Weak solutions of Stochastic Volterra Equations in convex domains with general kernels

2025/06/05 by Eduardo Abi Jaber, Jaber, Eduardo Abi, Aurélien Alfonsi +3 · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Nonlinear Differential Equations Analysis #Numerical methods in inverse problems #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.04911

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish new weak existence results for d-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional non-convolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in \citeAlfonsi23 to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a non-convolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result.

Citations

• State spaces of multifactor approximations of nonnegative Volterra processes
• On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients
• Polynomial Volterra processes
• Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation
• Inhomogeneous affine Volterra processes
• Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels
• Markovian lifts of positive semidefinite affine Volterra type processes
• Generalized Feller processes and Markovian lifts of stochastic Volterra\n processes: the affine case
• Markovian structure of the Volterra Heston model
• Multi-factor approximation of rough volatility models
• The characteristic function of rough Heston models
• Stochastic invariance of closed sets with non-Lipschitz coefficients
• Affine representations of fractional processes with applications in\n mathematical finance
• Rough fractional diffusions as scaling limits of nearly unstable heavy\n tailed Hawkes processes
• Uniqueness for Volterra-type stochastic integral equations
• Weak and strong solutions of general stochastic models
• Affine processes on symmetric cones
• Affine diffusions with non-canonical state space
• Fractional Brownian motion and the Markov Property

Cited by

• Optimal control of Volterra integral diffusions and application to contract theory

Related

• Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise
• Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation
• On the uniqueness of solutions of stochastic Volterra equations
• On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients
• Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations