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Singular paths spaces and applications

Carlo Bellingeri, Peter K. Friz, Máté Gerencsér

Abstract

Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modelled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime $κ< 1$. We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalisation methods.

Singular paths spaces and applications

Abstract

Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modelled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime . We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalisation methods.

Paper Structure

This paper contains 10 sections, 15 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Singular paths and spaces
  3. Hölder path spaces
  4. Improper Young integration
  5. Singular Besov paths
  6. Singular rough path spaces
  7. Application to Schramm-Loewner Evolution
  8. SLE trace as singular Hölder rough path
  9. Connection with regularity structures
  10. Application to Fractional Modelling and Rough Volatility

Key Result

Proposition 2.2

An $\alpha$-Hölder path $Y$ belongs to ${\mathcal{C}}^{\alpha,\eta}((0,T])$ if and only if $\| Y \|_{\alpha; [\varepsilon,T]} = {\mathrm O}( \varepsilon^{\eta-\alpha})$ as $\varepsilon\to 0^+$. Moreover one has the identity

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Remark 2.6
  • proof
  • ...and 32 more