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Polynomial Volterra processes

Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari, Sergio Pulido, Sara Svaluto-Ferro

Abstract

We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the moments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.

Polynomial Volterra processes

Abstract

We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the moments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.
Paper Structure (14 sections, 18 theorems, 157 equations)

This paper contains 14 sections, 18 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Definition and existence of polynomial Volterra processes
  3. Polynomial Volterra processes on the unit ball
  4. Moments of polynomial Volterra processes
  5. The main moment formula
  6. Moment formula using a variation of constants technique
  7. Moments of polynomial Volterra processes are polynomials
  8. Moments of the finite dimensional distributions and uniqueness in law
  9. Jump representation of polynomial Volterra processes
  10. One-dimensional case
  11. Multivariate case
  12. Volterra processes in the unit ball
  13. Proof of Theorem \ref{['T:invarianceball']}
  14. Well-posedness for Volterra-type integral equations

Key Result

Proposition 2.1

Let $X$ be a continuous solution to eq_CPVP. Then, for any $p\in\mathbb{N}$ and $T\geq 0$, for some constant $c$ which depends only on $\sup_{0\leq t\leq T}|g_0(t)|$, $p$, $K|_{[0,T]}$, $b_i$, $A_i,A_{ij}$, and $T$.

Theorems & Definitions (56)

  • Proposition 2.1
  • proof
  • Theorem 2.3: Existence of polynomial Volterra processes
  • proof
  • Example 2.4
  • Example 2.6
  • Theorem 2.7: Existence and uniqueness of polynomial Volterra processes in the unit ball
  • proof
  • Corollary 2.8
  • proof
  • ...and 46 more