Stochastic Volterra Equations with Jumps

• Stochastic Volterra equations with jumps are non-Markovian integral equations incorporating memory and jump components to model phenomena in finance, biology, and engineering.
• Their analysis hinges on detailed kernel properties, continuity conditions, and integrability of jump measures to ensure existence, uniqueness, and regularity of solutions.
• Numerical schemes, including small-jump truncation and series representation methods, are developed to approximate these equations under challenging infinite-activity noise regimes.

Stochastic Volterra equations with jumps are non-Markovian stochastic integral equations in which the evolution of a process depends on the weighted history of its own path, perturbed by both continuous martingale noise (such as Brownian motion) and discontinuous jump noise (such as compensated Poisson integrals or more general Lévy processes). Their mathematical structure and the irregularity induced by jumps—in both finite- and infinite-activity regimes—are central to a wide range of applications, from rough volatility modeling in finance to viscoelasticity and branching systems in biology and engineering.

1. Core Forms and General Frameworks

A prototypical Stochastic Volterra Equation with Jumps (SVEJ) is formulated as

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

where $K$ is a Volterra kernel encoding memory, $\mu$ and $\sigma$ are drift and diffusion coefficients, $W$ is Brownian motion, and $\widetilde{N}$ represents a compensated Poisson random measure corresponding to the jumps. Variations arise in infinite-dimensional and multidimensional settings, and generalizations are made to handle non-Lipschitz coefficients, non-Gaussian Lévy noise, and backward/controlled equations (Khalaf et al., 2020, Alfonsi et al., 2024, Jaber et al., 2019, Kovács et al., 2016, Bondi et al., 2022, Jaber, 2019, Popier, 2020, Agram et al., 2017).

The general theory encompasses both strong and weak solution frameworks, the former implying pathwise uniqueness and adaptedness, the latter dealing with existence of distributions and martingale problem representations. Equations can be posed in Hilbert spaces—critical for stochastic PDEs with memory and jumps (Kovács et al., 2016).

2. Existence, Uniqueness, and Regularity

Key Existence and Uniqueness Criteria

• Linear growth and continuity assumptions: For strong solutions, coefficients are required to have global linear growth and local regularity (Lipschitz or non-Lipschitz continuity, e.g., Osgood or Yamada–Watanabe-type) (Khalaf et al., 2020, Alfonsi et al., 2024).
• Kernel properties: $K$ is typically assumed continuous, non-increasing, and nonnegative. Complete monotonicity covers fractional and multi-factor kernels. For well-posedness under jumps, integrability and regularity of $K$ (often in Sobolev-Slobodeckij spaces) are imposed (Jaber, 2019, Jaber et al., 2019).
• Jump structure: Compensated Poisson random measures with Lévy measures satisfying $$\int_{U} (|\eta(x,u)| \wedge |\eta(x,u)|^2)\pi(du) < \infty$$ (finite second moment) are tractable. Extension to infinite-activity Lévy processes (e.g., $\alpha$-stable cases) requires sharper growth control and elaborate approximation arguments (Alfonsi et al., 2024).

Strong Solution Example

Given:

• $K \in C^2$, non-increasing, $K(0)>0$,
• $\sigma(0)=0$, $\mu(0)\ge0$, $\eta(0,u)=0$,
• For all $x$: $$|\mu(x)| + \sigma(x)^2 + \int_U (|\eta(x,u)| \wedge \eta(x,u)^2) \,\pi(du) \le L(1 + |x|)$$,
• Local Yamada–Watanabe regularity for coefficients.

There exists a pathwise unique strong nonnegative solution for the SVEJ (Alfonsi et al., 2024).

Weak Solution Theory

When only weak regularity (e.g., non-Lipschitz coefficients, singular kernels) is available, existence and uniqueness in law are established via $L^p$ estimates, compactness arguments, and martingale problems (Jaber et al., 2019). Notably, Sobolev-Slobodeckij a priori bounds guarantee tightness in $L^p$ spaces.

3. Stability, Approximation, and Numerical Simulation

Stability and Weak Approximation

• Functional stability is obtained via compactness/tightness for solutions under perturbations in data (kernels, coefficients, Lévy measures), which is critical for passing to limits in non-smooth regimes, e.g., $L^1$-kernel affine models and population genetics (Jaber, 2019, Jaber et al., 2019).
• Martingale problem equivalence is exploited for both analysis and approximation purposes, facilitating generic convergence and stability theorems reminiscent of classical SDE theory but adapted to the non-Markovian, jump-driven setting.

Numerical Schemes

• Truncation of small jumps: Only jumps larger than a threshold are simulated, directly reducing the equation to a finite-jump regime. Explicit $L^p$ and almost sure convergence rates are derived in terms of kernel singularity and jump-tail heaviness (Chen et al., 2015).
• Series representation methods: Use infinitely divisible representations (e.g., Rosinski/Lévy–Khintchine), controlling truncation error by controlling the tail of the infinite series (Chen et al., 2015).
• Simulation studies on, e.g., the stochastic heat equation illustrate that infinite-activity noise induces path irregularities: while surfaces $(t,x)\mapsto X(t,x)$ exhibit discontinuities almost everywhere, fixed-time or space sections admit continuous versions.

Scheme	Principle	Error Control
Small-jump truncation	Discard jumps below threshold	$r_2^N$: small-jump bias, $r_3^N$: drift
Series representation	Truncate sum in expansion	Tail integrals of jump representation

4. Affine, Rough, and Nonnegative Models

Affine Volterra Processes (AVP) with Jumps

Affine SVEs with jumps admit explicit Riccati–Volterra equations for their Fourier-Laplace transforms, yielding semi-explicit characteristic functions even in non-Markovian settings and for singular kernels (fractional, completely monotone) (Jaber, 2019, Bondi et al., 2022). The method extends to “hyper-rough” volatility models in finance (Hurst parameter $H<1/2$), as well as self-exciting Hawkes processes and population genetics scaling limits.

Nonnegative and $\alpha$-Stable CIR Volterra Models

The nonnegativity of solutions is guaranteed under additional structure:

• $\sigma(0)=0$, $\mu(0)\ge0$, $\eta(0,u)=0$,
• Nonnegativity-preserving kernel (as in Alfonsi's criterion), leading to the Volterra extension of the $\alpha$-stable Cox–Ingersoll–Ross process: $$X_t = X_0+\!\int_0^tK(t-s)(a-\kappa X_s)ds +\sigma\!\int_0^tK(t-s)\sqrt{X_s}\,dB_s +\eta\!\int_0^tK(t-s)X_{s-}^{1/\alpha}\,dL_s$$ where the jump term is driven by a spectrally positive $\alpha$-stable process (Alfonsi et al., 2024).

5. Backward and Controlled SVIEs With Jumps

Backward stochastic Volterra integral equations (BSVIEs) with jumps generalize the classical BSDE framework to incorporate memory and jump effects. Existence, uniqueness, and comparison principles have been extended to arbitrary filtrations supporting general martingales and random measures (Popier, 2020). Solutions are constructed in weighted $\mathfrak S^p$ spaces via fixed-point arguments and are compatible with $L^p$-data even in jump settings.

Optimal control of SVIEs with jumps is tractable via Hida-Malliavin calculus, allowing explicit maximum principles (both necessary and sufficient) in adjoint BSVIE form, with practical solutions for cash-flow and consumption models. The approach hinges on extended Clark-Ocone formulae for both Brownian and jump drivers (Agram et al., 2017).

6. Applications and Further Directions

• Financial mathematics: SVEs with jumps model rough volatility, multifactor affine term structures, and positive processes for risk and derivative pricing. The explicit transform techniques enable systematic model calibration and simulation (Jaber, 2019, Bondi et al., 2022, Alfonsi et al., 2024).
• Population genetics and biology: Volterra SVE limits arise in catalytic superprocesses, with existence and uniqueness vital for scaling limits and measure-valued population models (Jaber, 2019, Jaber et al., 2019).
• Engineering and physics: Viscoelasticity with memory and Lévy-type noises is modeled naturally in the semigroup-Volterra SPDE framework (Kovács et al., 2016).
• Insurance mathematics and risk theory: Jump-diffusive memory dynamics underpin claims reserving and risk aggregation with delay and catastrophic jumps (Popier, 2020).

The field is characterized by robust approximation schemes (via Markovian lifts, small-jump truncations, or series representations), a flexible framework for weak solutions with non-Markovian and jump noise, and a class of explicit or semi-explicit models suited to calibration and numerical simulation even in rough/singular regimes. Open directions include sharper regularity results for weakly singular kernels with jumps, more efficient numerical schemes adapted to the highest singularities, and the full integration of backward/forward SVIEs in high-dimensional or infinite-dimensional stochastic control problems.

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References:

(Khalaf et al., 2020, Alfonsi et al., 2024, Jaber, 2019, Bondi et al., 2022, Kovács et al., 2016, Jaber et al., 2019, Popier, 2020, Agram et al., 2017, Chen et al., 2015)