Exponential-Fractional Stochastic Volterra Equations

• Exponential-fractional SVIEs are integral equations with convolution kernels that exhibit singular behavior and exponential decay, capturing both rough and long-memory effects.
• Their analysis leverages Besov regularity, resolvent calculus, and stochastic integration techniques to ensure solution existence, uniqueness, and robust moment bounds.
• These equations underpin modern rough volatility models and facilitate numerical schemes that accurately approximate non-Markovian dynamics in financial and applied stochastic analysis.

Exponential-Fractional Stochastic Volterra Integral Equations (SVIEs) form a paradigmatic class of non-Markovian, non-local stochastic dynamical systems characterized by convolution kernels with both singularity and exponential decay. They model systems with memory and rough or non-semimartingale noise, and are central in modern stochastic analysis, mathematical finance, and the theory of rough paths. Their analysis combines tools from Besov regularity, resolvent calculus, stochastic calculus, and fractional integration. The following presents their rigorous definition, main analytical features, solvability, representation theory, kernel-dependent behavior (rough and long memory), approximation schemes, and functional extensions.

1. General Structure and Analytical Setting

Exponential-fractional SVIEs are defined by convolution equations of the form

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

where

• $$K(t) = \frac{t^{\alpha-1} e^{-\rho t}}{\Gamma(\alpha)}$$ for $\alpha>0, \rho\geq 0$ is the exponential-fractional kernel,
• $\phi$ is a deterministic modulation (with $\phi(0)=1$),
• $\mu$, $\lambda$, and $\sigma$ are coefficients, with $\sigma$ Lipschitz in $x$,
• $W$ is a standard Brownian motion or general semimartingale.

Under appropriate analytic assumptions—$K\in L^{2\beta}_{\mathrm{loc}}$ for some $\beta>1$, and Hölder continuity of $K$—one obtains unique pathwise continuous solutions with $\mathbb{E}[\sup_{s\le T}|X_s|^p]<\infty$ for all $p\ge1$ and $T<\infty$ (Gnabeyeu et al., 5 Nov 2025). The structure includes both finite-horizon and infinite-memory (stationary) settings; the latter requires refined weighted $L^p$-spaces and smallness/growth conditions on the kernel and coefficients (Chong, 2014).

Fractional Brownian motion and Lévy processes as drivers are included within this framework, via the apparatus of convolutional rough paths and paracontrolled distributions (Prömel et al., 2018). For affine coefficients, the resulting processes extend affine diffusions well outside the Markovian class and form the basis for rough volatility modeling (Jaber et al., 2017).

2. Exponential-Fractional and Power-Law Kernels

The exponential-fractional kernel,

$$K(t) = e^{-\lambda t} t^{\alpha-1} / \Gamma(\alpha),\quad \alpha>0,$$

interpolates between pure fractional ($\rho=\lambda=0$) and exponentially tempered memory. For $\alpha<1$, the kernel is singular at zero, representing memory with rough path effects; for $\alpha>1$ it is integrable at zero, producing long-range dependence and persistent covariance decay (Gnabeyeu et al., 5 Nov 2025).

The smoothness and decay of $K$ are captured precisely using Besov regularity:

• Near $t=0$, $K(t)\sim t^{\alpha-1}$ so $K\in B_{1,\infty}^\gamma$ for any $\gamma<\alpha$ (Prömel et al., 2018).
• The associated "resolvent" $R_\lambda$ satisfies $$R_\lambda(t)+\lambda \int_0^t K(t-s)R_\lambda(s)ds = 1$$.

When $X$ is driven by rough paths, such as fractional Brownian motion ($H>1/3$), the interpretation of the stochastic integral requires the paracontrolled distributions and convolutional rough path theory, which generalizes classical stochastic calculus to non-semimartingale settings (Prömel et al., 2018).

3. Existence, Uniqueness, and Moment Bounds

The existence-uniqueness theory for exponential-fractional SVIEs is formulated under regularity and smallness of coefficients and kernel (Prömel et al., 2018, Chong, 2014, Gnabeyeu et al., 5 Nov 2025):

• Analytic prerequisites: $K\in B_{1,\infty}^\gamma$, $L\in B_{1,\infty}^\delta$ (for deterministic drift/noise kernels).
• The driver $\xi$ (formally $dX/dt$) satisfies Besov regularity, and the resonant term $\pi(K*\xi,\xi)$ exists.
• For stationary solutions (infinite past), weighted $L^p$-spaces and explicit integrability/small-gain conditions on the kernel and Lévy characteristics $\zeta_p$, $|B_1|$ ensure solvability (Chong, 2014).

For the paracontrolled/rough path regime, the solution map is locally Lipschitz in the corresponding Besov topology, and the solution $u$ admits a decomposition of the form

$$u = T_{u^{(1)}}(K*\xi) + T_{u^{(2)}}(L*\xi_2) + u^\#,$$

where $T$ is Bony's paraproduct, and $u^\#\in B_{p/2,\infty}^{2\alpha}$ encapsulates remainders (Prömel et al., 2018).

A table summarizing key solvability domains:

Setting	Kernel/Noise	Solvability Condition
Finite horizon, Brownian	$K$ fractional, $W$ semimart.	Besov reg., $\Delta$ small
Infinite memory, Lévy	$G_{α,β}$, Lévy basis	$L^p$, small gain in $K$, $\nu$
Rough path	$K$ singular, fBM/Lévy	Paracontrolled, resonant μ

4. Representation, Resolvent, and Variation of Constants

The representation theory for exponential-fractional SVIEs draws on the resolvent approach and explicit variation of constants formulae (Hamaguchi, 2021):

• For deterministic linear kernel coefficients, the matrix or scalar resolvent is constructed via series

$$R(t,s) = \sum_{n=1}^\infty e^{\alpha(t-s)} \frac{(t-s)^{n\beta-1}}{\Gamma(n\beta)} = e^{\alpha(t-s)}(t-s)^{\beta-1} E_{\beta,\beta}((t-s)^\beta),$$

where $E_{\beta,\beta}$ is the Mittag–Leffler function.

• The mild solution is then written as

$$X(t) = g(t) + \int_0^t Q(t,s)g(s)ds + \int_0^t R(t,s)g(s)dW(s),$$

where $Q$ and $R$ are the deterministic and stochastic resolvents, respectively.

In the nonlinear setting with convolutional rough drivers, the solution is encoded via the paracontrolled ansatz and a fixed-point argument controlling all resonant terms (Prömel et al., 2018).

5. Stationarity and "Fake Stationarity" Phenomena

True strong stationarity (invariance of all finite-dimensional marginals) in exponential-fractional SVIEs generally fails except in degenerate or constant-kernel cases (Gnabeyeu et al., 5 Nov 2025). However, one can induce "fake stationarity," distinguished in two forms:

• Type I: All marginal distributions share constant mean and variance, achieved by selecting the volatility coefficient $\varsigma$ according to a Volterra–Wiener–Hopf equation,

$$c\lambda^2 (1-(\phi-f*\phi)^2(t)) = (f^2*\varsigma^2)(t), \quad c = v_0/\bar\sigma^2.$$

• Type II (Gaussian case): Marginals $X_t$ are distributionally invariant, $X_t\overset{d}{=}X_0$, for all $t$.

The resolvent $R_\lambda$ and its derivative $f(t) = -R_\lambda'(t)$ play a central role in both the mean/variance formulas and the construction of fake stationary regimes: $$\mathbb{E}[X_t] = x_\infty, \qquad \mathrm{Var}(X_t) = v_0,$$ where explicit expressions relate $x_\infty$ and $v_0$ to $\lambda, \mu_0, \bar\sigma^2$, and the initial law.

6. Memory Regimes: Roughness and Long-Range Dependence

The parameter $\alpha$ in $K$ controls both the local roughness and the decay of memory:

• $\alpha\le1$ (e.g., $\alpha=H+1/2\in(1/2,1)$) yields "rough path" regimes, short memory, and strong contraction (resolvent $R_\lambda(t)\to0$ as $t\to\infty$).
• $\alpha>1$ gives integrable kernels at the origin and slow polynomial (hyperbolic) decay of auto-covariance, characteristic of long-memory processes. In these cases, the process exhibits persistence and slow mixing, with a "weak" $L^2$-stationary limit (Gnabeyeu et al., 5 Nov 2025).

The exponential prefactor provides an interpolation between pure fractional and exponentially decaying memory, creating flexibility for modeling both rough volatility and mean-reverting long-memory phenomena.

7. Approximation, Simulation, and Numerical Schemes

Numerical treatment of exponential-fractional SVIEs relies on kernel approximation and time-stepping techniques:

• The exponential sum approximation of the kernel, $K_N(t)=\sum_{i=0}^N w_i e^{-x_i t}$, is constructed via Gaussian quadrature over a geometric mesh for optimal convergence, with superpolynomial ($O(e^{-c\sqrt N})$) rate in $N$ (Bayer et al., 2021).
• The original non-Markovian Volterra SDE is thus replaced by a coupled $(N+1)$-dimensional Markovian SDE for $X^N_t$, composed of weighted sums of Ornstein-Uhlenbeck processes.
• The strong error in $L^p(\Omega; C([0,T]))$ is $O(N^{(1-H)/2} e^{-\alpha\sqrt N/A})$, and in application to rough volatility models (e.g., rough Heston), this leads to uniformly negligible model error for $N\approx 50$–$200$ (Bayer et al., 2021).
• Higher-order time stepping methods, such as the Mittag-Leffler Euler integrator, leverage the explicit resolvent structure to achieve strong convergence of order $O(\Delta t)$, outperforming backward Euler–convolution quadrature methods in time-fractional stochastic PDEs (Kovács et al., 2018).

8. Applications and Connections

Exponential-fractional SVIEs are foundational in rough volatility modeling, where they capture empirically observed roughness in stochastic variance, as in the rough Heston model (Jaber et al., 2017, Bayer et al., 2021). The exponential-affine transform and convolutional Riccati-Volterra equations allow for tractable Laplace transforms and efficient Fourier pricing, despite the non-Markovian nature of the processes. Tractable fake-stationarity regimes underlie a new class of stabilized volatility models (Gnabeyeu et al., 5 Nov 2025). The analytical machinery applies broadly across rough path analysis, non-Markovian stochastic dynamics, and memory-driven SPDEs.

They also serve as testbeds for stationary theory in non-Markovian SVIEs, demonstrating the distinction between strong, fake, and weak stationary regimes and providing a flexible substrate for both applied probability and theoretical developments in stochastic analysis.