Extremes of $\gamma$-reflected Gaussian process with stationary increments

TL;DR

This paper derives asymptotic approximations for ruin probabilities and passage times of a gamma-reflected Gaussian process with stationary increments, with applications to fractional Brownian motion and integrated Gaussian processes, extending Piterbarg inequality.

Contribution

It provides new asymptotic results for ruin probabilities and passage times of gamma-reflected Gaussian processes, including extensions of Piterbarg inequality for threshold-dependent fields.

Findings

Asymptotic ruin probability approximations derived.

Joint distribution of passage times characterized.

Extension of Piterbarg inequality established.

Abstract

For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_\gamma(t)=X(t)-ct-\gamma\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $\gamma$-reflected process, where $\gamma\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex fractional Brownian motion and the integrated Gaussian processes. As a by-product we derive an extension of Piterbarg inequality \KD{for} threshold-dependent random fields.