Stochastic Volterra Equation

• Stochastic Volterra equation is a stochastic integral equation that generalizes SDEs by using convolution kernels to incorporate memory and hereditary effects.
• It replaces local Markovian dynamics with non-Markovian evolution, requiring specialized conditions on kernel integrability and coefficient regularity for solution existence and uniqueness.
• Applications span rough volatility, viscoelasticity, population genetics, and operator learning, offering a rigorous framework for modeling systems with long-memory effects.

A stochastic Volterra equation is a stochastic integral equation in which the future evolution of a process depends on its entire past history, through convolution with a deterministic “memory kernel.” SVEs generalize standard SDEs by replacing local, Markovian dynamics with path-dependent, non-Markovian evolution, providing a mathematically rigorous framework for modeling systems with hereditary, rough, or long-memory effects. Such equations underpin a wide variety of stochastic models, notably in rough volatility, viscoelasticity, population genetics, functional SPDEs, and modern data-driven operator learning architectures.

1. Mathematical Formulation and Canonical Examples

A typical scalar or $d$-dimensional SVE on a finite horizon $[0,T]$ driven by an $m$-dimensional Brownian motion $W$ is written in convolutional form: $$X_t = X_0 + \int_0^t K_{\mu}(t-s) \, b(X_s) \, ds + \int_0^t K_{\sigma}(t-s) \, \sigma(X_s) \, dW_s,\quad t \in [0, T],$$ Here:

• $X_0$ is the initial state, possibly random.
• $b:\R^d \to \R^d$, $\sigma: \R^d \to \R^{d\times m}$ are drift and diffusion coefficients.
• $K_\mu, K_\sigma: [0,T] \to \R$ are memory kernels, often of convolution type (i.e., $K(t,s) = K(t-s)$).

If $K_\mu \equiv K_\sigma \equiv 1$, this reduces to a classical SDE. If, for example, $K(t-s) = (t-s)^{-\alpha}$ $(0<\alpha<1/2)$, one obtains a “rough” SVE, which rigorously models non-Markovian, memory-dominated phenomena (Prömel et al., 28 Jul 2024, Fukasawa et al., 2021, Prömel et al., 2022).

Prototypical examples:

• Rough Heston model: Volterra kernel with $K(t-s) = (t-s)^{H-1/2}/\Gamma(H+1/2)$, $H\in(0,1/2)$, and square-root diffusion, yielding subdiffusive, rough volatility (Fukasawa et al., 2021, Prömel et al., 28 Jul 2024).
• Generalized Ornstein–Uhlenbeck: $K(t-s) = e^{-\theta(t-s)}$, leading to a Markovian process as a special case (Prömel et al., 28 Jul 2024).
• Local time dynamics for α-stable Lévy processes: SVE driven by a Poisson random measure characterizes the local time field (Xu, 2021).

2. Existence, Uniqueness, and Regularity Theory

General solvability: Existence and uniqueness of strong or weak solutions to an SVE depend on the growth, smoothness, and continuity of the kernel and coefficients. Standard sufficient conditions include:

• Integrability of kernel: $\|K_\mu\|_q<\infty$, $\|K_\sigma\|_{2\tilde q}<\infty$ for appropriate exponents.
• Drift and diffusion: Lipschitz continuity and linear growth, or Hölder continuity in $x$ of order at least $1/2$ for diffusion if $K$ is singular (Prömel et al., 28 Jul 2024, Prömel et al., 2022, Xu, 2021).
• Initial data: Sufficient regularity, often continuous or $\beta$-Hölder.

Pathwise uniqueness and strong existence extend to non-Lipschitz coefficients under generalized Yamada–