Affine Volterra Structure Overview

• Affine Volterra structure is a framework for modeling stochastic processes using Volterra integrals with affine state dependence that captures non-Markovian memory effects.
• Its exponential-affine transform properties yield explicit solutions via a Riccati–Volterra system, ensuring analytic tractability even in the presence of rough volatility and jumps.
• The approach finds applications in finance, insurance, and integrable systems by enabling efficient modeling and simulation of complex, memory-driven dynamics.

An affine Volterra structure describes a class of stochastic processes governed by Volterra-type integral equations with affine state dependence in drift, diffusion, and jump characteristics. This framework generalizes the classical affine Markov processes by incorporating convolutions against deterministic kernels—potentially singular—allowing for path regularity distinct from Brownian motion and capturing non-Markovian memory effects. Crucially, affine Volterra processes retain explicit exponential-affine transform formulas for their (conditional) Laplace–Fourier functionals, characterized by solutions to Riccati–Volterra equations. This property underpins both their analytic tractability and flexibility for modeling roughness, jumps, and memory in a wide range of applications.

1. Definition and Fundamental Structure

An affine Volterra process is defined as the (weak or strong) solution $X=(X_t)_{t\ge0}$, valued in a convex state space $E\subset\mathbb{R}^m$ (often $E=\mathbb{R}_+^m$ for square-root cases), to the stochastic Volterra equation

$$X_t = g_0(t) + \int_0^t K(t-s) b(X_s)\,ds + \int_0^t K(t-s)\,dM_s,$$

where:

• $$g_0 \in L^1_{\mathrm{loc}}(\mathbb{R}_+;\mathbb{R}^m)$$ is a deterministic inhomogeneity,
• $$K \in L^2_{\mathrm{loc}}(\mathbb{R}_+;\mathbb{R}^{m\times d})$$ is a deterministic Volterra kernel (often diagonal or positive, potentially singular),
• The local martingale $M$ is driven by affine dependence in $X$ in both continuous and jump components,

$$dM_s = \sigma(X_s)\,dW_s + \int_{\mathbb{R}^d} \xi\,\widetilde{N}(ds,d\xi),$$

where $W$ is Brownian motion, $\widetilde{N}$ is a compensated Poisson random measure, and

$$b(x) = b_0 + \sum_{k=1}^m x_k b_k, \quad \sigma(x)\sigma(x)^\top = A_0 + \sum_{k=1}^m x_k A_k, \quad \nu(x,d\xi) = \nu^0(d\xi) + \sum_{k=1}^m x_k \nu_k(d\xi).$$

These affine dependencies ensure that the characteristic functional of the process is explicitly governed by a generalized Riccati–Volterra system (Bondi et al., 2022, Jaber et al., 2017).

2. Exponential-Affine Transform: Riccati–Volterra System

The defining property of the affine Volterra structure is that, for each $u \in \mathbb{C}^m$ (in the analyticity strip), the (conditional) Laplace–Fourier transform of $X$ is exponential-affine: $$\varphi(t,u) := \mathbb{E}\left[e^{\langle u, X_t\rangle}\right] = \exp\left(\phi(t,u) + \langle \psi(t,u), X_0\rangle\right),$$ where $\psi(t,u)$ and $\phi(t,u)$ are determined as follows: \begin{align*} \psi(t,u) &= \int_0^t K(t-s) F\left(s,\psi(s,u)\right) ds, \ \phi(t,u) &= \int_0^t \left[ b_0^\top \psi(s,u) + \psi(s,u)^\top A_0 \psi(s,u) + \int_{\mathbb{R}^d}(e^{\xi^\top \psi(s,u)} - 1 - \psi(s,u)^\top \xi)\nu^0(d\xi)\right] ds. \end{align*} Here, the generalized Lévy-Khintchine exponent is

$$F_k(t,z) = f_k(t) + b_k^\top z + z^\top A_k z + \int_{\mathbb{R}^d}(e^{\xi^\top z} - 1 - z^\top \xi)\nu_k(d\xi),$$

with $f$ corresponding to possible exponential tilts. Jumps are incorporated directly at the level of the transform kernel via the jump measures $\nu^0$ and $\nu_k$.

This transform structure extends, for example, directly to square-root Volterra processes (fractional kernels), rough Heston, and multidimensional Wishart-Volterra examples, always yielding tractable transform relations in the exponential-affine form (Bondi et al., 2022, Friesen et al., 2022, Aichinger et al., 2021, Cuchiero et al., 2019).

3. Existence, Uniqueness, and Regularity

Existence and uniqueness of affine Volterra processes rely on:

• Measurable affine growth in $b,a,\nu$;
• Local $L^2$-integrability of $K$ (or $L^1$ for integrable drift);
• Kernel regularity: for instance, that $K$ is continuous and admits a nonnegative resolvent of the first kind (ensured by, e.g., $K$ being completely monotone or a power-law/fractional kernel);
• Nondegeneracy and positivity conditions on state-space and coefficients for invariance (e.g., for $\mathbb{R}_+^m$ valued processes, see (Friesen et al., 2022)).

In particular, uniqueness in law can be characterized via the exponential-affine transform, and non-explosion of the Riccati–Volterra system ensures global existence (i.e., solutions are defined for all $t\ge0$) (Bondi et al., 2022).

4. Resolvent Kernels and State Space Structure

Resolvent operators associated to $K$ play a central role:

• The first-kind resolvent $L$ is defined by $K * L = L * K = I$, and is crucial for expressing the solution and for explicit inversion of Volterra convolutions.
• The second-kind resolvent $R_B$ for a matrix $B$ is given by $R_B * K_B = K_B - R_B$, $K_B = -K B$, with $R_B, E_B$ (resolvent-correction) ensuring integrability and stationarity properties in some settings (Friesen et al., 2022).

In many constructions, especially infinite-dimensional Markovian lifts (see, e.g., (Cuchiero et al., 2018, Cuchiero et al., 2019)), the state space is defined via cones or positivity conditions induced by the resolvent structure, enabling the reduction of non-Markovian Volterra equations to Markov processes in Banach or measure spaces.

5. Stationarity, Ergodicity, and Long-Term Behavior

Affine Volterra processes generally lack genuine stationarity except in trivial (constant kernel) or fully degenerate cases. For square-root Volterra models, limiting distributions $\pi_{x_0}$ can exist and be characterized explicitly in terms of resolvent integrability and initial states (Friesen et al., 2022). Stationarity in the strict sense occurs only if the resolvent $E_\beta \in L^1(\mathbb{R}_+)$, and invariance with respect to $x_0$ further requires that $\int_0^\infty R_\beta(t)\,dt = I_m$.

For fractional (rough) Volterra kernels, “fake” stationarity—marginals with constant mean and variance but not full shift-invariance—is achievable by introducing deterministic stabilizers, yielding families of $L^2$-stationary processes in the sense of tightness and weak convergence (Pagès, 2024). Ergodicity and law of large numbers for Volterra CIR-type processes follow from fine analysis of Riccati–Volterra asymptotics: under mean-reversion $(\beta<0)$, the existence of a stationary law, asymptotic independence of marginals, and LLN in $L^p$ all hold (Alaya et al., 2024).

6. Applications: Jumps, Rough Volatility, and Integrable Systems

The affine Volterra paradigm encompasses a diverse range of models:

• Rough volatility: The rough Heston and square-root models with power-law kernels $K(t)=t^{\alpha-1}/\Gamma(\alpha)$, $\alpha \in (1/2,1)$, capture the stylized features of volatility in financial markets; affine Volterra transform structure ensures tractable pricing and calibration (Jaber et al., 2017, Chevalier et al., 2021).
• Jumps: By incorporating jump measures affine in the state, one obtains tractable extensions with jump discontinuities for stochastic volatility and insurance risk applications (Bondi et al., 2022, Cuchiero et al., 2018).
• Markovian lifts and high-dimensional SPDEs: Infinite-dimensional Markov processes in function or measure spaces provide tractable liftings for non-Markovian systems (Jaber et al., 2018, Cuchiero et al., 2019, Jacquier et al., 2022).
• Integrable discrete systems: In parallel, affine Volterra structures appear in birational Volterra maps and related recurrences, characterized by Lax pairs, first integrals, and connections to hyperelliptic Jacobians; order-$(2g+1)$ integrable recurrences and their algebraic geometry are encoded in the discrete affine Volterra maps (Svinin, 10 Feb 2025, Hone et al., 2023).
• Point processes and simulation: The affine Volterra structure underpins efficient simulation of Hawkes processes and self-exciting jump systems, including deterministic complexity algorithms based on the Volterra representation (Jaber et al., 17 Nov 2025).

7. Mathematical Significance and Further Directions

Affine Volterra structures fundamentally extend the classical affine Markov paradigm, trading semimartingale and Markovian properties for rich memory and roughness, while preserving the tractability of exponential-affine transform techniques through the Riccati–Volterra kernel machinery. This yields precise analytic control despite the non-Markovian path properties, enabling deep results on existence, uniqueness, stationarity, and ergodicity, and facilitating applications across stochastic analysis, mathematical finance, and integrable systems (Jaber et al., 2017, Bondi et al., 2022, Jacquier et al., 2022).

The interplay of convolution kernels, resolvent analysis, and affine transform techniques suggests further developments in high-dimensional extensions, robust simulation methods, extensions to SPDEs, and connections with algebro-geometric methods in integrable dynamics. The affine Volterra framework functions as a unifying structure across continuous, jump, and even discrete and algebraic systems with memory and rough path features.