Stochastic Volterra Equations

• Stochastic Volterra Equations are stochastic integral equations whose drift and diffusion coefficients are convoluted with deterministic kernels, capturing memory effects.
• They are applied in areas such as rough volatility in finance and control systems with delay, illustrating their broad practical significance.
• The analysis of SVEs involves addressing challenges in well-posedness, regularity, and numerical approximation due to their intrinsic non-Markovian, path-dependent nature.

A stochastic Volterra equation (SVE) is a stochastic integral equation in which both the drift and diffusion coefficients may be convoluted in time with deterministic kernels, often encoding memory or hereditary effects. SVEs generalize classical stochastic differential equations (SDEs), appearing pervasively in rough paths, turbulence, non-Markovian modeling, finance (e.g., rough Heston), and control of systems with delay or after-effects. The Volterra structure yields strong path-dependence, leading to significant mathematical challenges regarding well-posedness, regularity, Markovianity, and numerical analysis.

1. General Formulation and Kernel Structure

The prototypical SVE is

$$X_t = x_0 + \int_0^t K_\mu(s,t)\, \mu(s,X_s)\, ds + \int_0^t K_\sigma(s,t)\, \sigma(s,X_s)\, dB_s,$$

where $K_\mu,K_\sigma$ are deterministic kernels on the right triangle $\Delta_T = \{0 \le s \le t \le T\}$, $\mu,\sigma$ are measurable drift and diffusion coefficients, $B$ is standard Brownian motion, and $x_0$ is the (possibly path-dependent) initial datum. When $K_\mu(s,t)=\delta(t-s)$ and $K_\sigma(s,t)=\delta(t-s)$, the equation reduces to an Itô SDE. Nontrivial kernels naturally introduce time-nonlocality: $X_t$ depends on the entire trajectory $\{X_s: 0\le s < t\}$ (Prömel et al., 2022, Friesen et al., 25 Oct 2025).

Common kernel examples include:

• Smooth kernels: $K(t,s)\in C^2$, inducing "mild" memory.
• Completely monotone: $K(t) = \int_0^\infty e^{-x t} \nu(dx)$, e.g., $K(t)=t^{H-\frac{1}{2}}$, encoding power-law memory (Huber, 14 Jun 2024).
• Singular/fractional kernels: $K(t,s)=(t-s)^{H-\frac{1}{2}}$, $H\in(0,1/2]$, fundamental in rough volatility (Prömel et al., 2022, Bayer et al., 2021).

In the presence of jumps, the SVE generalizes to

$$X_t = X_0 + \int_0^t K(t-s)\, \mu(X_s) ds + \int_0^t K(t-s)\, \sigma(X_s) dB_s + \int_0^t\!\!\int_U K(t-s) \eta(X_{s-},u) N(ds,du),$$

where $N$ is a PRM, and $\eta$ models the jump amplitude (Alfonsi et al., 29 Feb 2024).

2. Existence, Uniqueness, and Regularity Theory

Well-posedness for SVEs is sharply governed by the regularity and integrability of $K_\mu,K_\sigma$, the growth/Hölder properties of $\mu,\sigma$, and the interaction with noise regularity.

Strong existence and uniqueness is established for one-dimensional SVEs with kernels satisfying explicit time regularity (e.g., $$\int_0^{t'} |K_\sigma(s,t')-K_\sigma(s,t)|^{2+\varepsilon}ds\le C|t'-t|^{\gamma(2+\varepsilon)}$$), nondegeneracy ($|K_\sigma(t,t)| \ge c > 0$), and drift/diffusion coefficients that are linear growth and locally $1/2+\xi$-Hölder in state (Prömel et al., 2022). Under these, for $p > \max\{1/\gamma, 1+2/\varepsilon\}$,

• There exists a unique strong $L^p$-solution.
• All moments are bounded: $\sup_t E|X_t|^q<\infty$.
• $X$ admits a modification with $\beta$-Hölder continuous paths for all $\beta < \gamma$.
• $X_t - x_0(t)$ is a semimartingale.

Weak existence for broader classes (including time-dependent, non-Lipschitz, and inhomogeneous coefficients) is established via the Volterra local martingale problem (Prömel et al., 2022). The key is approximating by Lipschitz coefficients, tightness via moment bounds, and limit identification through Skorokhod's representation. Weak solutions and the martingale problem are equivalent.

Pathwise uniqueness in the jump-diffusion setting is achieved when the kernel is nonnegativity-preserving ($K\ge0, K(0)>0, K\in C^2$), the coefficients satisfy local Yamada–Watanabe-type bounds (local Lipschitz in drift, square-root-Hölder in jump amplitude), and a jump monotonicity condition holds for $\eta$ (Alfonsi et al., 29 Feb 2024). The proof exploits time-grid discretization and a refined Itô/Yamada–Watanabe argument to control nonnegativity.

Importantly, for non-Markovian/memory equations, the Markov property almost always fails except when the kernel is precisely exponential $K(t)=c\,e^{-\lambda t}$, in which case the SVE reduces to a time-homogeneous Markovian SDE (Friesen et al., 25 Oct 2025).

3. Regularity, Invariant Measures, and Markovian Lifts

Sample-path regularity is governed by the interplay between kernel singularity and coefficients' Hölder indices. For SVEs in Hilbert spaces with additive local $L^2$-martingale noise, under sectoriality and $H^\infty$-calculus for the deterministic operator, if the kernel admits Laplace-bounded transform, the solution's path-regularity matches that of the driving noise (Schnaubelt et al., 2015). For completely monotone kernels (e.g., rough volatility), one constructs a Markovian lift in a weighted Sobolev space (e.g., dual of $W^{1,2}_w([0,\infty))$), pushing the Volterra memory into a nonlocal infinite-dimensional Markovian SEE (Huber, 14 Jun 2024, Benth et al., 2019). This allows leveraging SPDE machinery for existence, uniqueness, invariant measures, and Itô formulas.

For ergodicity and invariant measures, Lyapunov functionals on the lifted space and dissipativity conditions (on the generator and nonlinearities) ensure invariant probability measures for the lifted SEE and consequently for the original finite-dimensional SVE (Huber, 14 Jun 2024, Benth et al., 2019, Bianchi et al., 2023).

4. Numerical Analysis: Discrete Schemes and Markovian Approximations

Classical explicit Euler and Milstein schemes extend to SVEs with regular kernels and coefficients, yielding strong $L^p$-error rates dictated by the kernel's Hölder regularity: Euler converges at $O(\delta_n^{\alpha\wedge1})$ and Milstein attains $O(\delta_n^{2\alpha'\wedge1})$, where $\delta_n$ is the mesh size and $\alpha,\alpha'$ are regularity indices (Richard et al., 2020). When paired with multilevel Monte Carlo, complexity can be nearly optimal even for rough kernels.

For rough/fractional kernels, Markovian lifts via sum-of-exponentials approximations yield finite-dimensional SDE systems converging to the SVE at superpolynomial (Gaussian-in-$\sqrt{N}$) rate (Bayer et al., 2021). This underpins practical simulation of rough volatility models and option pricing, with Markovian schemes offering tractable and stable numerical workflows even when the exact process is not semimartingale.

Compound Poisson approximations further enable simulation in the presence of singular, discontinuous, or even discontinuous-in-time coefficients, maintaining strong convergence rates $O(\varepsilon^{\gamma/(2(2+\gamma))})$ where $\gamma$ is the kernel-regularity index and $\varepsilon$ the jump intensity parameter (Zhang et al., 31 Oct 2025).

High-order weak approximation is developed via cubature methods based on functional Itô–Taylor expansion and signature matching, yielding superior computational efficiency over Euler schemes in suitable (e.g., smooth, moderate-dimensional) contexts (Feng et al., 2021).

5. Mean-Field, Control, and Neural SVEs

Mean-field SVEs (MVSVE) have solution dynamics influenced by the distribution of the process, i.e.,

$$X_t = X_0 + \int_0^t K_\mu(s,t) b(s, X_s, \mathcal{L}(\phi(X_s))) ds + \int_0^t K_\sigma(s,t) \sigma(s, X_s, \mathcal{L}(\psi(X_s))) dW_s,$$

with kernels accommodating significant singularities. Well-posedness is established for multi-dimensional Lipschitz and one-dimensional Hölder cases, with propagation of chaos proven for the particle system with explicit rates (Prömel et al., 2023).

Stochastic control of SVEs—including optimal harvesting, LQ problems, and stochastic maximum principle—entails new technical ingredients. For example, time-changed Lévy SVEs require backward SDEs with non-anticipating stochastic derivatives and Hamiltonian analysis tailored to the path-dependence and filtrations (Nunno et al., 2020). In the LQ setting, SVEs with Laplace-representable kernels are lifted to infinite-dimensional Markovian systems, with the value function characterized via Banach-valued Riccati equations and the feedback law depending on state variables living in a space such as $L^1(\mu)$ (Jaber et al., 2019).

Neural SVEs generalize neural SDEs by learning both coefficients and time kernels—a necessity for modeling path-dependent data with memory. The architecture replaces the Markovian structure with parameterization of kernel and coefficient MLPs, leading to models that provably approximate arbitrary SVE law dynamics under moment and regularity constraints. Neural SVEs consistently outperform DeepONet and neural SDEs in data regimes requiring nonlocal memory (Prömel et al., 28 Jul 2024).

6. Singularities, Paracontrolled Analysis, and Limit Laws

Solving SVEs with singular ($K$ not in $L^1$) or rough inputs is analytically subtle. The paracontrolled distribution method treats convolutional rough paths (enhancements including "resonant terms" such as $\pi(K*\xi,\xi)$) and delivers existence and uniqueness in Besov-scale function spaces, under regularity conditions on both the driving noise and the kernel. The solution map is locally Lipschitz in these norms (Prömel et al., 2018). This theory encompasses delayed, fractional-derivative, and moving-average SVE models and provides a deterministic route to pathwise solutions even when classical stochastic calculus fails.

When the noise amplitude is small, a Malliavin-calculus-based fluctuation theory gives that the centered and rescaled solution converges to a Gaussian process satisfying a linear SVE, with $d_{TV} = O(\varepsilon)$ for the total variation distance, and second-order corrections quantified explicitly (Dung et al., 15 Nov 2025).

Local time functionals of Lévy processes can themselves be described by SVEs driven by Poisson random measures with affine dual representations, yielding Hölder regularity, moment bounds, and convex Laplace transforms amenable to duality theory (Xu, 2021).

7. Open Problems, Markovianity, and Applications

For general non-exponential kernels, SVEs fundamentally violate the time-homogeneous Markov property: their finite-dimensional law transitions cannot be captured by Chapman–Kolmogorov equations except when $K(t)\propto e^{-\lambda t}$ (Friesen et al., 25 Oct 2025). This precludes Markovian reduction unless entirely lifting the memory to an infinite-dimensional state. Such insights have driven both theoretical advances (e.g., rough volatility Markovian lifts) and efficient approximations for calibration and pricing (as in option pricing under rough Heston).

The applications of SVEs are vast: mathematical finance (rough volatility, rough Heston, fractional CIR), control of systems with memory, infinite-dimensional filtering, and stochastic modeling in biology or engineering. Efficient, stable simulation and inference in non-Markovian settings continues to drive theory, numerical analysis, and data-driven model design.