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From elephant to goldfish (and back): memory in stochastic Volterra processes

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Listed:
  • Ofelia Bonesini
  • Giorgia Callegaro
  • Martino Grasselli
  • Gilles Pag`es

Abstract

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" within the dynamics of the non-Markovian process. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we provide a financial application to volatility modeling. We also propose a numerical scheme for simulating the processes. The numerical scheme exhibits a strong convergence rate of 1/2, which is independent of the roughness parameter of the volatility process. This is a significant improvement compared to Euler schemes used in similar models. We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" (the goldfish) within the dynamics of the non-Markovian process (the elephant). Most notably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we propose a numerical scheme for simulating the processes, which exhibits a remarkable convergence rate of $1/2$. In particular, in the fractional kernel case, the strong convergence rate is independent of the roughness parameter, which is a positive novelty in contrast with what happens in the available Euler schemes in the literature in rough volatility models.

Suggested Citation

  • Ofelia Bonesini & Giorgia Callegaro & Martino Grasselli & Gilles Pag`es, 2023. "From elephant to goldfish (and back): memory in stochastic Volterra processes," Papers 2306.02708, arXiv.org, revised Sep 2023.
    • Handle: RePEc:arx:papers:2306.02708
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    References listed on IDEAS

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    1. Antonis Papapantoleon & Jasper Rou, 2024. "A time-stepping deep gradient flow method for option pricing in (rough) diffusion models," Papers 2403.00746, arXiv.org.

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