[Paper Review] Ruin probabilities and passage times of $\gamma$-reflected Gaussian processes with stationary increments

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

Financial Risk and Volatility Modeling20 references4 citations

TL;DR

This paper derives asymptotic approximations for ruin probabilities and joint passage time distributions in $γ$-reflected Gaussian processes with stationary increments, modeling tax-affected surplus processes. It extends Piterbarg's inequality and applies results to multiplex fractional Brownian motion and integrated Gaussian processes, offering precise tail behavior for risk modeling.

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 analyze ruin probabilities and first/last passage times in $γ$-reflected Gaussian processes with stationary increments.
• To model tax payments of loss-carry forward type in risk theory using $γ$-reflected processes.
• To derive asymptotic approximations for ruin probabilities and joint passage time distributions under general conditions.
• To extend Piterbarg's inequality to threshold-dependent random fields as a by-product.
• To apply theoretical results to specific processes, including multiplex fractional Brownian motion and integrated Gaussian processes.

Proposed method

• Define the $γ$-reflected process $W_\gamma(t) = X(t) - ct - \gamma \inf_{0\leq s\leq t}(X(s) - cs)$ for $\gamma \in (0,1)$, modeling surplus with tax.
• Use asymptotic analysis to approximate ruin probabilities as time $t \to \infty$ under heavy-tailed or heavy-tailed-like behavior.
• Apply sample path techniques and Pickands-type constants to derive tail asymptotics for first and last passage times.
• Leverage the stationarity of increments in $X(t)$ to simplify moment and dependence structure analysis.
• Extend Piterbarg's inequality to threshold-dependent Gaussian random fields using the derived extremal behavior.
• Apply results to concrete processes: multiplex fractional Brownian motion and integrated Gaussian processes, verifying applicability.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the ruin probability for $γ$-reflected Gaussian processes with stationary increments as time tends to infinity?
• RQ2 How do the joint distributions of first and last passage times behave, conditioned on ruin occurring?
• RQ3 What is the extension of Piterbarg's inequality for threshold-dependent Gaussian random fields in this context?
• RQ4 How do the results apply to specific processes like multiplex fractional Brownian motion and integrated Gaussian processes?
• RQ5 What are the precise asymptotic approximations for ruin probabilities and passage times under general Gaussian processes with stationary increments?

Key findings

• The paper derives precise asymptotic approximations for ruin probabilities in $γ$-reflected Gaussian processes with stationary increments, valid in the heavy-tailed regime.
• Joint distributions of first and last passage times given ruin are characterized asymptotically, providing insight into the temporal structure of ruin events.
• An extension of Piterbarg's inequality is established for threshold-dependent Gaussian random fields, enabling analysis of extremal events under non-stationary thresholds.
• For multiplex fractional Brownian motion, the asymptotic ruin probability is shown to follow a power-law decay determined by the self-similarity index and $γ$.
• In the case of integrated Gaussian processes, the asymptotic ruin probability is derived using the integrated covariance structure and Pickands-type constants.
• The results demonstrate that the $γ$-reflection mechanism significantly alters the tail behavior of ruin times, particularly in the joint passage time distribution.