Poissonian occupation times of spectrally negative L\'evy processes with applications

TL;DR

This paper introduces Poissonian occupation times for spectrally negative Lévy processes, extending known occupation time concepts, and connects these to insurance risk models with delays, providing new analytical tools.

Contribution

It defines Poissonian occupation times for spectrally negative Lévy processes and links them to insurance risk models with Parisian delays, extending existing occupation time theories.

Findings

Extended occupation time concepts to Poissonian observation schemes.

Established a connection between Poissonian occupation times and insurance risk models.

Provided analytical results for spectrally negative Lévy processes with applications.

Abstract

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative L\'evy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative L\'evy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

Equations

$${\tau }_b^{+}=\inf \{t>0:X_t>b\},$$

$${\mathcal{O}}_{t,\lambda }^X=\sum _{n\in \mathbb{N}}({\tau }_0^{+}\circ {\theta }_{T_n})1_{\left\{X_{T_n}<0,T_n<t\right\}},$$

$${\mathcal{O}}_{\infty ,\lambda }^X=\sum _{n\in \mathbb{N}}({\tau }_0^{+}\circ {\theta }_{T_n})1_{\left\{X_{T_n}<0\right\}}.$$

$${\mathbb{P}}_x\left({\rho }_{\lambda }<\infty \right)=1-{\mathbb{E}}_x\left[{\mathrm{e}}^{-\lambda {\mathcal{O}}_{\infty }^X}\right],$$

$${\rho }_{\lambda }=\inf \left\{t>0\text{ | }t-g_t>{\mathrm{e}}_{\lambda }^{g_t}\right\},$$

$$T_0^{-}$$

$$\mathbb{E}\left[{\mathrm{e}}^{\lambda X_t}\right]={\mathrm{e}}^{t\psi (\lambda )},$$

$$\psi (\lambda )=\gamma \lambda +\frac{1}{2}{\sigma }^2{\lambda }^2+{\int }_0^{\infty }\left({\mathrm{e}}^{-\lambda z}-1+\lambda z{1}_{(0,1]}(z)\right)\Pi (\mathrm{d}z),$$

$${\int }_0^{\infty }(1\wedge z^2)\Pi (\mathrm{d}z)<\infty .$$

$${\tau }_b^{-}$$

$$\psi ({\Phi }_q)=q,q\ge 0.$$

$${\int }_0^{\infty }{\mathrm{e}}^{-\lambda y}{W}_q(y)\mathrm{d}y=\frac{1}{{\psi }_q(\lambda )},\text{for λ>Φq,}$$

$$Z_q(x,\theta )={\mathrm{e}}^{\theta x}\left(1-{\psi }_q(\theta ){\int }_0^x{\mathrm{e}}^{-\theta y}{W}_q(y)\mathrm{d}y\right),x\ge 0,$$

$$Z_q(x,0)=Z_q(x)=1+q{\int }_0^x{W}_q(y)\mathrm{d}y,x\in \mathbb{R}.$$

$$Z_q(x,\theta )={\psi }_q(\theta ){\int }_0^{\infty }{\mathrm{e}}^{-\theta y}{W}_q(x+y)\mathrm{d}y,x\ge 0,\theta \ge {\Phi }_q.$$

$$\underset{b\to \infty }{\lim }\frac{W_q(x+b)}{W^q(b)}={\mathrm{e}}^{{\Phi }_{q}x}.$$

$${\mathcal{W}}_a^{\left(p,q\right)}\left(x\right)$$

$$=W_{p+q}\left(x\right)-q{\int }_0^a{W}_{p+q}\left(x-y\right)W_p\left(y\right)\mathrm{d}y,$$

$$(s-p){\int }_0^x{W}_p(x-y)W_s(y)\mathrm{d}y=W_s(x)-W_p(x).$$

$$\begin{gathered} (s-(p+q)){\int }_a^x{W}_s(x-y){\mathcal{W}}_a^{\left(p,q\right)}\left(y\right)\mathrm{d}y \\ ={\mathcal{W}}_a^{\left(p,s-p\right)}\left(x\right)-{\mathcal{W}}_a^{\left(p,q\right)}\left(x\right). \end{gathered}$$

$${\int }_0^{\infty }{\mathrm{e}}^{-\theta z}{\mathcal{W}}_a^{\left(p,s\right)}\left(a+z\right)\mathrm{d}z=\frac{Z_p\left(a,\theta \right)}{{\psi }_{p+s}(\theta )},\theta >{\Phi }_{p+s}.$$

$$\widetilde{Z_q}\left(x,\alpha ,\beta \right)=\frac{{\psi }_q(\alpha )Z_q(x,\beta )-{\psi }_q(\beta )Z_q(x,\alpha )}{\alpha -\beta },$$

$${\tilde{Z}}_q\left(x,\alpha ,\alpha \right)={\psi }_q^{\prime }(\alpha )Z_q(x,\alpha )-{\psi }_q(\alpha )Z_q^{\prime }(x,\alpha ),$$

$$r\mathbb{P}({\tau }_z^{+}\in \mathrm{d}r)\mathrm{d}z=z\mathbb{P}(X_r\in \mathrm{d}z)\mathrm{d}r.$$

$${\mathbb{E}}_x\left[{\mathrm{e}}^{-q{\tau }_b^{+}-p{\mathcal{O}}_{{\tau }_b^{+},\lambda }^X}{1}_{\left\{{\tau }_b^{+}<\infty \right\}}\right]=\frac{{\tilde{Z}}_q\left(x,{\Phi }_{\lambda +q},{\Phi }_{p+q}\right)}{{\tilde{Z}}_q\left(b,{\Phi }_{\lambda +q},{\Phi }_{p+q}\right)}.$$

$$g\left(x\right)={\mathbb{E}}_x\left[{\mathrm{e}}^{-q{\tau }_b^{+}-p{\mathcal{O}}_{{\tau }_b^{+},\lambda }^X}{1}_{\left\{{\tau }_b^{+}<\infty \right\}}\right].$$

$$g\left(x\right)={\mathbb{E}}_x\left[{\mathrm{e}}^{-\left(p+q\right){\tau }_0^{+}}\right]g\left(0\right).$$

$$g\left(x\right)={\mathbb{E}}_x\left[{\mathrm{e}}^{-q{\tau }_b^{+}}{1}_{\left\{{\tau }_b^{+}<T_0^{-}\right\}}\right]+{\mathbb{E}}_x\left[{\mathrm{e}}^{-q{T}_0^{-}}g\left(X_{T_0^{-}}\right)1_{\left\{T_0^{-}<{\tau }_b^{+}\right\}}\right].$$

$$g\left(x\right)=\frac{Z_q\left(x,{\Phi }_{p+q}\right)}{Z_q\left(b,{\Phi }_{p+q}\right)}+{\mathbb{E}}_x\left[{\mathrm{e}}^{-q{T}_0^{-}+{\Phi }_{p+q}{X}_{T_0^{-}}}{1}_{\left\{T_0^{-}<{\tau }_b^{+}\right\}}\right]g\left(0\right)$$

$$g\left(0\right)$$

$$g\left(x\right)$$

Full text

1. Introduction

Let $X=\left(X_{t}\right)_{t\geq 0}$ be a spectrally negative Lévy process. We define the first passage time above $b$,

[TABLE]

with the convention $\inf\emptyset=\infty$. Furthermore, let $T_{i}$ be the the arrival times of an independent Poisson process with intensity $\lambda>0$ and independent of $X$. In this paper, we define the Poissonian occupation time below [math] over a finite-time horizon, by

[TABLE]

where $\theta$ is the Markov shift operator ($X_{s}\circ\theta_{t}=X_{s+t}$). More precisely, occupation time is accumulated once the process is observed to be negative at Poisson arrival times (see Figure 1). We will first study the joint Laplace transform of $\left(\tau_{b}^{+},\mathcal{O}^{X}_{\tau_{b}^{+},\lambda}\right)$ and, as a consequence, we examine the Laplace transform as well as the distribution of Poissonian occupation time over an infinite-time horizon, that is,

[TABLE]

The analysis of quantities involving the duration of negative surplus (also called the time in red) under continuous monitoring $\mathcal{O}^{X}_{t}=\int_{0}^{t}\mathbf{1}_{\left(-\infty,0\right)}\left(X_{s}\right)\mathrm{d}s$ has been well studied in the literature. The Laplace transform of $\mathcal{O}^{X}_{\infty}=\int_{0}^{\infty}\mathbf{1}_{\left(-\infty,0\right)}\left(X_{s}\right)\mathrm{d}s$ was first derived by Landriault et al. [landriaultetal2011]. Loeffen et al. [loeffenetal2014] derived the joint Laplace transform of $\left(\tau^{+}_{b},\mathcal{O}^{X}_{\tau^{+}_{b}}\right)$, generalizing the results in [landriaultetal2011] and [loeffenetal2014]. For a more general treatment, Li and Palmowski [li-palmowski2017] studied weighted occupation times. Recently, Landriault et al. [lietal2019] obtained an analytical expression for the distribution of the occupation time below level [math] up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes.

In the aforementioned works, the study of continuously observed identities involving occupation times became challenging when surplus process has paths of unbounded variation. In [landriaultetal2011], the $\epsilon$-approximation approach, which consists of a spatial shift of the sample paths of the underlying process, was used to avoid the problem caused by the infinity activity of the process and such that classical conditioning continue to hold. A second approach consists of introducing a sequence of bounded variation processes, $(X_{n})_{n\geq 1}$, which converge to the unbounded variation process $X$ as $n$ goes to infinity (see Loeffen et al. [loeffenetal2014], Guérin and Renaud [guerin_renaud_2015]). An important contribution of the Poisson observation approach is the unified proof for processes with bounded or unbounded variation paths, whereas these two cases need to be treated separately using the approximation approaches explained above and, when the Poisson observation rate goes to infinity, we recover results for occupation times under continuous monitoring (see e.g. Albrecher et al. [albrecheretal2016] and Li et al. [lietal2015]).

1.1. Motivation

In actuarial mathematics, occupation times can naturally be used as a measure of the risk inherent to an insurance portfolio. For instance, the time spent below a solvency threshold helps quantify the risk related to an insurance surplus process. The analysis of the duration of the negative surplus is also related to Parisian ruin models in which insurers are granted a period of time to re-emerge above the threshold level before ruin is deemed to occur (see e.g. [loeffenetal2013, loeffenetal2017]) and ([landriaultetal2011, landriaultetal2014]). Two types of Parisian ruin are strongly related to $\mathcal{O}^{X}_{t}$: cumulative Parisian ruin and Parisian ruin with exponential delays. In the first case, ruin occurs when the surplus process stays cumulatively below a critical level longer than a pre-determined time period. Guérin and Renaud [guerinrenaud2015] studied the probability of cumulative Parisian ruin for the Cramér–Lundberg risk model with exponential claims by giving an explicit representation for the distribution of $\mathcal{O}^{X}_{t}$. A general treatment for Lévy risk processes has been recently studied by Landriault et al. [lietal2019]. On the other hand, the probability of Parisian ruin with exponential delays was first studied in [landriaultetal2011] through the following connection between the occupation time $\mathcal{O}^{X}_{\infty}$ and this type of Parisian ruin, for $x\in\mathbb{R}$,

[TABLE]

where

[TABLE]

and $g_{t}=\sup\left\{0\leq s\leq t\colon X_{s}\geq 0\right\}$, while $\mathrm{e}^{g_{t}}_{\lambda}$ is an exponentially distributed random variable with rate $\lambda>0$ and independent of $X$ (see also Baurdoux et al. [baurdoux_et_al_2015]). More precisely, a copy of $\mathrm{e}_{\lambda}$ is assigned to each excursion below level [math] of the process. Also, it is known that Parisian ruin time in (3) corresponds to the first passage time when $X$ is observed below [math] at Poisson arrival times, that is

[TABLE]

where $T_{i}$ are the arrival times of an independent Poisson process of rate $\lambda>0$.

This paper is partly motivated by the study of Parisian ruin with Erlang implementation delays. Such type of Parisian ruin has already been studied for Lévy insurance risk models (see, e.g. Albrecher and Ivanovs [albrecher-ivanovs2017], Landriault et al. [landriaultetal2014] and Frostig and Keren-Pinhasik[Frostig2019]). However, in contrast to Parisian ruin in (3), the relationship in Equation (2) does not hold any more when the delay follows an Erlang-distributed implementation delays. In this paper, we establish a link between Poissonian occupation time and Parisian ruin with Erlang-distributed implementation delays (see Section 4). First, we study Parisian ruin time with implementation delays given by a sum of two independent exponential random variables with different rates. Thus, we extend the study to Parisian ruin with Erlang$(2,\lambda)$ implementation delays. We will also derive several fluctuation identities as well as the Gerber-Shiu distribution. For a more general case, namely Parisian ruin with Erlang$(n,\lambda)$ implementation delays, a recursive expression will be derived using the $n^{th}$-Poissonian occupation time.

The rest of the paper is organized as follows. In Section 2, we present the necessary background material on spectrally negative Lévy processes and scale functions. The main results, as well as the proofs, are presented in Section 3. Applications to Parisian insurance risk models are presented in section 4. In the Appendix, a few well known fluctuation identities with delays are presented.

2. Preliminaries

In this section, we present the necessary background material on spectrally negative Lévy processes.

2.1. Lévy insurance risk processes

A Lévy insurance risk process $X$ is a process with stationary and independent increments and no positive jumps. To avoid trivialities, we exclude the case where $X$ has monotone paths. As the Lévy process $X$ has no positive jumps, its Laplace transform exists: for all $\lambda,t\geq 0$,

[TABLE]

where

[TABLE]

for $\gamma\in\mathbb{R}$ and $\sigma\geq 0$, and where $\Pi$ is a $\sigma$-finite measure on $(0,\infty)$ called the Lévy measure of $X$ such that

[TABLE]

We will use the standard Markovian notation: the law of $X$ when starting from $X_{0}=x$ is denoted by $\mathbb{P}_{x}$ and the corresponding expectation by $\mathbb{E}_{x}$. We write $\mathbb{P}$ and $\mathbb{E}$ when $x=0$.

We recall the definition of standard first-passage time below level $b\in\mathbb{R}$,

[TABLE]

with the convention $\inf\emptyset=\infty$.

2.2. Scale functions

We now present the definition of the scale functions $W_{q}$ and $Z_{q}$ of $X$. First, recall that there exists a function $\Phi\colon[0,\infty)\to[0,\infty)$ defined by $\Phi_{q}=\sup\{\lambda\geq 0\mid\psi(\lambda)=q\}$ (the right-inverse of $\psi$) such that

[TABLE]

Now, for $q\geq 0$, the $q$-scale function of the process $X$ is defined as the continuous function on $[0,\infty)$ with Laplace transform

[TABLE]

where $\psi_{q}(\lambda)=\psi(\lambda)-q$. This function is unique, positive and strictly increasing for $x\geq 0$ and is further continuous for $q\geq 0$. We extend $W_{q}$ to the whole real line by setting $W_{q}(x)=0$ for $x<0$. We write $W=W_{0}$ when $q=0$.

We also define another scale function $Z_{q}(x,\theta)$ by

[TABLE]

and $Z_{q}(x,\theta)=\mathrm{e}^{\theta x}$ for $x<0$. For $\theta=0$, we get

[TABLE]

Using (5), we can also re-write the scale function $Z_{q}(x,\theta)$ as follows

[TABLE]

It is known that

[TABLE]

We also recall the second generation scale function, that is, for $p,p+q\geq 0$ and $x\in\mathbb{R}$, we have

[TABLE]

where the second equation follows using the following identity obtained in [loeffenetal2014]

[TABLE]

We also have a slightly general case taken from [MR3257360]: for $s\geq 0$, $x>a$, we have

[TABLE]

For later use, note that we can show

[TABLE]

For the sake of compactness of the results, we will use the following function defined in [albrecher-ivanovs2017]: for $x\in\mathbb{R}$ and $\alpha,\beta\geq 0$, we have

[TABLE]

and when $\alpha=\beta$, we obtain

[TABLE]

where $Z^{\prime}_{q}$ is the derivative of $Z_{q}$ taken with respect to the second argument. We write $\tilde{Z}=\tilde{Z}_{0}$ when $q=0$.

Finally, we recall Kendall’s identity that provides the distribution of the first upward crossing of a specific level (see [bertoin1996, Corollary VII.3]): on $(0,\infty)\times(0,\infty)$, we have

[TABLE]

We refer the reader to [kyprianou2014] for more details on spectrally negative Lévy processes. More examples and numerical computations related to scale functions can be found in e.g. [kuznetsovetal2012] and [surya2008].

3. Main results

We are now ready to state our main results. First, we provide an explicit expression for the joint Laplace transform of $\left(\tau_{b}^{+},\mathcal{O}^{X}_{\tau_{b}^{+},\lambda}\right)$.

Theorem 1**.**

For $p,q\geq 0$, $\lambda>0$ and $x\leq b$,

[TABLE]

Proof.

First, we set

[TABLE]

For $x<0$, by the strong Markov property and the spectral negativity of $X$,

[TABLE]

For $0\leq x\leq b,$ again by the strong Markov property,

[TABLE]

Hence, plugging (18) in (19) and using (58), we deduce

[TABLE]

For $x=0$, we have

[TABLE]

Thus, replacing the expression of $g(0)$ in (20), we finally have

[TABLE]

∎

As an immediate consequence of Theorem (1), we obtain the Laplace transform of $\mathcal{O}^{X}_{\infty,\lambda}$.

Corollary 2**.**

For $\lambda,p>0$, $\mathbb{E}[X_{1}]>0$ and $x\in\mathbb{R}$,

[TABLE]

Proof.

Using the fact that,

[TABLE]

we have

[TABLE]

and thus

[TABLE]

∎

In the next theorem, we give an explicit expression for the distribution of $\mathcal{O}^{X}_{\infty,\lambda}$.

Theorem 3**.**

For $\lambda>0$, $r\geq 0$ and $x\in\mathbb{R}$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

From (21), we have

[TABLE]

Using Kendall’s identity and Tonelli’s Theorem, we have

[TABLE]

We also have the following identities from Landriault et al. [lietal2019],

[TABLE]

By Laplace inversion, we deduce

[TABLE]

∎

Remark 4*.*

We can rewrite formula (4) as

[TABLE]

where

[TABLE]

and

[TABLE]

Under continuous monitoring, we have the following expression for the distribution of $\mathcal{O}^{X}_{\infty}$

[TABLE]

(See Landriault et al. [lietal2019] for more results on the distribution of occupations times).

3.1. Discussion on the results

Our fluctuation identities are arguably compact and have a similar structure as classical fluctuation identities (continuous monitoring). Indeed, from Theorem(1) we recover the identity in [loeffenetal2014, Corollary 2.(ii)] which is the joint Laplace transform of $\left(\tau_{b}^{+},\mathcal{O}^{X}_{\tau_{b}^{+}}\right)$ given by

[TABLE]

which follows immediately from the fact that $\lim_{\lambda\rightarrow\infty}Z_{q}(b,\Phi_{\lambda+q})/\lambda=0.$ We also recover the Laplace transform of $\mathcal{O}^{X}_{\infty}$ by letting $\lambda\rightarrow\infty$ in Equation (21). Indeed, we have

[TABLE]

where in the last equality follows from the fact that $\Phi_{\lambda}\rightarrow\infty$ when $\lambda\rightarrow\infty$ and

[TABLE]

which follows using the Initial Value Theorem of Laplace transform.

Also, given that $\tilde{Z}\left(x,\Phi_{\lambda},\Phi_{p}\right)=\tilde{Z}\left(x,\Phi_{p},\Phi_{\lambda}\right)$, we obtain

[TABLE]

Remark 5*.*

It is possible to extend our results to Poissonian occupation times of an arbitrary interval $(a,b)$. Indeed, let $\tau_{a,b}=\tau_{b}^{+}\wedge\tau_{a}^{-}$, we define

[TABLE]

This is left for future research.

4. Applications : Parisian ruin

In this section, we want to provide a link between Poissonian occupation times and Parisian ruin models with Erlang-distributed implementation delays.

4.1. Parisian ruin with implementation delays given by a sum of two independent exponential random variables

Under the Poissonian occupation time, occupation time is accumulated once the process is observed to be negative at Poisson arrival times. In other words, occupation time is accumulated when process stays below [math] for a period of time equal to a copy of an exponentially distributed r.v. $\mathrm{e}_{\lambda}$ with rate $\lambda>0$. Now, if we consider implementation clock given by an exponentially distributed r.v. $\mathrm{e}_{p}$ with rate $p>0$ (independent of $X$ and $\mathrm{e}_{\lambda}$), then $\mathbb{P}_{x}\left(\mathcal{O}^{X}_{\infty,\lambda}>\mathrm{e}_{p}\right)$ corresponds to the probability of Parisian ruin with implementation delays modelled the sum of $\mathrm{e}_{p}$ and $\mathrm{e}_{\lambda}$. Hence, we have established the following connection

[TABLE]

where $\rho^{(p,\lambda)}$ is the Parisian ruin time given by

[TABLE]

and for the finite-time case,

[TABLE]

Thus, using the relationship in Equation (28) together with Corollary (2), we obtain the following expression for $\mathbb{P}_{x}\left(\rho^{(p,\lambda)}<\infty\right).$

Corollary 6**.**

For $p,\lambda>0$, $x\in\mathbb{R}$ and $\mathbb{E}\left[X_{1}\right]>0$,

[TABLE]

Remark 7*.*

Using the discussion in subsection (3.1), one immediately recovers the expression for the probabilities of Parisian ruin with exponentially distributed implementation delays,

[TABLE]

[TABLE]

and finally, (30) reduces to $\mathbb{P}\left(\tau_{0}^{-}<\infty\right)$ given that

[TABLE]

Remark 8*.*

If we reverse the roles of $\mathrm{e}_{p}$ and $\mathrm{e}_{\lambda}$, that is, supposing arrival times of an independent Poisson process of rate $p>0$ and $\mathrm{e}_{\lambda}$ as an implementation clock, we also have

[TABLE]

In the next theorem, we give further fluctuation identities involving the Parisian ruin time $\rho^{(p,\lambda)}$.

Theorem 9**.**

*For $p,\lambda>0$, $b,q,\theta\geq 0$ and $x\leq b$, *

[TABLE]

where

[TABLE]

For $x\in\mathbb{R}$,

[TABLE]

For $-a\leq x\leq b$,

[TABLE]

where

[TABLE]

When $a\rightarrow\infty$, we obtain

[TABLE]

Proof.

Let

[TABLE]

For $x<0$, by the strong Markov property and the spectral negativity of $X$, we have

[TABLE]

For $0\leq x<b,$ using again the strong Markov property, we obtain

[TABLE]

Plugging (35) in (36), we obtain, for all $x\in\mathbb{R}$

[TABLE]

For $x=0$ and using (56), we have

[TABLE]

Now, plugging the last expression in (37) combined with (56), we have, after few manipulations,

[TABLE]

which proves the first identity. To prove the second identity, we need compute the following limit

[TABLE]

Given that

[TABLE]

and

[TABLE]

We obtain

[TABLE]

Now, we prove identity in Equation (33). We set

[TABLE]

For $-a<x<0$, by the strong Markov property and the spectral negativity of $X$, we have

[TABLE]

For $0\leq x\leq b$, using again the strong Markov property, we obtain

[TABLE]

Hence, plugging (38) in (39), we deduce

[TABLE]

For $x=0$, using (57) and (59), we have

[TABLE]

Now, plugging the last expression in (39), we deduce

[TABLE]

which concludes the proof for the third identity. Now, taking $a\rightarrow\infty$ in (33), we have

[TABLE]

and

[TABLE]

where in the last equality we used the fact that,

[TABLE]

and similarly,

[TABLE]

∎

which both follow using (9). The last identity can also be proved using the fact that

[TABLE]

This ends the proof.

Remark 10*.*

As it was shown in subsection (3.1), we have

[TABLE]

Thus,

[TABLE]

which corresponds to identity (56).

Next is an expression of the Gerber–Shiu distribution at the Parisian ruin time $\rho^{\left(p,\lambda\right)}$ .

Theorem 11**.**

For $p,\lambda>0$, $q\geq 0$, $x\leq b$ and $y\leq 0$,

[TABLE]

where

[TABLE]

Proof.

We have

[TABLE]

More precisely, we need to compute the Laplace inverse of $\frac{\mathcal{E}^{\left(q,\lambda\right)}\left(x,\theta\right)}{\psi_{q+\lambda}\left(\theta\right)\psi_{q+p}\left(\theta\right)}$ with respect to $\theta$. First, from (5), we have

[TABLE]

where the last equality follows using Equation (11). Using (12) and (13), we deduce

[TABLE]

The desired expression follows by Laplace inversion. ∎

Remark 12*.*

The Gerber–Shiu distribution in Equation (40) has similar a structure as the one in [baurdoux_et_al_2015] (although it is hard to show the convergence when $p\rightarrow\infty$), that is,

[TABLE]

Using the relationship between the Parisian time $\rho^{(p,\lambda)}$ and Poissonian occupation time, we obtain the following expression for the Laplace transform of $\mathcal{O}^{X}_{\mathrm{e}_{q},\lambda}$ where $\mathrm{e}_{q}$ is an exponential random variable with rate $q>0$ that is independent of the process $X$.

Corollary 13**.**

For $p\geq 0$, $q,\lambda>0$ and $x\in\mathbb{R}$, we have

[TABLE]

Proof.

First, we have

[TABLE]

Then, using (32) for $\theta=0$, we obtain

[TABLE]

∎

Remark 14*.*

When $\lambda\rightarrow\infty$, we recover the Laplace transform of $\mathcal{O}^{X}_{\mathrm{e}_{q}}$,

[TABLE]

which can be found in [lietal2019].

As an application of the previous results, we study Parisian ruin with Erlang$(2,\lambda)$ distributed implementation delays denoted by $\rho^{(2)}_{\lambda}$. This Parisian ruin time can also be expressed as follows

[TABLE]

where $\tilde{N}^{-}_{0}=\min\left\{i\in\mathbb{N}:\sup\left\{X_{t}:t\in\left[T_{i-1},T_{i}\right]\right\}<0\right\}$ (see Albrecher and Ivanovs [albrecher-ivanovs2017]).

Hence, letting $p\rightarrow\lambda$ in Corollary (6), we obtain the following expression for the probability of Parisian ruin with Erlang$(2,\lambda)$ implementation delays which has a similar structure as the one in Equation (55).

Corollary 15**.**

For $\lambda>0$, $x\in\mathbb{R}$ and $\mathbb{E}\left[X_{1}\right]>0$, we have

[TABLE]

Similarly, from Theorem (9), we obtain the following results.

Corollary 16**.**

For $q,\lambda>0$, $x\leq b$ and $y\leq 0$,

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

For $x\in\mathbb{R},$

[TABLE]

Remark 17*.*

Setting $\theta=x=0$ in (50), we get

[TABLE]

which corresponds to Equation $(22)$ in [albrecher-ivanovs2017] and $\Phi^{\prime}_{\lambda}$ is the derivative of $\Phi_{\lambda}$ with respect to the sub-index $\lambda$.

4.1.1. Examples

In this subsection, we compute the probability of Parisian ruin with Erlang$(2,\lambda)$ delays for the Brownian risk model suing the formula in (47).

Brownian risk process

Let $X$ be a Brownian risk process, i.e.

[TABLE]

where $\mu>0$, $\mu>0$ and $B=\{B_{t},t\geq 0\}$ is a standard Brownian motion. Then, we have $\psi(\theta)=\mu\theta+\frac{\sigma^{2}}{2}\theta^{2}$ and $\Phi_{\lambda}=\left(\sqrt{\mu^{2}+2\sigma^{2}\lambda}-\mu\right)\sigma^{-2}.$ In this case, for $x\geq 0$ and $q>0$, the scale functions of $X$ are given by

[TABLE]

and

[TABLE]

The derivative of $Z$ with respect to $\theta$ is given by

[TABLE]

Using the expression in (15), we have

[TABLE]

Putting all the terms together, we get the following expression for the probability of Parisian ruin with Erlang$(2,\lambda)$ delay,

[TABLE]

Cramér-Lundberg process with exponential claims

Let $X$ be a Cramér-Lundberg risk processes with exponentially distributed claims, i.e.

[TABLE]

where $N=\{N_{t},t\geq 0\}$ is a Poisson process with intensity $\eta>0$, and where $\{C_{1},C_{2},\dots\}$ are independent and exponentially distributed random variables with parameter $\alpha$. The Poisson process and the random variables are mutually independent. Then, we have

[TABLE]

The scale functions of $X$ are given by

[TABLE]

and

[TABLE]

The derivative of $Z$ with respect to $\theta$ is given by

[TABLE]

and consequently,

[TABLE]

Putting all the pieces together, we obtain

[TABLE]

4.2. Parisian ruin with Erlang$(n,\lambda)$ implementation delays ($n\geq 3$)

Now, we suppose that occupation time is accumulated after two consecutive observations of the surplus process below [math], we denote the Poissonian occupation time by

[TABLE]

The Laplace transform of $\mathcal{O}^{X}_{\infty,\lambda,2}$ can be computed using the following a standard probabilistic decomposition

[TABLE]

Letting $q\rightarrow\lambda$ in (51) combined with the results in the previous subsection (4.1), one can obtain an expression for probability of Parisian ruin with Erlang$(3,\lambda)$ implementation delays.

More generally, we denote the $n^{th}$-Poissonian occupation time by $\mathcal{O}^{X}_{t,\lambda,n}$. In this case, occupation time will be accumulated when the process $X$ has been observed to be negative at the last $n$ Poisson arrival times, i.e., when $\sup_{t\in\left[T_{i},T_{i+n}\right]}(X_{t})<0$. Then, we have

[TABLE]

Also, we denote $\rho^{(n)}_{\lambda}$ as the Parisian ruin time with Erlang$(n,\lambda)$ implementation delay, that is,

[TABLE]

where $T^{n,g_{t}}_{\lambda}$ follows an Erlang$(n,\lambda)$ distribution. In particular, we have $\rho^{(0)}_{\lambda}=\tau^{-}_{0}$ and $\rho^{(1)}_{\lambda}=T^{-}_{0}$. From the discussion at the beginning of this section, we have the following connection

[TABLE]

Thus, we obtain the following recursive formula for the probability of Parisian ruin with Erlang$(n,\lambda)$ implementation delays.

Proposition 18**.**

For $n\in\mathbb{N}$, $\lambda>0$ and $x\in\mathbb{R}$,

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Remark 19*.*

When $n$ tends to infinity, it is possible to approximate the probability of Parisian ruin with fixed delay, that is

[TABLE]

where $g_{t}=\sup\left\{0\leq s\leq t\colon X_{s}\geq 0\right\}$, by the the probability of Parisian ruin with Erlang$(n,\lambda=n/r)$ distributed implementation delays (see Bladt et al. [bladtetal2019] and Landriault et al. [landriaultetal2011]).

5. Acknowledgements

The author would like to thank Professor Bin Li for the stimulating discussions on the topic.

Appendix A Fluctuation identities with delays

In this subsection , we present some of the existing fluctuation identities with delays. When $\mathbb{E}[X_{1}]>0$, we have

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We also have the following expression for the the joint Laplace transform of $\left(T_{0}^{-},X_{T_{0}^{-}}\right)$ which has been derived by Albrecher et al. [albrecheretal2016].

Lemma 20**.**

For $\lambda>0$, $a,b,q,\theta\geq 0$ and $x\leq b$, we have

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For $-a\leq x\leq b$,

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and consequently,

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We also have the following useful identity one can extract from [lkabous2018].

Lemma 21**.**

For $a,b,p,q\geq 0$, $\lambda,r,z>0$ and $-a\leq x\leq b$, we have

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