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Sojourns of locally self-similar Gaussian processes

Svyatoslav M. Novikov

Abstract

Given a Gaussian risk process $R(t)=u+c(t)-X(t),t\ge 0$, the cumulative Parisian ruin probability on a finite time interval $[0,T]$ with respect to $L \geq 0$ is defined as the probability that the sojourn time that the risk process $R$ spends under the level 0 on this time interval $[0,T]$ exceeds $L$. In this contribution we derive exact asymptotic approximations of the cumulative Parisian ruin probability for a general class of Gaussian processes introduced in [9] assuming that $X$ is locally self-similar. We illustrate our findings with several examples. As a byproduct we show that Berman's constants can be defined alternatively by a self-similar Gaussian process which could be quite different to the fractional Brownian motion.

Sojourns of locally self-similar Gaussian processes

Abstract

Given a Gaussian risk process , the cumulative Parisian ruin probability on a finite time interval with respect to is defined as the probability that the sojourn time that the risk process spends under the level 0 on this time interval exceeds . In this contribution we derive exact asymptotic approximations of the cumulative Parisian ruin probability for a general class of Gaussian processes introduced in [9] assuming that is locally self-similar. We illustrate our findings with several examples. As a byproduct we show that Berman's constants can be defined alternatively by a self-similar Gaussian process which could be quite different to the fractional Brownian motion.
Paper Structure (7 sections, 11 theorems, 48 equations)

This paper contains 7 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Examples
  4. Auxiliary lemmas
  5. Proofs
  6. Proofs of auxiliary results
  7. Proof of main results

Key Result

Theorem 2.1

If $\widehat{Y} \in \mathbf{S}\left(\kappa,\kappa,c_{\widehat{Y}}\right)$ and $x \geq 0$, then

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 9 more