Fake Stationary Volterra Processes

• Fake stationary Volterra processes are stochastic processes driven by SVIEs that achieve constant mean and variance through deterministic stabilization despite lacking full stationarity.
• The methodology uses resolvent equations and memory kernels to counteract time inhomogeneity, ensuring moment invariance in non-Markovian systems.
• Applications in rough volatility and rough Heston models demonstrate how empirical moment stability can mask the true non-stationary structure in complex dynamical systems.

A fake stationary Volterra process is a class of stochastic process—typically governed by stochastic Volterra integral equations (SVIEs) with memory kernels—that, while lacking true strong (distributional) stationarity, can be engineered to possess constant mean and variance for all time by suitable deterministic stabilization of the volatility. This regime is of particular importance in the context of non-Markovian systems, rough volatility modeling, and the rigorous characterization of apparent versus genuine stationarity in systems with memory. The concept is motivated by the proof that—for nontrivial Volterra kernels—no strong stationary regime exists, yet observable marginals can be made indistinguishable from stationary in first and second moments via the introduction of a deterministic stabilizer function. This regime is known as “fake stationarity,” coined to distinguish it from true stationarity, where all finite-dimensional distributions are invariant under time-shift.

1. SVIEs, Stationarity Impossibility, and the Need for Stabilizers

Let $X$ be the solution to a general stochastic Volterra integral equation with memory kernel $K$,

$$X_t = X_0\phi(t) + \int_0^t K(t-s)\bigl(\mu(s) - \lambda X_s\bigr)\,ds + \int_0^t K(t-s)\sigma(s,X_s)dW_s,$$

where $W$ denotes Brownian motion, $\phi$ and $\mu$ are deterministic, and $\sigma$ is Lipschitz in its second argument. Such SVIEs, including those with mean-reverting affine drift, fundamentally lack strong stationary solutions unless $K$ is constant. The crux is that the convolutional memory $K$ prevents the covariance structure from reducing to a function of lag; even under standard (e.g., $L^2$-bounded) kernel conditions, any attempt to find time-invariant finite-dimensional laws is obstructed except in the Markovian (trivial) case.

To impose constant mean and variance on the solution—thus producing a “fake stationary” regime—the volatility function is decomposed into a time-dependent stabilizer $\varsigma$ and a state function $\sigma(x)$,

$\sigma(t,x) = \varsigma(t)\,\sigma(x).$

The stabilizer $\varsigma$ is then chosen to precisely offset the time-inhomogeneity introduced by the kernel $K$, thereby freezing the first two moments at constant values.

2. Mathematical Construction: Resolvent and Stabilizer Equations

The construction relies critically on the deterministic resolvent $R_\lambda$ of $K$, defined by

$$R_\lambda(t) + \lambda \int_0^t K(t-s)R_\lambda(s)ds = 1, \quad R_\lambda(0)=1,$$

with

$f_\lambda(t) = -R'_\lambda(t).$

The SVIE can be rewritten via the Wiener–Hopf trick as

$$X_t = X_0\left[\phi(t) - (f_\lambda*\phi)(t)\right] + \frac{1}{\lambda}(f_\lambda*\mu)(t) + \frac{1}{\lambda} \int_0^t f_\lambda(t-s) \sigma(s,X_s)dW_s.$$

To achieve constant mean $x_\infty$ and constant variance $v_0$, the following system is imposed:

• Mean: For $X_0 = x_\infty$ and

$$\phi(t) = 1 - \lambda \int_0^t K(t-s)\left(\frac{\mu(s)}{\lambda x_\infty} - 1\right)ds,$$

the mean remains fixed.

• Variance: To force $\operatorname{Var}(X_t) \equiv v_0$ and $\mathbb{E}[\sigma^2(X_t)] \equiv \bar{\sigma}^2$, one solves the stabilizer equation

$$c\lambda^2\left[1 - \left(\phi - f_\lambda*\phi\right)^2(t)\right] = (f_\lambda^2 * \varsigma^2)(t), \qquad c = \frac{v_0}{\bar{\sigma}^2}.$$

Under conditions such as $f_\lambda \in L^2$, this Volterra equation has a unique bounded, integrable solution for $\varsigma^2$.

In the affine case, one recovers the reduced resolvent form

$$X_t = X_0 R_b(t) + \frac{a}{b}(1-R_b(t)) + \frac{1}{b}\int_0^t f_b(t-s) \varsigma(s)\sigma(X_s) dW_s,$$

and builds the stabilizer via

$(f_b^2 * \varsigma^2)(t) = c b^2 (1 - R_b(t)^2).$

3. Characteristics of the Fake Stationary Regime

The fake stationary regime of order two (type I) is realized when, for all $t\ge 0$,

$$\mathbb{E}[X_t] = x_\infty, \qquad \operatorname{Var}(X_t) = v_0, \qquad \mathbb{E}[\sigma^2(X_t)] = \bar{\sigma}^2,$$

even though the full finite-dimensional distributions are not time-invariant, except in the degenerate Markov case. The covariance, unless $K$ is constant, depends separately on $s$ and $t$: $$\operatorname{Cov}(X_s, X_t) = v_0 R_b(s)R_b(t) + \frac{\bar{\sigma}^2}{b^2} \int_0^s f_b(t-u)f_b(s-u)\varsigma(u)^2 du,$$ so that only the first and second moments are frozen; higher-order dependencies remain non-stationary.

A type II fake stationary regime would additionally have $X_t \stackrel{d}{=} X_0$ for all $t$; this only occurs in trivial cases.

The implication is twofold:

• Processes governed by non-Markovian memory kernels may exhibit empirical moment-stationarity on any finite window, misleading finite-sample statistical testing based solely on moment matching.
• Only verification of the underlying integrability and the structure of the memory kernel distinguishes genuine stationary regimes from fake stationary ones.

4. Long-Run Asymptotic Behavior: Weak Stationarity and $L^p$-Confluence

Even though solutions lack strong stationarity, long-time shifted versions of the process display convergence in law toward $L^2$-stationary limiting processes. The time-shifted process $X^{(t)}_u = X_{t+u}$, for fixed $u$, is tight in $C([0, U], L^2)$ and, as $t \to \infty$, its finite-dimensional distributions converge to a limit $\Xi$ with

$$\mathbb{E}[\Xi_u] = x_\infty, \qquad \operatorname{Cov}(\Xi_{u_1}, \Xi_{u_2}) = C_{\Xi}(u_2-u_1)$$

with $C_\Xi$ universal and computable from $f_\lambda$ or $f_b$. This limit is $L^2$-stationary, i.e. all moments of the same lag coincide, but the original process retains its non-stationary structure on all finite intervals.

An important property, $L^p$-confluence, ensures that starting from different initial distributions, the processes’ marginals converge in $L^p$ norm as $t \to \infty$, i.e., independence from initial conditions in long-time law (Gnabeyeu et al., 5 Nov 2025, Pagès, 26 Jan 2024).

5. Examples and Applications: Stabilized Rough Heston and Volatility Models

Applications include the “fake stationary quadratic rough Heston model” (Pagès, 26 Jan 2024), where the memory kernel is fractional: $$K(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)},\quad R_b(t) = E_{\alpha}(-bt^{\alpha}),\quad f_b(t) = -\frac{d}{dt}R_b(t),$$ and the stabilizer $\varsigma(t)$ has explicit series representations. In this setting, despite the lack of strong stationarity, all marginals have constant mean and variance, and long-time behavior converges to a universal $L^2$-stationary limit with Mittag–Leffler covariance.

This construction is crucial in financial mathematics (notably in rough volatility and rough Heston models) where empirical stationarity of increments on observed horizons may obscure structural non-stationarity in the model. The modeler can enforce moment stability via the stabilizer, producing processes with desired long-run covariance while retaining rough or long-memory structure.

6. Model Diagnostics and Statistical Implications

A key modeling implication is that fake stationarity allow processes to mimic stationary structure in empirical analysis over finite samples, yet lack a true invariant law—especially dangerous for statistical procedures relying solely on constancy of first two moments. Only integrability checks or analysis of higher-order cumulants and covariance structures distinguish fake stationary from strictly stationary processes (Čoupek, 2017).

For pseudo-stationary processes driven by Volterra noise (such as fBm or Rosenblatt) where the tail integral condition

$$\int_0^\infty \|S(r)\Phi\|_{L_2}^{2/(1+2\alpha)}dr<\infty$$

fails, the process can appear empirically stationary in finite windows, but possesses no proper limiting law as $t\to\infty$. In physically modeled systems, this distinction is not just theoretical: it governs the existence of invariant distributions, ergodic behavior, and the reliability of bootstrapping and statistical estimation schemes.

7. Comparison with True Stationarity and Volterra Bootstrap

True stationary Volterra processes—those with absolutely summable and fading-memory Volterra kernels in univariate time series—admit rigorous cumulant-matching and bootstrap schemes capable of preserving higher moments under resampling (Sirotko-Sibirskaya et al., 2020). In contrast, fake stationary regimes for continuous-time SVIEs can never be realized as time-shift invariant solutions except in the constant kernel case; their higher moments and dynamical dependence always betray the memory-induced time inhomogeneity unless further degeneracies are imposed.

In summary, fake stationary Volterra processes are a structural tool to engineer (or deconstruct) constant-moment processes in systems ruled by memory kernels, both revealing fundamental obstructions to stationarity and enabling controlled stabilization of mean–variance in modeling contexts that demand marginal stationarity under inescapable long-range dependence.