Paper: arXiv 2306.02708

Authors: Ofelia Bonesini, Giorgia Callegaro, Martino Grasselli, Gilles Pagès

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 $H$.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an innovative theoretical framework transforming non-Markovian Volterra SDEs into Markovian diffusions, backed by rigorous mathematical proofs of convergence rates. However, it lacks empirical validation with real market data or explicit code implementations, focusing primarily on theoretical and numerical analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Efficient Simulation of Volterra-Type SDEs] --> B[Input: Fractional Kernel with Hurst Coefficient H ∈ 0, 0.5]
    B --> C[Framework: Reversible Convolution Kernel Transformation]
    C --> D[Transform Non-Markovian Process to Markovian Diffusion]
    D --> E[Develop New Numerical Simulation Scheme]
    E --> F[Outcome: Strong Convergence Rate of Order 1/2 vs Standard Rate H]