Stochastic Volterra Equations (SVEs)

• Stochastic Volterra Equations (SVEs) are defined as stochastic integral equations with non-Markovian memory effects introduced via deterministic Volterra kernels.
• They model path-dependent dynamics and are applied in areas such as rough volatility modeling in finance and anomalous diffusion in physics.
• Their formulation using convolution integrals with deterministic kernels presents unique mathematical challenges and insights in stochastic analysis.

Stochastic Volterra Equations (SVEs) are a class of stochastic integral equations in which memory effects are encoded via deterministic Volterra kernels, resulting in fundamentally path-dependent dynamics. Their non-Markovian structure, profound mathematical challenges, and wide-ranging applications—from rough volatility modeling in finance to anomalous diffusion in physics—position SVEs as central objects in the modern theory of stochastic processes with memory.

1. Definition and Canonical Framework

A general $\mathbb{R}^d$-valued SVE on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge0}, P)$ with an $m$-dimensional Brownian motion $W$ is given by

$$X_t = g(t) + \int_0^t K^b(t-s) b(s,X_s)\,ds + \int_0^t K^\sigma(t-s) \sigma(s,X_s)\,dW_s, \quad t\ge0.$$

Here:

• $X_t \in \mathbb{R}^d$ is the primary unknown;
• $g: [0, \infty) \to \mathbb{R}^d$ encodes initial history or input;
• $$K^b, K^\sigma \in L^2_{\rm loc}(\mathbb{R}_+; \mathbb{R}^{d\times d})$$ are deterministic “Volterra kernels” dictating the memory structure;
• $$b: \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^d$$ and $$\sigma: \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^{d\times m}$$ are drift and diffusion coefficient functions.

The SVE reduces to a classical Itô SDE if $K^b$ and $K^\sigma$ are simply the identity times indicator on $[0, t]