Parisian Pickands-type Constants

• Parisian Pickands-type constants are generalized extremal constants that account for a Parisian delay, modeling persistent ruin events over a finite time window.
• They extend classical Pickands constants by replacing the sup-functional with a sup-inf structure, thereby providing a correction factor in Gaussian risk and ruin asymptotics.
• Computation techniques, including Monte Carlo simulations and discretization methods, help estimate these constants, with explicit cases (e.g., for α=1) illustrating their behavior.

Parisian Pickands-type constants are generalized extremal constants for Gaussian-functionals with a Parisian delay constraint, meaning that the relevant extremal event is not a pointwise crossing but the persistence of an excursion over a nonzero time window. In the self-similar Gaussian risk model of Dȩbicki, Hashorva, Ji, and Tabiś, this role is played by the constant $F_\alpha(T)$, introduced as the generalized Pickands constant for Parisian ruin and satisfying $F_\alpha(0)=H_\alpha$, where $H_\alpha$ is the classical Pickands constant (Dȩbicki et al., 2014). Subsequent work formulates analogous constants for broader Parisian functionals, including locally self-similar Gaussian processes and multidimensional Parisian ruin models, while simulation, discretization, and continuity results for Pickands-type constants provide part of the methodological background for their analysis (Novikov, 1 Apr 2026).

1. Emergence from Parisian ruin theory

The original Parisian Pickands-type constant in this line of work arises in the Gaussian risk process

$R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$

where $X_H$ is a centered self-similar Gaussian process with Hurst/self-similarity index $H\in(0,1)$, under the nonlinear premium regime

$\beta>H.$

The classical ruin time is

$\tau_u=\inf\{t\ge0:R_u(t)<0\},$

whereas the Parisian ruin time is

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$

Equivalently,

$$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$

In this formulation, Parisian ruin occurs only if the process stays below $F_\alpha(0)=H_\alpha$0 for a continuous delay of length $F_\alpha(0)=H_\alpha$1 (Dȩbicki et al., 2014).

This change from a pointwise crossing criterion to a window-persistence criterion is exactly what forces the replacement of the classical Pickands constant by a new constant. The classical object controls asymptotics of ordinary Gaussian extrema, while the Parisian object controls asymptotics of extrema with an additional $F_\alpha(0)=H_\alpha$2-over-window structure. A plausible implication is that Parisian Pickands-type constants should be understood as extremal constants for constrained suprema, rather than as direct analogues of ordinary maximum-growth constants.

2. Definition of the generalized Pickands constant $F_\alpha(0)=H_\alpha$3

The classical Pickands constant recalled in the Parisian ruin paper is

$F_\alpha(0)=H_\alpha$4

where $F_\alpha(0)=H_\alpha$5 is fractional Brownian motion with Hurst index $F_\alpha(0)=H_\alpha$6 (Dȩbicki et al., 2014).

The Parisian extension is

$F_\alpha(0)=H_\alpha$7

The authors refer to $F_\alpha(0)=H_\alpha$8 as the generalized Pickands constant. It is finite and positive, and it is the factor that replaces $F_\alpha(0)=H_\alpha$9 when one considers the Parisian version of the problem (Dȩbicki et al., 2014).

The identity

$H_\alpha$0

shows that the new constant is not merely analogous to the classical one but genuinely extends it. The structural difference is transparent in the definition: $H_\alpha$1 is built from a $H_\alpha$2-functional, whereas $H_\alpha$3 is built from a $H_\alpha$4-functional over a window of length $H_\alpha$5. This suggests that the Parisian constant encodes local persistence of high excursions, not just their occurrence.

For $H_\alpha$6, the constant is explicitly computable: $H_\alpha$7 This special case is important because it gives a direct check on the generalized definition and illustrates how the Parisian window modifies the classical constant (Dȩbicki et al., 2014).

3. Exact asymptotic role in Parisian ruin probabilities

To state the asymptotics, the Parisian ruin paper introduces the standardized process

$H_\alpha$8

Its variance attains a unique maximum at

$H_\alpha$9

and near $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$0,

$R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$1

where

$R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$2

The analysis also assumes a local stationarity condition for $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$3: there exists a regularly varying function $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$4 at $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$5 with index $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$6 and $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$7 such that

$R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$8

With the asymptotic inverse of $R_u(t)=u+ct^\beta-X_H(t), \qquad t\ge 0,$9, the constant

$X_H$0

enters the effective Parisian Pickands term (Dȩbicki et al., 2014).

Under the scaling condition

$X_H$1

the exact Parisian ruin asymptotic is

$X_H$2

This is the basic Parisian Pickands-type relation: the Parisian ruin probability is asymptotically the classical ruin probability multiplied by the correction factor

$X_H$3

Thus the exponential decay rate remains that of classical ruin, while the delay constraint modifies only the leading constant (Dȩbicki et al., 2014).

The paper explicitly frames $X_H$4 as the classical Pickands constant from Gaussian extremes and $X_H$5 as the Parisian or generalized Pickands constant encoding the extra requirement that the process remain below $X_H$6 for a window of length $X_H$7. When $X_H$8, the correction factor is $X_H$9, so the Parisian and classical asymptotics coincide.

4. Fractional Brownian motion, random delays, and ruin-time asymptotics

For the special case $H\in(0,1)$0 and

$H\in(0,1)$1

the Parisian ruin paper derives an explicit asymptotic formula in which the leading constant is $H\in(0,1)$2 multiplied by the standard prefactor for the classical fractional Brownian ruin problem (Dȩbicki et al., 2014). A particularly important consequence is the identity

$H\in(0,1)$3

which yields

$H\in(0,1)$4

when $H\in(0,1)$5. The paper further states that for $H\in(0,1)$6, this asymptotic equivalence holds even when $H\in(0,1)$7, provided $H\in(0,1)$8 in the paper’s scaling. This is the stated asymptotic relation between Parisian and classical ruin in the long-range dependent fBm case (Dȩbicki et al., 2014).

The same paper also studies a random Parisian delay $H\in(0,1)$9 independent of the risk process. If

$\beta>H.$0

then

$\beta>H.$1

Under this condition, random-delay Parisian ruin is asymptotically equivalent to classical ruin (Dȩbicki et al., 2014).

The paper also derives a conditional Gaussian approximation for the Parisian ruin time: $\beta>H.$2 where $\beta>H.$3, and

$\beta>H.$4

Conditional on ruin, the Parisian ruin time and classical ruin time are therefore asymptotically indistinguishable at the fluctuation scale, both centered near $\beta>H.$5 with Gaussian fluctuations (Dȩbicki et al., 2014). This helps explain why the Parisian delay changes the asymptotic constant through $\beta>H.$6 without changing the dominant exponential scale.

5. Later formulations and extensions

A later extension studies Parisian ruin for locally self-similar Gaussian processes with a power-type deterministic trend and introduces a more general family of Parisian Pickands-type constants (Novikov, 1 Apr 2026). For a centered Gaussian process $\beta>H.$7, $\beta>H.$8, and a continuous function $\beta>H.$9, the paper defines

$\tau_u=\inf\{t\ge0:R_u(t)<0\},$0

When $\tau_u=\inf\{t\ge0:R_u(t)<0\},$1, the superscript is omitted, and the Parisian Pickands constant is

$\tau_u=\inf\{t\ge0:R_u(t)<0\},$2

with existence, finiteness, and positivity established in the relevant cases (Novikov, 1 Apr 2026).

For fractional Brownian motion $\tau_u=\inf\{t\ge0:R_u(t)<0\},$3,

$\tau_u=\inf\{t\ge0:R_u(t)<0\},$4

These constants are explicitly described as the Parisian counterparts of the classical Pickands constant, with the extra $\tau_u=\inf\{t\ge0:R_u(t)<0\},$5 encoding a continuous-time Parisian delay (Novikov, 1 Apr 2026).

A structural theorem in that paper states that if $\tau_u=\inf\{t\ge0:R_u(t)<0\},$6 belongs to $\tau_u=\inf\{t\ge0:R_u(t)<0\},$7, then for every $\tau_u=\inf\{t\ge0:R_u(t)<0\},$8,

$\tau_u=\inf\{t\ge0:R_u(t)<0\},$9

Up to scaling, the Parisian constant for a general self-similar limiting field is therefore the same as the corresponding fractional-Brownian constant (Novikov, 1 Apr 2026).

In a different direction, simultaneous Parisian ruin for insurer and reinsurer under a quota-share treaty introduces both Parisian Piterbarg-type and Parisian Pickands-type constants (Jasnovidov et al., 2021). For Brownian motion, the main constant is

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$0

with

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$1

for the piecewise linear drift

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$2

For the fBm case, the paper uses

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$3

states that these are finite positive constants, and records the explicit Brownian formula

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$4

(Jasnovidov et al., 2021). This formula coincides with the special case $$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$5 given in the 2014 Parisian ruin paper, reinforcing the interpretation of the constant as the canonical one-dimensional Parisian extremal constant.

6. Computation, discretization, and relation to generalized Pickands theory

Parisian Pickands-type constants are defined through limits of expectations of constrained suprema, so their computation is closely tied to the broader theory of generalized Pickands constants. For a large class of processes $$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$6, generalized Pickands constants are written as

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$7

and, under appropriate assumptions, admit a Dieker–Yakir-type representation

$$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$8

where $$\tau_u^*=\inf\{t\ge T_u:\; t-\kappa_{t,u}\ge T_u\},\qquad \kappa_{t,u}=\sup\{s\in[0,t]:R_u(s)\ge0\}.$$9 is the supremum of $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$0 over the grid and $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$1 is the corresponding discrete or continuous exponential sum or integral (Dębicki et al., 2016). The paper further shows that such generalized Pickands constants coincide with normalization constants arising in mixed moving maxima representations of stationary max-stable processes (Dębicki et al., 2016). This suggests that Parisian analogues may also admit non-limit representations when an appropriate max-stable or spectral structure is available.

Continuity and discretization results for classical and generalized Pickands constants form part of the methodological backdrop. For nonnegative separable random fields $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$2, the constants

$$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$3

exist, are finite, and satisfy

$$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$4

under the assumptions of stochastic continuity, separability, unit mean, and a shift-invariance condition (Dȩbicki et al., 2021). The same paper gives a Dieker–Yakir-type representation for $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$5, which is important because it makes the constants accessible by simulation and finite-dimensional approximation (Dȩbicki et al., 2021).

For the classical fBm-based Pickands constants, the discretization error has been quantified sharply: $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$6

$$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$7

with exact limits for $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$8 and $$\mathbb P\{\tau_u^*<\infty\} =\mathbb P\Bigl\{\inf_{t\ge0}\sup_{s\in[t,t+T_u]}R_u(s)<0\Bigr\}.$$9 (Bisewski et al., 2021). That paper explicitly states that the same strategy may apply to Parisian Pickands constants, so the discretization theory of the classical constants is directly relevant to numerical work on their Parisian counterparts (Bisewski et al., 2021).

The quota-share Parisian ruin paper proposes concrete Monte Carlo approximations for the Parisian Piterbarg-type constant $F_\alpha(0)=H_\alpha$00 and the fBm Parisian Pickands-type constant $F_\alpha(0)=H_\alpha$01 (Jasnovidov et al., 2021). For $F_\alpha(0)=H_\alpha$02, it uses the Dieker–Yakir-style representation

$F_\alpha(0)=H_\alpha$03

and approximates it by discrete truncation on $F_\alpha(0)=H_\alpha$04 (Jasnovidov et al., 2021). The same paper reports numerical observations that $F_\alpha(0)=H_\alpha$05 is strictly decreasing in $F_\alpha(0)=H_\alpha$06 for all $F_\alpha(0)=H_\alpha$07, and that $F_\alpha(0)=H_\alpha$08 is decreasing in $F_\alpha(0)=H_\alpha$09 and converges to $F_\alpha(0)=H_\alpha$10 as $F_\alpha(0)=H_\alpha$11 (Jasnovidov et al., 2021). These are reported numerical findings rather than general theorems.

7. Conceptual interpretation and place in the literature

The conceptual role of Parisian Pickands-type constants is clearest in the original self-similar Gaussian risk setting. There, $F_\alpha(0)=H_\alpha$12 is the classical Pickands constant from Gaussian extremes, $F_\alpha(0)=H_\alpha$13 is the generalized Pickands constant for Parisian ruin, and the ratio

$F_\alpha(0)=H_\alpha$14

is the Parisian correction to the classical ruin asymptotic (Dȩbicki et al., 2014). This ratio isolates the effect of the delay window from the underlying large-deviation geometry of the ruin event.

Later work shows that this pattern persists in broader settings. In locally self-similar Gaussian risk models, the asymptotic constants appear either as integrals over families of Parisian Pickands constants with varying window length or as a single leading Parisian constant in the boundary regime (Novikov, 1 Apr 2026). In two-dimensional simultaneous ruin, the analogue may be a Parisian Piterbarg-type constant in Brownian or critical cases and a Parisian Pickands-type constant in fractional-Brownian regimes (Jasnovidov et al., 2021).

A recurrent misconception is to identify Parisian Pickands-type constants with ordinary Pickands constants evaluated at a different scale. The 2014 Parisian ruin paper explicitly avoids this simplification: $F_\alpha(0)=H_\alpha$15 extends $F_\alpha(0)=H_\alpha$16 through the identity $F_\alpha(0)=H_\alpha$17, but for $F_\alpha(0)=H_\alpha$18 it is a different functional, built from $F_\alpha(0)=H_\alpha$19 rather than from $F_\alpha(0)=H_\alpha$20 alone (Dȩbicki et al., 2014). A second misconception is that the Parisian delay necessarily alters the exponential rate of ruin. In the principal self-similar Gaussian regime, it does not; instead, it modifies the leading constant by the factor $F_\alpha(0)=H_\alpha$21 (Dȩbicki et al., 2014).

Taken together, these results place Parisian Pickands-type constants at the intersection of Gaussian extreme-value theory, Parisian ruin asymptotics, and the broader theory of generalized Pickands constants. Their distinguishing feature is the encoding of persistence over a time window, and their main mathematical function is to convert local constrained-extremal behavior into explicit leading constants in high-threshold asymptotics.