Parisian ruin of locally self-similar Gaussian processes

Abstract: We derive exact tail asymptotics of the Parisian ruin probability for Gaussian risk models driven by locally self-similar Gaussian processes with a power-type deterministic trend. The considered setting includes non-stationary Gaussian processes whose local correlation structure is governed by a self-similar limiting process, extending classical fractional Brownian motion models. The asymptotic behaviour is shown to depend on the interplay between the local variance decay, the self-similarity index, and the trend exponent, leading to several distinct regimes. In each regime, the ruin probability admits an explicit asymptotic representation involving Parisian Pickands-type constants. The analysis relies on a uniform Pickands lemma allowing for families of limiting Gaussian fields, extending existing double-sum techniques and enabling the treatment of locally self-similar Gaussian risk models.

• The paper presents exact asymptotic expressions for Parisian ruin probabilities for locally self-similar Gaussian risk models.
• It introduces a uniform Pickands lemma to manage non-stationary Gaussian processes and derive explicit constants in various parameter regimes.
• The findings generalize classical risk models, enhancing precise risk quantification in insurance, finance, and network applications.

Parisian Ruin Asymptotics for Locally Self-Similar Gaussian Risk Models

Introduction and Problem Framework

The paper "Parisian ruin of locally self-similar Gaussian processes" (2604.00916) addresses the asymptotic analysis of Parisian ruin probabilities for Gaussian risk processes driven by locally self-similar Gaussian processes with power-type trends. Classic Gaussian risk models typically model the surplus process via Brownian motion or more generally, a continuous Gaussian process $X(t)$ with deterministic drift. The considered extension involves non-stationary Gaussian processes whose local behavior is governed by a self-similar Gaussian process in the limit, thereby generalizing risk modeling beyond the fractional Brownian motion (fBm) framework. The primary goal is the exact asymptotic evaluation of

$$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$

where the inner $\inf$ is the Parisian functional and $d(t)^\gamma$ is a deterministic trend, with Parisian ruin corresponding to excursions above level $u$ that persist over a time window of length $L$.

Main Asymptotic Regimes and Parisian Pickands-Type Constants

The central results characterize the precise tail asymptotics for the Parisian ruin probabilities across several parameter regimes, depending on the scaling indices of the local variance decay ($\beta$), the local self-similarity parameter ($\alpha$), the trend exponent ($\gamma$), and the self-similarity index of the limiting process ($\kappa$). The asymptotics take the explicit form: $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$0 where $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$1 is the standard Gaussian survival function, and both the power $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$2 and constant $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$3 are given in terms of the interplay between process and trend regularity parameters. The constant $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$4 involves Parisian Pickands-type constants, which generalize Pickands’ classical constant for suprema over shrinking intervals in the space of non-stationary and locally self-similar processes.

The author systematically derives three distinct parameter regimes characterized by the relationship between $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$5, $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$6, $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$7, and $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$8. For each, the asymptotic expressions involve:

• Integration over an auxiliary variable $$\mathbb{P} \left\{ \sup_{t \in [0,T]} \inf_{s \in [0,L]} \left( X(t+s) - d(t+s)^\gamma \right) > u \right\}, \quad u \to \infty$$9 in case $\inf$0 and $\inf$1, with the integral kernel containing exponential terms from the deterministic trend and the local variance.
• A direct expression in terms of the Parisian Pickands-type constant in regime $\inf$2, which includes degenerate cases where the contribution comes from the Parisian mean functional alone.

For precise formulation, the ruin probability asymptotics reduce to

$\inf$3

with constants $\inf$4 explicitly involving objects such as

$\inf$5

where $\inf$6 is a self-similar Gaussian process and $\inf$7 encodes the power-type trend and variance decay terms.

Uniform Pickands Lemma for Non-Unique Limits

A principal technical advancement is the development of a uniform Pickands lemma for families of limiting Gaussian fields. In contrast to previous work (e.g., Piterbarg's approach for stationary and locally stationary processes), the locally self-similar setup necessitates asymptotic analysis for a whole family of limit fields—rather than a single one—depending on the local behavior of the process around the variance maximizing point.

The uniform Pickands lemma enables handling sequences of functionals and processes, providing uniform tail asymptotics for extended classes of non-stationary Gaussian processes. This methodological contribution directly facilitates the main asymptotic expansion results in the Parisian context and is of independent value for problems involving local self-similarity and non-uniqueness of limiting maxima.

Representative Examples

The paper verifies the general theory via a range of illustrative examples:

• Fractional Brownian motion and its generalizations, including sub-fractional and negative sub-fractional Brownian motions, where closed-form parameters for the Parisian constants are established.
• Weighted fBm, integrated fBm, and dual fBm, covering processes with various smoothness and dependence properties.
• In each instance, asymptotic constants and exponents are computed explicitly, with careful tracking of how local covariance and trend competition determine the leading order behavior.

Theoretical and Practical Implications

The paper extends the theory of extremes for Gaussian risk models to processes with non-homogeneous, locally self-similar structures and non-trivial trend functions. This generalization is highly relevant in insurance mathematics, mathematical finance, and queueing theory, where risk processes often exhibit local scaling properties and trends. The main results allow for asymptotically accurate assessment of Parisian ruin risk in settings well beyond standard Brownian or fBm risk models, enabling analysis for insurance portfolios or stochastic networks with memory, roughness, or changing local dynamics.

The principal theoretical contribution is the establishment of exact, non-logarithmic asymptotic expressions for Parisian ruin probabilities in the locally self-similar regime, with all constants explicitly identified. The work also resolves some limitations of earlier tools (such as Slepian inequality and classic Pickands lemma), making the methods applicable to vector-valued and non-stationary Gaussian fields.

Practically, these asymptotics enable precise calibration of Parisian risk-related functionals for actuaries, financial engineers, and other applied probabilists, facilitating robust risk quantification in environments where intrinsic local variability is present.

Conclusion

This paper systematically resolves the asymptotic analysis of Parisian ruin probabilities for a broad class of locally self-similar Gaussian risk processes with deterministic trends. The main results include sharp asymptotic regimes, a novel uniform Pickands lemma for families of limit Gaussian fields, and explicit calculation of Parisian Pickands-type constants. These advances extend both the theoretical apparatus for extremes of Gaussian processes and the applied toolkit for risk modeling in non-standard contexts. The techniques developed herein suggest further investigation into Parisian-type extremal phenomena in multivariate, vector-valued, and anisotropic locally self-similar setups.