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Efficient simulation of a new class of Volterra-type SDEs

Ofelia Bonesini, Giorgia Callegaro, Martino Grasselli, Gilles Pagès

TL;DR

The paper introduces a convolution-kernel framework that lifts Volterra-type path-dependent SDEs to standard Markovian diffusions via a memory process $\xi$, with a reversible mapping back to the original process $X$. It proves strong existence, uniqueness, and path regularity for the coupled system, and shows that a carefully designed Euler-based scheme (Smart-Euler) achieves a strong convergence rate of $1/2$ for fractional kernels with $H\in(0,\tfrac{1}{2})$, independent of $H$. Specializing to fractional kernels highlights connections to rough volatility and the fractional Heston model, while maintaining finite-dimensional tractability for simulation. The work combines theoretical analysis with numerical illustrations, establishing both the feasibility and efficiency of the proposed method for memory-laden SVEs. Overall, the approach provides a robust, flexible toolkit for accurately simulating Volterra SDEs with memory and rough-path behavior, with significant implications for applications in biology, physics, and finance.

Abstract

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations. In the fractional kernel case, when $H \in (0,\frac12)$, where $H$ is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a remarkable strong convergence rate of order $1/2$, which constitutes a bold improvement when compared with the performance of available Euler schemes, whose strong rate of convergence is $H$.

Efficient simulation of a new class of Volterra-type SDEs

TL;DR

The paper introduces a convolution-kernel framework that lifts Volterra-type path-dependent SDEs to standard Markovian diffusions via a memory process , with a reversible mapping back to the original process . It proves strong existence, uniqueness, and path regularity for the coupled system, and shows that a carefully designed Euler-based scheme (Smart-Euler) achieves a strong convergence rate of for fractional kernels with , independent of . Specializing to fractional kernels highlights connections to rough volatility and the fractional Heston model, while maintaining finite-dimensional tractability for simulation. The work combines theoretical analysis with numerical illustrations, establishing both the feasibility and efficiency of the proposed method for memory-laden SVEs. Overall, the approach provides a robust, flexible toolkit for accurately simulating Volterra SDEs with memory and rough-path behavior, with significant implications for applications in biology, physics, and finance.

Abstract

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations. In the fractional kernel case, when , where is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a remarkable strong convergence rate of order , which constitutes a bold improvement when compared with the performance of available Euler schemes, whose strong rate of convergence is .
Paper Structure (34 sections, 13 theorems, 168 equations, 6 figures, 5 tables)

This paper contains 34 sections, 13 theorems, 168 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and prerequisites
  3. Notation
  4. Convolution transforms
  5. First existence and path-regularity results
  6. From path-dependent to standard SDEs
  7. The $\rho$-pseudo-inverse co-kernel
  8. Transformation into a Markovian process (and back)
  9. An informal intuition
  10. Euler schemes: $L^p(\mathbb P)$-convergence rate
  11. Genuine (continuous) Euler scheme for $\xi$.
  12. Euler schemes for $X$
  13. The case of fractional kernels: a relevant example
  14. Well-posedness results
  15. Numerical analysis: Euler scheme
  16. ...and 19 more sections

Key Result

Lemma 2.2

Let $\xi^0 \in L^p(\mathbb{P})$, with $p\geq 2$, and assume that the following two conditions are satisfied: Then, there exists a unique ${(\mathcal{F}_t)}_{t \in [0,T]}$-adapted process $X$ such that, for almost all $t \ge 0$, Moreover, if there exist $\theta \in (0,1)$ and $p > \frac{1}{\theta \wedge \frac{\beta - 1}{2 \beta}}$, such that: then the stochastic process $X$ has a path-wise conti

Figures (6)

  • Figure 1: Simulation of $10$ trajectories of the processes $\xi$ (on the left hand side) and $X$ (on the right hand side), for $\alpha=0.9$ and $\sigma_1$ unbounded.
  • Figure 2: Simulation of $10$ trajectories of the processes $\xi$ (on the left hand side) and $X$ (on the right hand side), for $\alpha=0.6$ and $\sigma_1$ unbounded.
  • Figure 3: Strong convergence rate in log-log plot for $H=0.1$ (up) and $H=0.4$ (down) for the process $X$ compared with the theoretical convergence rate of $1/2$ (dashed line).
  • Figure 4: Convergence rate at the end-point in log-log plot for $H=0.4$ (up) and $H=0.1$ (down) for the processe $X$ compared with the theoretical convergence rate of $1/2$ (dashed line).
  • Figure 5: Projected scheme: strong rate of convergence in $\log-\log$ scale against the reference rate $H$ (dashed line) for $H=\{0.1,0.2,0.3,0.4\}$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 2.1
  • Lemma 2.2
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.4
  • Theorem 4.1
  • Remark 4.2
  • Corollary 4.3: Step-wise constant Euler scheme $\overline{\xi}_{\underline t}$: $L^r(dt)$-$L^p(\mathbb{P})$-error
  • Remark 4.4: Complexity of the schemes
  • Theorem 4.5
  • ...and 19 more