---
title: Truncated Variation Functional in fBm
url: https://www.emergentmind.com/topics/truncated-variation-functional-of-fractional-brownian-motions
type: topic
---

# Truncated Variation Functional in fBm

The truncated variation functional offers a critical tool for quantifying the regularity of sample paths of fractional Brownian motion (fBm) by controlling the magnitude and frequency of path oscillations at a fixed truncation scale. For fBm—a family of centered Gaussian processes parameterized by the Hurst index $H\in(0,1)$—the truncated variation functional admits a precise probabilistic and analytic description, including sharp concentration, moment, and asymptotic results. These properties exhibit a phase transition at $H=1/2$ and connect to broader stochastic path properties such as $1/H$-variation and local times.

## 1. Formal Definition and Properties

Let $B^H=(B_t^H)_{t\ge0}$ be an fBm of Hurst index $H\in(0,1)$. The truncated variation on $[0,T]$ at truncation level $\delta>0$ is defined as
$$
TV^\delta(B^H)_{0}^{T} := \sup_{\Pi:0=t_0<\cdots<t_n=T} \sum_{i=1}^n \left(|B_{t_i}^H - B_{t_{i-1}}^H| - \delta \right)_+,
$$
where $(x)_+ = \max\{x,0\}$ and the supremum is over all partitions $\Pi$ of $[0,T]$. 

This functional generalizes total variation, as it measures the sum of those increments of $B^H$ in the partition whose magnitude exceeds the threshold $\delta$. Unlike total variation, $TV^\delta(B^H)$ remains finite for typical sample paths of fBm for all $H$.

For any real-valued function $f:[0,T]\to\mathbb{R}$, the truncated variation also admits the characterization
$$
TV^c(f;[0,T]) = \inf\left\{ TV(g;[0,T]) : \|f-g\|_\infty \leq \frac{c}{2} \right\}
$$
with the infimum attained, where $TV(g;[0,T])$ denotes the total variation of $g$ on $[0,T]$ and $\|\cdot\|_\infty$ the uniform norm. This definition is equivalent to the partition supremum for $fBm$ and, in particular, for any continuous $f$.

One-sided truncated variations are given by
$$
UTV^\delta(B^H)_0^T := \sup_{\Pi} \sum_{i=1}^n \left(B^H_{t_i} - B^H_{t_{i-1}} - \delta\right)_+,
$$
$$
DTV^\delta(B^H)_0^T := UTV^\delta(-B^H)_0^T,
$$
with $TV^\delta(B^H)_0^T = UTV^\delta(B^H)_0^T + DTV^\delta(B^H)_0^T$.

## 2. Exponential Integrability and Moment Generating Functions

The law of $TV^\delta(B^H)_{0}^{T}$ possesses distinctly different tail behaviors depending on the value of $H$.

For $fBm$ with $H\in(0,1)$, there exist constants $A_H,B_H,C_H<\infty$ such that for all $u\geq0$ and all $\delta,T>0$,
$$
\Pr\Bigl(TV^\delta(B^H)_{0}^{T} \geq \delta^{\frac{H-1}{H}}T(A_H+B_Hu) \Bigr) \leq C_H \exp(-u^{2H}).
$$
This "stretched exponential" decay for deviations implies the following moment-generating behavior:
- For $H > 1/2$, $2H > 1$, hence the decay is faster than pure exponential, and $\mathbb{E}[\exp(\alpha\, TV^\delta(B^H))] < \infty$ for all $\alpha > 0$.
- For $H = 1/2$, corresponding to standard Brownian motion, the tail decays like $\exp(-u)$, so the exponential moment exists only below a critical threshold of $\alpha$.
- For $H < 1/2$, $2H < 1$, so the decay is not integrable at infinity, and $\mathbb{E}[\exp(\alpha\, TV^\delta(B^H))] = \infty$ for every $\alpha > 0$ [1211.3870].

## 3. Concentration Inequalities and Large Deviations

Sharp deviation inequalities for the truncated variation center around the conditional mean $\mu_T(\delta) = \mathbb{E}[TV^\delta(B^H)_0^T]$.

**For $H\in(0, \tfrac12)$** (negatively correlated increments), with $\delta \leq T^H$:
$$
\Pr\bigl(|TV^\delta(B^H)_0^T - \mu_T(\delta)| > x\bigr) \leq \bar A_H \exp\left( - \bar B_H T^{-1}\delta^{1/H - 2} x^{1+2H} \min\{T\delta^{1-1/H}, x^{1-2H}\} \right)
$$
for constants $\bar A_H \leq 36$, $\bar B_H>0$ depending on $H$.

**For $H\in[\tfrac12,1)$** (positively correlated increments), with $\delta \leq T^H$:
$$
\Pr(|TV^\delta(B^H)_0^T - \mu_T(\delta)| > x) \leq \bar A \exp\left(-\bar B T^{-2H} x^2 \right)
$$
with universal constants $\bar A \leq 4$, $\bar B \geq 2/\pi^2$.

These results establish sub-Gaussian concentration (i.e., Gaussian-type tails) for $H\geq 1/2$ and heavier-tailed, stretched-exponential concentration for $H<1/2$ [2512.14021], [1211.3870]. The bounds are optimal up to constants for large deviation scales.

## 4. Proof Techniques: Chaining, Gaussian Suprema, and Metric Entropy

The probabilistic analysis of truncated variation leverages a combination of chaining techniques, metric entropy, and Gaussian supremum concentration:

- The truncated variation (and one-sided variants) is represented as a supremum over finitely many affine functions of a centered Gaussian vector.
- Fresen's Gaussian-supremum concentration inequality is employed for finite-dimensional approximations.
- Sharp control requires norming by increment variances, and the analysis differentiates regimes with positive versus negative increment correlation (i.e., $H \ge 1/2$ versus $H < 1/2$).
- Passage to the continuum involves a discretization limit, exploiting the self-similarity and stationary increments of fBm [2512.14021], [1211.3870].

Corollary results elucidate the influence of domain length and truncation parameter on concentration and scaling.

## 5. $1/H$-Variation and Crossing Functional

For $fBm$ on $[0,T]$, the $1/H$-variation along Lebesgue partitions is analyzed in terms of strip (level) crossings. Define
$$
K_{s,t}(c, f) := \sum_{p\in\mathbb{Z}} N_{s,t}(c, f - p c),
$$
where $N_{s,t}(c, g)$ counts up- or down-crossings of the strip $\{y \in [0,c]\}$. For $c \to 0$,
$$
V^{(1/H)}(B^H)_0^T := \lim_{c\to 0} c^{1/H} K_{0,T}(c, B^H) = \mathfrak c_H T \quad \text{a.s.}
$$
with constant $\mathfrak c_H > 0$. Tail probabilities for $K_{s,t}(c, B^H-\rho)$ derive from the sharp concentration theory, with stretched exponential decay in $v$, reflecting the underlying roughness of $B^H$ sample paths [2512.14021].

## 6. Asymptotic Behavior, Mean Scaling, and Law of Large Numbers

Self-similarity and scaling properties of $B^H$ directly yield the asymptotic mean behavior for small truncation:
$$
\mathbb{E}\,TV^\delta(B^H;[0,T]) \asymp T\,\delta^{(H-1)/H} \qquad \text{as }\delta\to 0
$$
with constants dependent only on $H$. For $H = 1/2$,
$$
\mathbb{E}\,TV^\delta(B^{1/2}) \sim T/\delta
$$
and $c\,TV^c(B^{1/2}) \to T$ almost surely as $c\downarrow 0$, reflecting the law-of-large-numbers for partition-wise sum functionals of Brownian motion [1211.3870].

## 7. Connection to Local Times

The strip crossing counts for small $c$ recover the local time density $L^H(T,a)$ of $B^H$ at time $T$. For a bounded test function $g$,
$$
\int_{\mathbb{R}} c^{1/H-1} U_{0,T}(c, B^H - a) g(a) da \longrightarrow \frac{\mathfrak c_H}{2} \int_{\mathbb{R}} L^H(T,a) g(a) da \quad \text{almost surely as } c\to 0,
$$
and the random measures
$$
c^{1/H-1} U_{0,T}(c, B^H - \cdot)\, da
$$
converge almost surely (weakly in $L^1$) to $(1/2)\mathfrak{c}_H L^H(T, \cdot) da$ [2512.14021]. This establishes a direct pathwise link between the truncated variation, crossing counts, and local time structure of fBm.

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These results collectively provide an essentially optimal description of the distributional and almost-sure behavior of the truncated variation functional of fBm, including precise large deviation bounds, moment asymptotics, and ergodic-type scaling limits. The technical framework, utilizing Gaussian concentration, chaining, and partition-supremum methods, is robust for extensions to related Gaussian processes and sample path functionals.

Source: https://www.emergentmind.com/topics/truncated-variation-functional-of-fractional-brownian-motions