[Paper Review] Extremes of $\gamma$-reflected Gaussian process with stationary increments

Krzysztof Dȩbicki, Enkelejd Hashorva+1 more|arXiv (Cornell University)|Nov 30, 2015

Probability and Risk Models3 citations

TL;DR

This paper establishes asymptotic expansions for the tail probability of the supremum of γ-reflected Gaussian processes with stationary increments, using extended versions of Pickands and Piterbarg inequalities. It derives exact asymptotics for ruin probabilities in risk theory and first/last passage time approximations, showing that generalized Piterbarg constants govern the asymptotic tax equivalence between models with and without tax, particularly for fractional Brownian motion and integrated Gaussian processes.

ABSTRACT

For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_\gamma(t)=X(t)-ct-\gamma\inf_{0\leq s\leq t}\left(X(s)-cs ight), \quad t\geq 0$$ 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.

Motivation & Objective

• To derive exact asymptotic expansions for the tail probability $ \mathbb{P}(\sup_{0\leq t\leq T} W_\gamma(t) > u) $ as $ u \to \infty $, where $ W_\gamma $ is a γ-reflected Gaussian process with stationary increments.
• To investigate the limiting behavior of first and last passage times for large thresholds $ u $, particularly in the context of ruin time approximations.
• To extend classical extreme value theory results—specifically Pickands and Piterbarg inequalities—for threshold-dependent Gaussian random fields arising in non-self-similar settings.
• To characterize the asymptotic ruin probability $ \psi_{\gamma,\infty}(u) $ in terms of generalized Piterbarg constants, establishing a precise asymptotic tax equivalence between models with and without tax.
• To apply the results to concrete processes such as fractional Brownian motion and integrated Gaussian processes, revealing dependence structure-driven trichotomy in tail behavior based on the limit $ \phi = \lim_{u\to\infty} \sigma^2(u)/u $.

Proposed method

• Introduces a γ-reflected process $ W_\gamma(t) = X(t) - ct - \gamma \inf_{0\leq s\leq t} (X(s) - cs) $, where $ X $ is a centered Gaussian process with stationary increments and $ \gamma \in (0,1) $, modeling risk processes with tax or fluid queues.
• Applies a uniform double-sum method to analyze the supremum of the threshold-dependent random field $ Y_u(s,t) = \frac{X(tu) - \gamma X(su)}{1 + ct - c\gamma s} $, which arises from scaling the original process.
• Develops a uniform version of the Pickands-Piterbarg lemma (Lemma 5.3) and extends Piterbarg inequality (Lemma 5.1) to threshold-dependent Gaussian fields, enabling asymptotic analysis under non-self-similar conditions.
• Imposes regular variation conditions on the variance function $ \sigma^2(t) $, assuming $ \sigma^2(t) \sim t^{2\alpha_0} $ near zero and $ \sigma^2(t) \sim t^{2\alpha_\infty} $ at infinity, to derive the asymptotic expansion.
• Identifies three distinct asymptotic regimes for $ \psi_{\gamma,\infty}(u) $ based on the limit $ \phi = \lim_{u\to\infty} \sigma^2(u)/u \in [0,\infty] $, corresponding to short-range, Brownian, and long-range dependence.
• Uses the generalized Piterbarg constant $ P_a^Z $, defined as $ \mathbb{E}\left[\sup_{t\in[0,\infty)} \exp\left(\sqrt{2}Z(t) - (1+a)\mathrm{Var}(Z(t))\right)\right] $, to express the asymptotic equivalence between ruin probabilities with and without tax.

Experimental results

Research questions

• RQ1 How does the asymptotic behavior of the supremum of γ-reflected Gaussian processes depend on the dependence structure of the underlying process $ X $, particularly through the limit $ \phi = \lim_{u\to\infty} \sigma^2(u)/u $?
• RQ2 What is the precise asymptotic expansion of the infinite-horizon ruin probability $ \psi_{\gamma,\infty}(u) = \mathbb{P}(\sup_{t\geq 0} W_\gamma(t) > u) $ as $ u \to \infty $, and how does it relate to the corresponding ruin probability without tax?
• RQ3 How do the first and last passage times $ \tau_1(u) $ and $ \tau_2(u) $ behave asymptotically in distribution when the ruin event occurs, and does the tax rate $ \gamma $ affect their limiting distribution?
• RQ4 Can classical extreme value results like the Pickands-Piterbarg lemma and Piterbarg inequality be extended to threshold-dependent Gaussian random fields arising from non-self-similar processes?
• RQ5 What is the role of the generalized Piterbarg constant $ P_a^Z $ in governing the asymptotic equivalence between ruin probabilities in models with and without tax, and how does it vary with the Hurst index $ H $ in fractional Brownian motion?

Key findings

• The asymptotic behavior of $ \psi_{\gamma,\infty}(u) $ exhibits a trichotomy determined by $ \phi = \lim_{u\to\infty} \sigma^2(u)/u $, with distinct asymptotic regimes for $ \phi \in (0,\infty) $, $ \phi = 0 $, and $ \phi = \infty $, corresponding to Brownian, short-range dependent, and long-range dependent processes respectively.
• For fractional Brownian motion with Hurst index $ H \in (0,1) $, the asymptotic ruin probability satisfies $ \psi_{\gamma,\infty}(u) \sim P_\gamma^{V_\phi} \psi_{0,\infty}(u) $, where $ \gamma := (1-\gamma)/\gamma $, and $ V_\phi $ is a constant depending on $ \phi $, establishing an asymptotic tax equivalence.
• The generalized Piterbarg constant $ P_a^Z $ governs the scaling between ruin probabilities with and without tax, with explicit expressions derived for $ H = 1/2 $ and $ H = 1 $, and bounds provided for general $ H \in (0,1) $.
• The first and last passage times $ \tau_1(u) $ and $ \tau_2(u) $, conditionally on ruin, have the same limiting distribution as $ u \to \infty $, regardless of $ \gamma $, extending known results from the $ \gamma = 0 $ case.
• The paper establishes a uniform extension of the Pickands-Piterbarg lemma and a threshold-dependent version of Piterbarg inequality, which are general tools applicable beyond the current setting to other families of threshold-dependent Gaussian fields.
• For the finite-horizon case $ \psi_{\gamma,T}(u) $, the supremum is asymptotically concentrated near $ t = T $, and the maximum of the variance function $ \sigma_{1,u}(s,t) $ is attained in a small neighborhood of $ (0,T) $, with the maximum value approaching $ \sigma^2(T) $ as $ u \to \infty $.