Parisian Ruin Probability

• Parisian ruin probability is defined as the chance that a surplus remains continuously below zero for a prescribed delay, contrasting classical ruin which only requires a brief breach.
• The framework includes various formulations such as deterministic delay, exponential clocks, and cumulative occupation times, applied to Gaussian, Brownian, spectrally negative Lévy, and discrete-time models.
• Methodological advancements involve asymptotic analysis, scale function techniques, and stochastic control, highlighting how delay mechanisms modify ruin probabilities and optimal strategies.

Parisian ruin probability is the probability that a surplus or reserve process exhibits a continuous excursion below a prescribed level for at least a specified delay, rather than merely crossing the level instantaneously. In the literature, the delay may be deterministic, exponentially distributed, or replaced by a cumulative occupation-time requirement, and the underlying risk process may be Gaussian, Brownian, spectrally negative Lévy, discrete-time, or multidimensional. The subject therefore sits at the intersection of fluctuation theory, Gaussian extremes, occupation-time asymptotics, and stochastic control (Peng et al., 2016, Loeffen et al., 2011, Liang et al., 2021, Ji, 2018).

1. Formal definition and principal variants

A standard deterministic-delay formulation declares ruin when there exists a time interval of fixed length during which the reserve remains below zero. In one dimension, a representative event is

$$\mathbb{P}\Big\{\exists\, t\ge 0:\forall s\in[t,t+T]\ \ R(s)<0\Big\},$$

while on a finite horizon one often studies

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$

These formulations make the contrast with classical ruin explicit: classical ruin requires a single hit below zero, whereas Parisian ruin requires a sustained subzero excursion (Jasnovidov et al., 2021, Peng et al., 2016).

A second major variant replaces the deterministic delay by an exponential clock. In the lifetime exponential Parisian formulation, if $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$ is the last time wealth was nonnegative, then ruin occurs when the current negative excursion length exceeds an independent exponential clock; one form is

$\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$

with the clock reset at each return above zero. A closely related discounted formulation uses

$\tau_P=\inf\{t>0:t-g_t>T_p\},$

where $T_p$ is exponential with hazard rate $p$ (Liang et al., 2021, Liang et al., 2020).

A third variant is cumulative Parisian ruin. Here the relevant object is not a single uninterrupted excursion but the total time spent in the ruin set. In the multi-line Brownian setting,

$$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$

with the inequality understood componentwise. In two-dimensional Brownian models over $[0,1]$, the cumulative condition is often scaled as

$$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$

This replaces excursion length by occupation time as the operative persistence variable (Ji, 2018, Kriukov, 2020, Krystecki, 2021).

The literature also includes a drawdown-based formulation for spectrally negative Lévy insurance risk processes. If $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$0 is the drawdown from the last record maximum, then Parisian drawdown ruin occurs when $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$1 stays above a fixed drawdown level $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$2 for at least $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$3 consecutive units of time. This is distinct from fixed-level ruin because the barrier is measured relative to the running maximum rather than an absolute solvency threshold (Surya, 2018).

2. Gaussian and Brownian asymptotics

For finite-horizon Gaussian risk models, the principal asymptotic structure is governed by the local variance maximum and local correlation decay near the most likely ruin point. In the model

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$4

with $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$5 centered Gaussian, one assumes that the standard deviation $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$6 attains its unique maximum at the horizon endpoint $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$7, with local variance decay exponents $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$8, and local correlation roughness exponent $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$9. The exact asymptotics of

$g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$0

then split into three regimes. If $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$1 and $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$2, generalized Pickands constants appear together with a polynomial factor in $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$3; if $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$4, generalized Piterbarg-type constants arise; if $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$5 and $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$6 vanishes sufficiently fast, then

$g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$7

Thus the Parisian delay may alter the leading constant and even the power prefactor, while leaving the Gaussian tail form intact (Debicki et al., 2015).

The integrated Gaussian risk model introduces discounting directly into the reserve: $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$8 with $g_t:=\sup\{s\in[0,t]:W_s\ge 0\}$9 centered Gaussian and nonnegative covariance. In this setting, the variance function

$\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$0

is strictly increasing. For any bounded $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$1, the Parisian and classical ruin probabilities are asymptotically identical on the log-scale: $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$2 If $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$3, then the equivalence is sharper: $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$4 and the conditional ruin time satisfies

$\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$5

This identifies a boundary-localized exponential limit law for the Parisian ruin time (Peng et al., 2016).

Self-similar Gaussian risk processes

$\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$6

with $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$7 exhibit a related but infinite-horizon picture. Under local stationarity of $\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$8 near the unique maximizer

$\kappa^\pi:=\inf\{t>0:t-g_t>e_\rho^{g_t}\},$9

the Parisian ruin probability satisfies

$\tau_P=\inf\{t>0:t-g_t>T_p\},$0

where $\tau_P=\inf\{t>0:t-g_t>T_p\},$1 is the asymptotic delay scale, $\tau_P=\inf\{t>0:t-g_t>T_p\},$2 is the Pickands constant, and $\tau_P=\inf\{t>0:t-g_t>T_p\},$3 is its Parisian analogue. The Parisian and classical ruin times have the same Gaussian limit after centering and scaling, and their difference is negligible on that scale (Dȩbicki et al., 2014).

Brownian models with constant force of interest provide explicit finite- and infinite-horizon analogues. For

$\tau_P=\inf\{t>0:t-g_t>T_p\},$4

finite-horizon asymptotics take the form

$\tau_P=\inf\{t>0:t-g_t>T_p\},$5

with $\tau_P=\inf\{t>0:t-g_t>T_p\},$6, whereas the infinite-horizon model yields

$\tau_P=\inf\{t>0:t-g_t>T_p\},$7

In both cases the Parisian factor is a generalized Parisian/Piterbarg constant, and the conditional ruin time is asymptotically exponential after $\tau_P=\inf\{t>0:t-g_t>T_p\},$8-scaling (Bai et al., 2016, Bai, 2017).

Recent work extends this framework to locally self-similar Gaussian drivers with power-type trend. The asymptotic form remains $\tau_P=\inf\{t>0:t-g_t>T_p\},$9, but the constants and exponents depend on the interaction between the local self-similarity index, the variance-drop exponent, the trend exponent, and the local roughness parameter. Parisian Pickands-type constants continue to govern the leading term, now for families of limiting Gaussian fields rather than a single limiting process (Novikov, 1 Apr 2026).

3. Spectrally negative Lévy and drawdown formulations

For spectrally negative Lévy processes, Parisian ruin admits compact fluctuation-theoretic expressions in terms of scale functions. If $T_p$0 is a spectrally negative Lévy process with $T_p$1, the deterministic-delay Parisian ruin time is

$T_p$2

and the ruin probability satisfies

$T_p$3

This formula is valid for all $T_p$4, involves only the scale function $T_p$5 and the law of $T_p$6, and yields the correct limiting cases: $T_p$7 The monotonicity in both $T_p$8 and $T_p$9 is immediate from the same representation (Loeffen et al., 2011).

An earlier deterministic-delay analysis for spectrally negative Lévy insurance risk processes expresses Parisian ruin through classical ruin, the undershoot distribution, and the probability that a negative excursion survives longer than the delay. Its central decomposition is

$p$0

Within this framework the paper derives both Cramér-type and convolution-equivalent asymptotics. In the Cramér regime,

$p$1

while in the convolution-equivalent regime the asymptotic scale is the integrated tail of the Lévy measure. This shows that the deterministic Parisian delay modifies constants but preserves the principal exponential or heavy-tail scale determined by the underlying Lévy fluctuations (Czarna et al., 2010).

The drawdown formulation changes the geometry of ruin. For the drawdown process

$p$2

Parisian drawdown ruin above level $p$3 over delay $p$4 is defined by the first excursion of $p$5 above $p$6 that lasts at least $p$7. The resulting transform identities are semi-explicit in terms of the scale function and the one-time law of the Lévy process, and the joint Laplace transform of ruin time and position at ruin can be written in closed transform form. The most striking feature is qualitative rather than algebraic: $p$8 This contrasts sharply with fixed-level Parisian ruin, where ruin probability can be strictly less than one under positive drift (Surya, 2018).

4. Discrete-time and grid-based ruin

In discrete time, Parisian ruin requires the surplus to remain nonpositive for a prescribed number of consecutive periods. For the classical discrete-time reserve

$p$9

with $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$0, the Parisian ruin time with delay $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$1 is

$$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$2

Finite-time survival admits an exact decomposition through the first classical ruin epoch, the deficit at ruin, and Kendall’s identity for subsequent recovery. Infinite-time ruin is then expressed in terms of classical ruin probabilities and the probability of returning to $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$3 within $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$4 steps after entering the nonpositive region. On the asymptotic side, light-tailed claims yield

$$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$5

so the Parisian delay changes the prefactor but not the Cramér exponent. In heavy-tailed regimes, the effect depends on the tail class: for $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$6, the leading asymptotic is unchanged from classical ruin, whereas for $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$7, $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$8, the delay modifies the leading constant (Czarna et al., 2014).

The discrete-time dual risk model reverses the drift structure: $$\tau_r(u)=\inf\Big\{t>0:\int_0^t \mathbf{1}\{U(s)<0\}\,ds>r\Big\},$$9 with $[0,1]$0. Here classical ruin is hitting $[0,1]$1, and Parisian ruin requires the process to remain strictly negative for $[0,1]$2 consecutive periods. The finite-time Parisian ruin probability is obtained recursively from classical dual ruin probabilities. In infinite time, the model has an especially simple structure: if $[0,1]$3 is the unique solution of

$[0,1]$4

where $[0,1]$5 is the claim-size pgf, then the classical dual ruin probability is $[0,1]$6, and the Parisian probability factorizes as

$[0,1]$7

for a constant $[0,1]$8 depending on the delay and the gain distribution but not on $[0,1]$9. The paper works out this factorization explicitly for the Binomial/Geometric model and for the Gambler’s ruin problem (Palmowski et al., 2017).

A third discrete framework retains continuous-time Brownian dynamics but allows ruin observation only on a uniform grid $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$0. In this model,

$$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$1

and grid-based Parisian ruin with delay $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$2 has the asymptotic form

$$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$3

Thus the grid affects only the multiplicative constant, through a discrete Parisian Pickands-type constant, while the leading exponential scale remains $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$4. The associated ruin time, conditionally on ruin, has the same Gaussian limit as in the continuous model after centering by $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$5 and scaling by $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$6 (Jasnovidov, 2020).

5. Multidimensional Brownian and reinsurance geometries

Multidimensional Parisian ruin introduces both persistence and geometry. In the quota-share insurer–reinsurer model, both companies share a single Brownian or fractional Brownian loss stream, so simultaneous Parisian ruin reduces to crossing a one-dimensional piecewise-linear barrier: $$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$7 The asymptotic behavior depends critically on the intersection time

$$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$8

relative to the variance-maximizing points

$$\int_0^1 \mathbf{1}\{R_1(t)<0,\ R_2(t)<0\}\,dt>\frac{L}{u^2}.$$9

and on the scaling

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$00

In the outer regimes $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$01, the asymptotics are controlled by one side of the barrier; in the intersection regime $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$02, two-sided Piterbarg-type constants enter. For $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$03, the growth condition on $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$04 is necessary; keeping $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$05 produces substantially smaller probabilities (Jasnovidov et al., 2021).

In two-dimensional Brownian risk models with correlated marginals, simultaneous Parisian and cumulative Parisian ruin over a finite horizon typically live on the $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$06 time scale. For

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$07

with correlation $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$08, simultaneous Parisian ruin with delay $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$09 satisfies

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$10

and cumulative Parisian ruin with threshold $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$11 has

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$12

The conditional cumulative ruin time then has an exponential limit, with rate depending on $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$13 in the genuinely two-dimensional regime and equal to $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$14 in the effectively one-dimensional regime (Kriukov, 2020).

The non-simultaneous Brownian Parisian model replaces a common time window by separate windows for each component. The asymptotics are then conditional on non-simultaneous classical ruin and exhibit a phase transition at

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$15

When $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$16, the limiting conditional Parisian probability is a ratio of two-dimensional Parisian constants $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$17; when $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$18 or $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$19, mixed one-dimensional constants $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$20 and $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$21 appear, and the geometry of the optimizer changes from boundary to interior. If $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$22, the limit collapses to a one-dimensional constant, showing that the second component becomes asymptotically negligible under the conditioning (Krystecki, 2021).

The cumulative version of this conditional problem has an analogous five-regime structure. For occupation thresholds $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$23 and $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$24, the limit of

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$25

is given by ratios or products of occupation-time constants $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$26, $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$27, and $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$28, again organized by the sign of $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$29 and the special symmetric case $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$30. If $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$31, the conditional cumulative Parisian probability tends to $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$32, so the occupation-time requirement becomes asymptotically void given ruin (Krystecki, 2021).

For genuinely $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$33-dimensional Brownian models, cumulative Parisian ruin has a general exact asymptotic form

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$34

where $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$35 is the essential index set of the associated quadratic program, $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$36, and $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$37 is a cumulative Pickands-type constant. The same essential-index-set geometry determines the conditional limit law of the ruin time. In two dimensions, the active face switches according to explicit thresholds in the correlation parameter, producing different polynomial exponents and different Gaussian versus non-Gaussian limit laws (Ji, 2018).

6. Stochastic control and optimization

Parisian ruin is not only a passive diagnostic; it can also be the target of optimization. In the Black–Scholes lifetime model, wealth evolves as

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$38

death occurs at rate $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$39, and Parisian ruin is triggered when a negative wealth excursion outlasts an exponential clock with hazard $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$40. The minimal probability

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$41

is characterized by an HJB equation with a Parisian penalty term active only on $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$42: $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$43 The optimal feedback rule satisfies

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$44

For $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$45, $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$46 coincides with the classical lifetime-ruin minimizer and is independent of $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$47. For $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$48, $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$49 is strictly larger than the classical rule, increases with the excursion hazard $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$50, and also increases with the mortality rate $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$51. For small $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$52,

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$53

so the minimal probability of lifetime exponential Parisian ruin is asymptotically proportional to the minimum expected occupation time below zero (Liang et al., 2021).

A reinsurance analogue is developed for the discounted probability of exponential Parisian ruin. In the classical risk model controlled by per-loss reinsurance $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$54, with mean-variance reinsurance premium principle, the value function

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$55

solves an integro-differential HJB equation of the form

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$56

with boundary conditions

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$57

The analysis uses stochastic Perron’s method and proves that $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$58 is the unique continuous viscosity solution despite the discontinuity of the Hamiltonian at $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$59. The optimal control has a sharp structural split: full retention is optimal for $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$60, while for $$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$61 the optimal retention is characterized implicitly by the first-order condition

$$\mathcal{P}_S(u,T_u)=\mathbb{P}\Big\{\inf_{t\in[0,S]}\sup_{s\in[t,t+T_u]}R_u(s)<0\Big\}.$$62

whenever an interior solution exists. This yields excess-of-loss behavior under the expected-value principle and proportional behavior under the variance principle as special cases (Liang et al., 2020).

Taken together, these control results show that Parisian ruin has become a genuine optimization criterion. A plausible implication is that the delay mechanism is not merely a technical modification of classical ruin, but a state-dependent objective that can reverse comparative statics when the process is already below zero: in both investment and reinsurance formulations, the negative region induces qualitatively different optimal behavior than the solvent region (Liang et al., 2021, Liang et al., 2020).