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On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

Emmanuel Gnabeyeu, Gilles Pagès

Abstract

The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the $L^p$ setting, $p>0$, locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated $K-$integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the $L^p$ norm.

On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

Abstract

The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the setting, , locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the norm.
Paper Structure (30 sections, 21 theorems, 221 equations, 3 figures)

This paper contains 30 sections, 21 theorems, 221 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Our contribution
  4. Plan of the paper and Notations
  5. Path-dependent Stochastic Volterra Equations with explicit kernels.
  6. Existence and pathwise uniqueness of the solution.
  7. Pathwise Regularity, Moments Control and Maximal Inequality
  8. Representation of the solution of the path-dependent Volterra equation
  9. Stochastic Approximation of Path-dependent Volterra Equations
  10. Temporal discretization by an interpolated K-integrated discrete time Euler-Maruyama scheme.
  11. Some Properties of the continuous extension process of the interpolated K-integrated Euler scheme and $L^p$-convergence at fixed times.
  12. The convergence rate of the genuine interpolated $K$-integrated Euler scheme
  13. Applications and Numerical Simulations with Fractional Kernels
  14. Numerical Illustration
  15. Numerical Verification of the Strong Convergence Rate
  16. ...and 15 more sections

Key Result

Lemma 2.3

Under Assumption assump:kernelVolterra, the coefficient functions $b$ and $\sigma$ have a linear growth in $x \in \mathbb{X}$ in the sense that there exists a constant $C_{b,\sigma,T}$ such that for every $t \in [0, T]$ and $x \, \in \mathbb{X}$,

Figures (3)

  • Figure 1: Simulation of $5$ trajectories of the process $X$ for $\alpha=0.9$ (left) and $\alpha=0.6$ (right). Number of time steps: $n = 400$.
  • Figure 2: Strong convergence rate of the mean error in log-log plot with 1000 Monte Carlo iterations for $H = 0.1$ (left) and $H = 0.4$ (right) for the process $X$.
  • Figure 3: Strong convergence rate of the end point error in log-log plot with 1000 Monte Carlo iterations for $H = 0.1$ (left) and $H = 0.4$ (right) for the process $X$.

Theorems & Definitions (27)

  • Example 2.1: The case of a convolutive kernel
  • Example 2.2: Examples of kernels
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6: Well-posedness and Lipschitzianity of $\mathcal{T}_c$
  • Theorem 2.7: Well-posedness results: Strong Existence and uniqueness
  • Theorem 2.8: Pathwise Regularity and Maximal Inequality
  • Lemma 2.9
  • Theorem 2.10
  • ...and 17 more