the roughest noise: mean-square error is $O(\Delta + \Delta^{2H-1})$, where $\Delta$ is the step-size. The phase transition at $H=3/4$ demarcates Brownian- versus fractional-dominated convergence regimes (Mishura et al., 2011).
• Infinite-dimensional analogues (Hilbert space-valued SDEs/SPDEs) have been developed, using Karhunen–Loève expansions, and possess maximum likelihood estimators with consistency and asymptotic normality under natural identifiability and growth conditions (Rao, 2021).

Moderate deviation principles for slow-fast systems driven by mfBm are established via variational representations and Khasminskii averaging. Moderate fluctuations are quantified at the intermediate (non-Gaussian, non-LDP) scale (Yang et al., 2023).

6. Long-Range Dependence, Time-Changes, and Generalized Constructions

Long-range dependence (LRD) of mfBm and its variants is governed by the largest Hurst index present. For mixed or generalized mixtures,

• The increment covariance decays as $n^{2 H_\text{sup} - 2}$ for large lag $n$, yielding LRD if any $H_k > 1/2$ (Almani et al., 2021, Mliki et al., 2021).
• Time changes (subordination) by inverse stable, tempered stable, or gamma subordinators preserve or enhance LRD, giving new decay exponents governed by both the subordinator index and the Hurst parameters. The LRD criterion becomes $0 < 2 \alpha H_1 - \alpha H_2 < 1$, where $\alpha$ is the subordinator index, and $H_1<H_2$ are the Hurst exponents of the two components (Mliki, 2023, Rao, 2023).
• Completely correlated versions (ccmfBm) generated from a single Brownian motion via Volterra kernels have short-time Brownian and long-time fBm character. The transfer principle (invertibility between $X$ and $W$ via Volterra operators) allows explicit simulation and likelihood calculation (Dufitinema et al., 2021).

7. Applications and Modeling Implications

Mixed and multi-mixed fractional Brownian motions serve as flexible models for phenomena exhibiting both short-term randomness and persistent memory:

• In financial mathematics, they model rough volatility, long-memory interest rates, and arbitrage-free price evolution, benefiting from properties like conditional full support and the existence of Gaussian shifts (Almani et al., 2021, Mliki et al., 2021, Rao, 2021, Li et al., 2023).
• In stochastic control, mfBm-driven systems require Malliavin-calculus-based maximum principles; adjoint backward SDEs incorporate both classical and fractional Brownian parts (Li et al., 2023).
• Mixed models allow precise tuning of regularity and memory, facilitate simulation via spectral and Volterra methods, and offer a testing ground for statistical inference tools in non-Markovian, non-semimartingale regimes.

The explicit covariance and spectral representations of mfBm and its generalizations ensure analytical tractability, while the hierarchy of path and dependence properties provides a calibrated framework for advanced stochastic modeling in both finite and infinite dimensions.