On occupation times in the red of L\'evy risk models

TL;DR

This paper derives explicit formulas for the distribution of occupation times below zero in spectrally negative Lévy risk models, enhancing understanding of risk processes and enabling new applications in finance and insurance.

Contribution

It provides the first analytical expressions for occupation times in these models, extending beyond Laplace transform results to explicit distributions.

Findings

Explicit distribution formulas for occupation times in Lévy risk models

Applications to drawdown, ruin, and maximum-related quantities

Distribution results for refracted Brownian and Cramér-Lundberg models

Abstract

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level $0$) up to an (independent) exponential horizon for spectrally negative L\'{e}vy risk processes and refracted spectrally negative L\'{e}vy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum. By a further Laplace inversion to our results, we obtain the distribution of the occupation time up to a finite time horizon for refracted Brownian motion risk process and refracted Cram\'{e}r-Lundberg risk model with exponential claims.

Equations

$${\mathcal{O}}_t^X={\int }_0^t{1}_{\left(-\infty ,0\right)}\left(X_s\right)\mathrm{d}s,$$

$$\mathbf{1}_{A}\left(x\right)=\left\{\begin{array}[]{ll}1,\text{ }x\in A,&\\ 0,\text{ otherwise.}&\end{array}\right.$$

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

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

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

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

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

$$\underset{q\to 0}{\lim }\frac{q}{{\Phi }_q}={\psi }^{\prime }(0+)=\mathbb{E}[X_1].$$

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

$$\begin{array}{rl}{W}_q(0+)& =\{\begin{array}{ll}1/c& \text{when X has bounded variation;,}\\ 0& \text{when X has unbounded variation,}\end{array}\end{array}$$

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

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

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

$${\Lambda }^{\left(q\right)}\left(x,r\right)={\int }_0^{\infty }{W}_q\left(x+z\right)\frac{z}{r}\mathbb{P}\left(X_r\in \mathrm{d}z\right),$$

$${\Lambda }^{\left(q\right)}\left(0,r\right)={\mathrm{e}}^{qr}.$$

$${\Lambda }^{(q{)}^{\prime }}(x,r)=\frac{\partial {\Lambda }^{(q)}}{\partial x}(x,r)={\int }_0^{\infty }{W}_q^{\prime }\left(x+z\right)\frac{z}{r}\mathbb{P}\left(X_r\in \mathrm{d}z\right),$$

$${\Lambda }^{\prime }\left(x,r\right)={\Lambda }^{\left(q\right)\prime }\left(x,r\right)-q{\int }_0^r{\Lambda }^{\left(q\right)\prime }\left(x,s\right)\mathrm{d}s-q{W}_q(x).$$

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

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

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

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

$${\kappa }^q=\inf \left\{t>0:t-g_t>{\mathrm{e}}_q^{g_t}\right\},$$

$${\mathbb{E}}_x\left[{\mathrm{e}}^{-\lambda {\kappa }^q}\right]=\frac{q{Z}_{\lambda }(x)}{q+\lambda }-\lambda \frac{{\Phi }_{q+\lambda }-{\Phi }_{\lambda }}{\left(\lambda +q\right){\Phi }_{\lambda }}{Z}_{\lambda }\left(x,{\Phi }_{\lambda +q}\right).$$

$${\mathbb{E}}_x\left[{\mathrm{e}}^{-q{\mathcal{O}}_{{\mathrm{e}}_{\lambda }}^X}\right]=1-{\mathbb{E}}_x\left[{\mathrm{e}}^{-\lambda {\kappa }^q}\right]=\lambda \frac{{\Phi }_{q+\lambda }-{\Phi }_{\lambda }}{\left(\lambda +q\right){\Phi }_{\lambda }}{Z}_{\lambda }\left(x,{\Phi }_{\lambda +q}\right)-\frac{q{Z}_{\lambda }(x)}{q+\lambda }+1.$$

$${\mathbb{P}}_x\left({\mathcal{O}}_{{\mathrm{e}}_{\lambda }}^X\in \mathrm{d}y\right)$$

$${\mathcal{B}}^{\left(\lambda \right)}\left(x,s\right)=\frac{{\Lambda }^{\left(\lambda \right)\prime }\left(x,s\right)}{{\Phi }_{\lambda }}-{\Lambda }^{\left(\lambda \right)}\left(x,s\right),$$

$$\mathbb{P}\left({\mathcal{O}}_{{\mathrm{e}}_{\lambda }}^X\in \mathrm{d}y\right)=\frac{\lambda }{{\Phi }_{\lambda }}\left(W_{\lambda }\left(0\right){\delta }_0\left(\mathrm{d}y\right)+{\mathrm{e}}^{-\lambda y}{\Lambda }^{\prime }\left(0,y\right)\mathrm{d}y\right).$$

$${\mathbb{P}}_x\left({\mathcal{O}}_{{\mathrm{e}}_{\lambda }}^X=0\right)={\mathbb{P}}_x\left({\tau }_0^{-}>{\mathrm{e}}_{\lambda }\right)=1-Z_{\lambda }\left(x\right)+\frac{\lambda }{{\Phi }_{\lambda }}{W}_{\lambda }\left(x\right)\text{,}$$

$${\mathbb{E}}_x\left[{\mathrm{e}}^{-q{\mathcal{O}}_{{\mathrm{e}}_{\lambda }}^X}\right]$$

Full text

1. Introduction

Occupation times measure the amount of time a stochastic process stays in a certain region. It is a long-standing research topic in applied probability and has wide applications in many fields. In finance, occupation-time-related financial derivatives were studied under various dynamics for the underlying asset (e.g., Cai et al. [caietal2010], Cai and Kou [caietal2011] and Linetsky [linetsky1999]). In actuarial mathematics, occupation times can naturally be used as a measure of the risk inherent to an insurance portfolio. The occupation time of an insurer’s surplus process below a given threshold level (often chosen to be the “symbolic” level [math]) is particularly critical in the assessment of an insurer’s solvency risk (e.g., Landriault et al. [landriaultetal2014] and Guérin and Renaud [guerinrenaud2015]). With this application in mind, we define the occupation time in the red of a risk process $X$ in the time interval $(0,t)$ as

[TABLE]

where

[TABLE]

Its infinite-time counterpart $\mathcal{O}^{X}_{\infty}$ is defined as $\mathcal{O}^{X}_{\infty}=\int_{0}^{\infty}\mathbf{1}_{\left(-\infty,0\right)}\left(X_{s}\right)\mathrm{d}s$.

There exists a number of results on the occupation time $\mathcal{O}^{X}_{t}$ in the literature. For the standard Brownian motion, the distribution of $\mathcal{O}^{X}_{t}$ appeared in Lévy’s [levy1940] famous arc-sine law. This formula was generalized by Akahori [akahori1995] and Takács [takacs1996] to a Brownian motion with drift. For the classical compound Poisson process with some special jump distributions, Dos Reis [dosreis1993] obtained the moment generating function of $\mathcal{O}^{X}_{\infty}$ using a martingale approach. Zhang and Wu [zhangwu2002] further solved the Laplace transform of $\mathcal{O}^{X}_{\infty}$ by considering a compound Poisson process perturbed by an independent Brownian motion.

Spectrally negative Lévy process is a wide used more general risk model, which includes Brownian motions (with drift) and compound Poisson processes (with diffusion) as special cases. Under this framework, Landriault et al. [landriaultetal2011] first derived the Laplace transform of $\mathcal{O}^{X}_{\infty}$ in terms of the process’ scale function. Loeffen et al. [loeffenetal2014] generalized the results in [landriaultetal2011] by characterizing the joint Laplace transform of $\left(\tau_{b}^{+},\mathcal{O}^{X}_{\tau_{b}^{+}}\right)$ where $\tau_{b}^{+}=\inf\{t>0\colon X_{t}>b\}$. Li and Palmowski [li-palmowski2017] considered a further extension by studying weighted occupation times. Also, potential measures involving occupation times were also analyzed by Guérin and Renaud [guerin_renaud_2015] and Li et al. [lietal2015].

Admittedly, it is most desirable to solve the distribution of $\mathcal{O}^{X}_{t}$, the occupation time up to a finite time horizon, but it is a highly challenging problem. To the best of our knowledge, the distribution of $\mathcal{O}^{X}_{t}$ has been found for only two types of processes: Brownian motions with drift by Akahori [akahori1995] and compound Poisson with exponential jumps by Guérin and Renaud [guerinrenaud2015].

The main theoretical result of this paper is that, for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes, we obtain an analytical expression for the distribution of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$, where $\mathrm{e}_{\lambda}$ denotes an independent exponential time horizon with rate $\lambda>0$. Note that refracted spectrally negative Lévy risk processes was introduced by Kyprianou and Loeffen [kyprianouloeffen2010]). This class of processes is of interest in a number of insurance applications. This includes dividend payouts under a threshold strategy (e.g., Hernández-Hernández [HH2016] and Czarna et al. [czarna2018]) and variable annuities with a state-dependent fee structure (e.g., Bernard et al. [bernard_hardy_mackay_2014]). Results on occupation times on the refracted spectrally negative Lévy process can be found in Kyprianou et al. [kyprianou2014occupation], Renaud [renaud2014] and Li and Zhou [li-zhou2018]. Note that in these papers, the aim was to identify the Laplace transform of some occupation times while in this paper we partially generalize their results by solving the distribution.

The main contributions of our paper are summarized below. First, since the Laplace transform of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$, namely $\mathbb{E}\left[\mathrm{e}^{-q\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}}\right]$, is known in the literature (see Lemma 1 below), one may obtain numerically the distribution of $\mathcal{O}^{X}_{t}$ via a double Laplace transform inversion (with respect to $q$ and $\lambda$). However, it is well known that numerical Laplace inversion methods suffer from various stability issues, that may lead to significant computational errors. By deriving a general expression for the distribution of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$, we explicitly invert one Laplace transform, providing further structure to the problem in addition to saving one round of (numerical) Laplace inversion. Second, since occupation time is closely related to many other quantities, we provide a few applications of our results in Section 2.3 including future drawdown, inverse occupation time, Parisian ruin with exponential delays, and the last time at running maximum. Third, in addition to aforementioned Brownian motions with drift and compound Poisson with exponential jumps, we obtain the distribution of $\mathcal{O}^{X}_{t}$ for two more models in Section 3: refracted Brownian motion risk process and refracted Cramér-Lundberg risk model with exponential claims.

The rest of the paper is organized as follows. In Section 2, we first present the necessary background material on spectrally negative Lévy processes and scale functions. We then derive the main results of this paper and consider some relevant applications. We conclude this section by providing some examples of Lévy risk processes. In Section 3, we extend our study in parallel to the class of refracted spectrally negative Lévy process.

2. Occupation times of spectrally negative Lévy processes

2.1. Preliminaries

First, we present the necessary background material on spectrally negative Lévy 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]

Throughout, 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 definitions of standard first-passage stopping times : for $b\in\mathbb{R}$,

[TABLE]

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

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]

When $\mathbb{E}[X_{1}]>0$, we have

[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 $q=0$. The initial values of $W_{q}$ are known to be

[TABLE]

where $c:=\gamma+\int^{1}_{0}z\Pi(\mathrm{d}z)>0$ is the drift of $X$.

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 $\theta=0$,

[TABLE]

For $\theta\geq\Phi_{q}$, using (2), the scale function $Z_{q}(x,\theta)$ can be rewritten as

[TABLE]

We recall the delayed $q$-scale function of $X$ introduced by Loeffen et al. [loeffenetal2017] defined as

[TABLE]

and we write $\Lambda=\Lambda^{(0)}$ when $q=0$. Note that we can show

[TABLE]

We also denote the partial derivative of $\Lambda^{\left(q\right)}$ with respect to $x$ by

[TABLE]

For $x\leq 0$, we also have

[TABLE]

Also, we recall the following identity stated in [loeffenetal2014] and related to scale functions : for $p,q,x\geq 0$,

[TABLE]

Finally, we recall Kendall’s identity that provides the distribution off 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 and fluctuation identities. More examples and numerical computations related to scale functions can be found in e.g., [kuznetsovetal2012] and [surya2008].

2.2. Main results

2.2.1. Distribution of occupation times

This subsection presents our main result for the density of the occupation time $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$ for a spectrally negative Lévy process $X$, where, throughout this paper, $\mathrm{e}_{\lambda}$ denotes an exponential random variable with rate $\lambda>0$ that is independent of the process $X$. We first give the following lemma on the Laplace transform of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$. Note that this result is essentially known in the literature (one may integrate the expression of $\mathbb{E}_{x}\left[\mathrm{e}^{-q\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}};X_{\mathrm{e}_{\lambda}}\in\mathrm{d}y\right]$ (with respect to $y)$, see, e.g., Corollary 1 of Guérin and Renaud [guerinrenaud2015]), but an alternative and shorter proof is provided her.

Lemma 1**.**

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

[TABLE]

Proof.

Using Proposition $3.4$ in Guérin and Renaud [guerinrenaud2015], one can deduce that

[TABLE]

where $\kappa^{q}$ is the time of Parisian ruin with exponential delays defined as

[TABLE]

where $g_{t}=\sup\left\{0\leq s\leq t\colon X_{s}\geq 0\right\}$ and $\mathrm{e}_{q}^{g_{t}}$ is an exponentially distributed random variable with rate $q>0$ (independent of $X$). The Laplace transform of $\kappa^{q}$ can be extracted from Bardoux et al. [baurdoux_et_al_2015] (see also Albrecher et al. [albrecheretal2016]) and it is given by

[TABLE]

Therefore,

[TABLE]

∎

The following main theorem presents the density of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$, which is derived from (11) using the Laplace inversion technique.

Theorem 2**.**

For $\lambda>0$, $x\in\mathbb{R}$ and $y\geq 0$,

[TABLE]

where

[TABLE]

and $\delta_{0}(\cdot)$ is the Dirac mass at [math]. In particular, when $x=0$, equation (13) reduces to

[TABLE]

Proof.

Given that

[TABLE]

which is [math] if $x<0$, we first rewrite (11) as

[TABLE]

By simple manipulations, the above expression can also be rewritten as

[TABLE]

Using the following identities

[TABLE]

and

[TABLE]

which can be proved using Kendall’s identity (10) and Tonelli’s Theorem, leads to

[TABLE]

Hence, by Laplace inversion, we obtain

[TABLE]

Equation (13) reduces to equation (14) when $x=0$. This ends the proof. ∎

Note that the expression of $\mathbb{P}_{x}\left(\mathcal{O}_{\mathrm{e}_{\lambda}}^{X}\in\text{d}y\right)$ in (13) only relies on the scale function $W_{\lambda}(x)$ and the law $\mathbb{P}\left(X_{r}\in\mathrm{d}z\right)$. Hence, one can obtain a closed-form expression for the density of $\mathcal{O}_{\mathrm{e}_{\lambda}}^{X}$ as long as $W_{\lambda}(x)$ and $\mathbb{P}\left(X_{r}\in\mathrm{d}z\right)$ are explicit. In Section 2.4, we will provide a few examples of the underlying process $X$ such that these two quantities are explicit.

Remark 3*.*

To better understand the formula (13), we can rewrite it as

[TABLE]

where

[TABLE]

and

[TABLE]

Remark 4*.*

The time horizon in Theorem (2) can be easily extended to a hypoexponential distribution (also called generalized Erlang distribution). The class of hypoexponential distributions is known to be dense within the class of continuous nonnegative distributions in terms of weak convergence; see, e.g., Botta and Harris [botta1986approximation]. Let $\tilde{\mathrm{e}}_{n}$ be a hypoexponential-distributed time horizon given by $\tilde{\mathrm{e}}_{n}=\mathrm{e}_{1}+\mathrm{e}_{2}+\cdots+\mathrm{e}_{n}$, where $\mathrm{e}_{1},\mathrm{e}_{2},\ldots,\mathrm{e}_{n}$ are mutually independent and exponentially distributed with distinct means $1/\lambda_{1},1/\lambda_{2},\ldots,1/\lambda_{n}$ , respectively. Since the density of $\tilde{e}_{n}$ is given by

[TABLE]

where $\lambda_{k}>0$ and $a_{k}^{n}=\prod\limits_{j=1,j\neq k}^{n}\frac{\lambda_{j}}{\lambda_{j}-\lambda_{k}}$ with $\sum_{k=1}^{n}a_{k}^{n}=1$, it follows that

[TABLE]

For instance, by setting

[TABLE]

the variance of $\tilde{\mathrm{e}}_{n}$ became $\dfrac{2n(2n+1)}{3n(n+1)}k\left(\mathbb{E}\left[\tilde{\mathrm{e}}_{n}\right]\right)^{2}$ which converges to [math] when $n\rightarrow\infty$. Thus, it is possible to use the Erlangization method to approximate the fixed time horizon case (see Klugman et al. [klugmanetal2012]).

Letting $\lambda\rightarrow 0$ in (13), we obtain the following expression for the distribution of $\mathcal{O}^{X}_{\infty}$, the occupation time $X$ stays below level [math] up to infinity.

Corollary 5**.**

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

[TABLE]

Again we can rewrite (18) as

[TABLE]

where

[TABLE]

and

[TABLE]

2.3. Applications

This section is devoted to introduce a few applications of Theorem 2.

2.3.1. Future drawdown

Drawdown is used as a dynamic risk metric to measure the magnitude of the decline of insurance surplus from its maximum. Interested readers are referred to Zhang [hongzhong2018stochastic] for more theoretical results and applications of drawdown in insurance and finance. Recently, Baurdoux et al. [baurdouxetal2017] introduced the future drawdown extreme defined as

[TABLE]

where $s,t>0$. The infinite-horizon version is denoted by

[TABLE]

From Corollary 5.2 (ii) of Baurdoux et al. [baurdouxetal2017], we have

[TABLE]

where the last equality is due to Corollary 1 of Landriault et al. [landriaultetal2011]. By (18), we conclude that

[TABLE]

2.3.2. Inverse occupation time

The occupation time $\mathcal{O}^{X}_{t}$ certainly consists of some information on how long a surplus process may stay in the red zone up to time $t$. But it fails to provide a solvency early warning mechanism (in the form of a stopping time or others) that the insurer can act on in periods of financial distress. This motivates us to consider the inverse occupation time, that is the first time the accumulated duration of all periods of financial distress (periods in which the risk process is below the solvency threshold level) exceeds a deterministic tolerance level. Specifically, the inverse occupation time with parameter $r>0$ is defined as

[TABLE]

The stopping time $\sigma_{r}$ is deemed to occur at the first time the process $X$ cumulatively stays below level [math] in excess of $r$. Here, the parameter $r$ can represent the insurer’s tolerance level for the surplus process to cumulatively stay below threshold [math]. Note that, in actuarial ruin terminology, the inverse occupation time is also known as the * cumulative Parisian ruin* time (see Guérin and Renaud [guerinrenaud2015]).

The finite-time probability of inverse occupation time is given by

[TABLE]

while in the infinite-time horizon case

[TABLE]

The Laplace transform of $\sigma_{r}$ is given by

[TABLE]

which can be readily obtained from Theorem 2 as below.

Theorem 6**.**

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

[TABLE]

Using (18), we obtain the following expression for the probability that the inverse occupation time ever occurs.

Corollary 7**.**

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

[TABLE]

Finally, we point out that (24) reduces to $\mathbb{P}\left(\tau_{0}^{-}<\infty\right)=1-\mathbb{E}[X_{1}]W\left(x\right)$ when $r\rightarrow 0$ and using also the fact that the stopping time $\sigma_{r}$ converges $\mathbb{P}_{x}-$a.s. to the time of classical ruin $\tau_{0}^{-}$ (see Proposition $3.3$ in [guerinrenaud2015]).

2.3.3. Parisian ruin with exponential delay

Another type of ruin in actuarial science with strong ties to the distribution of $\mathcal{O}^{X}_{t}$ is the time of Parisian ruin with exponential delays $\kappa^{q}$ defined in (12). An expression for the probability of Parisian ruin with exponential delays was first given in [landriaultetal2011] through the relation between the occupation time $\mathcal{O}^{X}_{\infty}$ and $\kappa^{q}$, that is, for $\mathbb{E}[X_{1}]>0$, $q>0$ and $x\in\mathbb{R}$,

[TABLE]

We can readily recover (25) using our result. Given that from Proposition $3.4$ in [guerinrenaud2015], it is known that $\kappa^{q}$ and $\sigma_{\mathrm{e}_{q}}$ have the same distribution. Replacing the delay $r$ by an exponential random time $\mathrm{e}_{q}$ in (24),

[TABLE]

where the last equality follows from identity (17).

2.3.4. Last time at running maximum

Denote the last time $X$ was at its peak by

[TABLE]

where $\bar{X}_{t}=\sup_{s\leq t}X_{s}$ is the running maximum of $X$. The quantity $t-G_{t}$ is so-called the duration of drawdown at time $t$ (see Landriault et al. [landriault2017magnitude]). We also denote the occupation time of $X$ in the positive half-line by

[TABLE]

Since $X_{0}=0$, by the Sparre Andersen’s identity (see Lemma VI.15 of Bertoin [bertoin1996]), we know that, for every $t>0$,

[TABLE]

Then we have

[TABLE]

where the second last equality is due to (14).

2.4. Examples

This subsection is devoted to provide some examples of the spectrally negative Lévy process $X$ for the main results in Theorem 2, i.e., the law of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$. For cases of Brownian risk process and Cramér-Lundberg process with exponential claims, we will obtain the law of $\mathcal{O}^{X}_{t}$ by a further inversion. We assume $X_{0}=0$ in the following examples for simplicity.

2.4.1. Brownian risk process

Let $X_{t}=\mu t+\sigma B_{t},$ where $\mu>0$, $\sigma>0$, and $\{B_{t}\}_{t\geq 0}$ is a standard Brownian motion. For this process, the scale function and the right-inverse of the Laplace exponent are given by

[TABLE]

and

[TABLE]

respectively. Also, since $X_{s}$ has a normal distribution with mean $\mu s$ and variance $s\sigma^{2}$,

[TABLE]

and consequently,

[TABLE]

where $\mathcal{N}=1-\bar{\mathcal{N}}$ is the cumulative distribution function of the standard normal distribution. One can easily check that

[TABLE]

Since $X$ has paths of unbounded variation (i.e., $W_{\lambda}(0)=0$) and from (26) at $x=0$, we have

[TABLE]

Using Laplace inversion, we finally obtain

[TABLE]

which is consistent with the result in Akahori [akahori1995]. But we point out that our approach is under a more general framework of spectrally negative Lévy process, while Akahori [akahori1995] uses the specific Feynman–Kac formula for Brownian motions. In particular, letting $\sigma=1$, $\mu=0$, and integrating the law of $\mathcal{O}^{X}_{t}$ over $[r,\infty)$, one obtains the famous Paul Lévy’s arcsine law, that is,

[TABLE]

2.4.2. Cramér-Lundberg process with exponential claims

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

[TABLE]

where $\{N_{t}\}_{t\geq 0}$ is a Poisson process with intensity $\eta>0$, and $\{C_{1},C_{2},\dots\}$ are independent and exponentially distributed random variables with parameter $\alpha$, also independent of $N$. The scale function of $X$ is known to be

[TABLE]

and the right-inverse has the closed-form expression

[TABLE]

As noted in Loeffen et al. [loeffenetal2013], we have

[TABLE]

and consequently

[TABLE]

where $\Gamma(a,x)=\int_{0}^{x}\mathrm{e}^{-t}t^{a-1}\mathrm{d}t$ is the incomplete gamma function, and

[TABLE]

Then,

[TABLE]

and

[TABLE]

Since $X$ is of bounded variation paths (i.e., $W_{\lambda}(0)>0$), we have

[TABLE]

As shown in Guérin and Renuad [guerinrenaud2015], we have

[TABLE]

where

[TABLE]

Then, we also have

[TABLE]

We then obtain the following expression for the distribution of the occupation time $\mathcal{O}^{X}_{t}$ which is more compact than the one in [guerinrenaud2015]: for $t>0$,

[TABLE]

For the next two examples, we aim to provide a characterization of $\mathcal{O}^{X}_{\mathrm{e}_{\lambda}}$ (rather than $\mathcal{O}^{X}_{t}$). As shown in Equation (13) (and (14)), it is sufficient to identify the scale function $W_{\lambda}(x)$ and the density of $X_{t}$. For completeness, we recall known results pertaining to these quantities.

2.4.3. Jump diffusion risk process with phase-type claims

As a generalization of the previous two examples, we consider a jump diffusion risk process with phase-type claims, that is,

[TABLE]

where $\sigma\geq 0$, $\{B_{t}\}_{t\geq 0}$ is a standard Brownian motion, $\{N_{t}\}_{t\geq 0}$ is a Poisson process with intensity $\eta>0$, and $\{C_{1},C_{2},\dots\}$ are independent random variables with common phase-type distribution with with the minimal representation $(m,\mathbf{T},\boldsymbol{\alpha})$, i.e. its cumulative distribution function is given by $F(x)=1-\boldsymbol{\alpha}\mathrm{e}^{\mathbf{T}x}\mathbf{1}$, where $\mathbf{T}$ is an $m\times m$ matrix of a continuous-time killed Markov chain, its initial distribution is given by a simplex $\boldsymbol{\alpha}=[\alpha_{1},...,\alpha_{m}]$, and $\mathbf{1}$ denotes a column vector of ones. All of the aforementioned objects are mutually independent (for more details we refer to Egami and Yamazaki [egamiyamazaki2014]).

The Laplace exponent of $X$ is known to be of the form

[TABLE]

where $\mathbf{t}=-\mathbf{T}\mathbf{1}$. Let $\rho_{j,\lambda}$ be the roots with negative real parts of the equation $\theta\mapsto\psi(\theta)=\lambda$. Since we assume the net profit condition $\mathbb{E}[X_{1}]>0$, from Proposition 5.4 in Kuznetsov et al. [kuznetsovetal2012], we have that the $\rho_{j,\lambda}$’s are distinct roots. Then, from Proposition 2.1 in [egamiyamazaki2014], we have

[TABLE]

where $A_{j,\lambda}=\frac{1}{\psi^{\prime}(\rho_{j,\lambda})}$ and $\mathcal{I}_{\lambda}$ is the set of indices corresponding to the $\rho_{j,\lambda}$’s. Moreover,

[TABLE]

where $\mathcal{N}$ is the cumulative distribution function of a standard normal random variable, $F^{\ast k}$ is the $k$-th convolution of $F$ and for $k=0$ we understand $F^{\ast 0}(\mathrm{d}y)=\delta_{0}(\mathrm{d}y)$.

2.4.4. Stable risk process

We suppose that $X$ is a spectrally negative $\alpha$-stable process with $\alpha=3/2$. In this case, the Laplace exponent of $X$ is given by $\psi(\lambda)=\lambda^{3/2}$. Then, for $q,x\geq 0$, we have

[TABLE]

where $E_{3/2}=\sum_{k\geq 0}z^{k}/\Gamma(1+3k/2)$ is the Mittag-Leffler function of order $3/2$. As noted in Loeffen et al. [loeffenetal2013], we have

[TABLE]

where $u=\frac{4}{27}t^{9/2}|y|^{3}$ and $W_{\kappa,\mu}$ is Whittaker’s W-function (see Lebedev [lebedev]). The density of $X_{t}$ is readily obtained by a simple change of variable.

3. Occupation times of the Refracted Lévy process

We now extend our results to a refracted spectrally negative Lévy process $U=\{U_{t}\}_{t\geq 0}$ at level [math] defined as

[TABLE]

where $\delta\geq 0$ is the refraction parameter. As discussed in Kyprianou and Loeffen [kyprianouloeffen2010], such process exists and it is a skip-free upward strong Markov process. Above [math], the surplus process $U$ evolves as $Y=\{Y_{t}=X_{t}-\delta t\}_{t\geq 0}$ for which the Laplace exponent is given by

[TABLE]

with right-inverse $\varphi_{q}=\sup\{\lambda\geq 0:\psi(\lambda)-\delta\lambda=q\}$. Then, for each $q\geq 0$, we define the scale functions of $Y$, namely $\mathbb{W}_{q}$ and $\mathbb{Z}_{q}$, by

[TABLE]

and

[TABLE]

We also have

[TABLE]

We denote the delayed $q$-scale function of $Y$ by

[TABLE]

In [kyprianouloeffen2010] and [renaud2014], many fluctuation identities for the refracted process are expressed in terms of the scale function of $U$, that is, for $q\geq 0$ and for $x,z\in\mathbb{R}$, set

[TABLE]

Note that when $x<0$, we have

[TABLE]

and when $q=0$, we will write $w^{(0)}(x;z)=w(x;z)$. First, for $a\in\mathbb{R}$, we define the following first-passage stopping times:

[TABLE]

Since $Y$ is also a spectrally negative Lévy process, the identities for $X$ also hold for $Y$. For example, for $x\in\mathbb{R}$,

[TABLE]

We denote by $\kappa_{U}^{q}$ the time of Parisian ruin with exponential delays for the refracted Lévy process $U$

[TABLE]

We have the following new results for the Laplace transforms of $\kappa_{U}^{q}$ and $\mathcal{O}^{U}_{\mathrm{e}_{\lambda}}$.

Lemma 8**.**

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

[TABLE]

and consequently,

[TABLE]

Proof.

For $x<0$, using the strong Markov property of $U$ and the fact that $U_{\kappa_{0}^{+}}=0$ on $\left\{\kappa_{0}^{+}<\infty\right\}$, we have

[TABLE]

Since $\left\{X_{t},t<\tau_{0}^{+}\right\}$ and $\left\{U_{t},t<\kappa_{0}^{+}\right\}$ have the same distribution with respect to $\mathbb{P}_{x}$ when $x<0$, we further have

[TABLE]

For $x\geq 0$, using the strong Markov property of $U$, we get

[TABLE]

where in the last equality we used the fact that $\left\{Y_{t},t<\nu_{0}^{-}\right\}$ and $\left\{U_{t},t<\kappa_{0}^{-}\right\}$ have the same distribution under $\mathbb{P}_{x}$. Note that the above expression holds for all $x\in\mathbb{R}$.

Now, we assume $X$ and $Y$ have paths of bounded variation. Solving for $\mathbb{E}\left[\mathrm{e}^{-\lambda\kappa_{U}^{q}}\right]$ and using (30), we get

[TABLE]

Substituting (30) and (33) into (32), we have

[TABLE]

The case where $X$ has paths of unbounded variation follows using the same approximating procedure as in [kyprianouloeffen2010] (see also [guerin_renaud_2015]).

Finally, Equation (31) is immediate using again the following identity from Proposition $3.4$ in [guerinrenaud2015], namely

[TABLE]

∎

Using similar techniques as in the proof of Theorem 2, we obtain the following expression for the distribution of $\mathcal{O}^{U}_{\mathrm{e}_{\lambda}}$. The result is stated without proof. We point out that Equations (31) and (34) generalize Corollary 2 of Kyprianou et al. [kyprianou2014occupation] in which the occupation time is up to an infinite time horizon.

Theorem 9**.**

For $\lambda>0$, $x\in\mathbb{R}$ and $y\geq 0$,

[TABLE]

where

[TABLE]

We denote the inverse occupation time of the refracted process $U$ by

[TABLE]

and for which we obtain the following Laplace transform.

Theorem 10**.**

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

[TABLE]

We also obtain the following expression of the probability of inverse occupation time for the refracted process $U$.

Corollary 11**.**

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

[TABLE]

It is a trivial exercise to show that when $\delta=0$, the results reduced to those given in Section 2.

Remark 12*.*

The above expression can also be expressed as follows,

[TABLE]

which is due to the following useful identity relating different scale functions and taken from [renaud2014], that is, for $p,q\geq 0$ and $x\in\mathbb{R}$,

[TABLE]

and for which we consider $p=q=0$. Note that when $\delta=0$, we recover the spectrally negative analogue in (9). Letting $r\rightarrow 0$ in (37), we recover the classical probability of ruin of $U$

[TABLE]

3.1. Examples

As mentioned in the introduction, to the best of our knowledge, the distribution of the finite horizon occupation time is known only for Brownian motions with drift and Cramér-Lundberg process with exponential claims in the literature. In this section, by a further inversion of (34), we are able to derive the distribution of the finite horizon occupation time formula for refracted Brownian risk process and refracted Cramér-Lundberg process with exponential claims. Both formulas are new in the literature.

3.1.1. A refracted Brownian risk process

Let $X$ and $Y$ be two Brownian risk processes, defined as

[TABLE]

where $B=\{B_{t},t\geq 0\}$ is a standard Brownian motion. In this case, the Laplace exponent of $X$ is given by

[TABLE]

Then, for $x\geq 0$, we have

[TABLE]

and $\varphi_{\lambda}=\sigma^{-2}\left(\sqrt{\left(\mu-\delta\right)^{2}+2\lambda\sigma^{2}}-\left(\mu-\delta\right)\right).$ Using the fact that

[TABLE]

and

[TABLE]

we obtain

[TABLE]

Hence,

[TABLE]

We also have

[TABLE]

Using Laplace inversion, we finally obtain

[TABLE]

3.1.2. A refracted Cramér-Lundberg process with exponential claims

Let $X$ and $Y$ be two Cramér-Lundberg risk processes with exponentially distributed claims, then they are defined as

[TABLE]

where $N=\{N_{t},t\geq 0\}$ is a Poisson process with intensity $\eta>0$, and $\{C_{1},C_{2},\dots\}$ are independent and exponentially distributed random variables with mean $1/\alpha$, independent of $N$. In this case, the Laplace exponent of $X$ is given by

[TABLE]

and the net profit condition is given by $\mathbb{E}\left[Y_{1}\right]=c_{\delta}-\eta/\alpha\geq 0$ where $c_{\delta}=c-\delta$. Then, for $x\geq 0$, we have

[TABLE]

The right-inverse is given by

[TABLE]

We have

[TABLE]

where

[TABLE]

Then,

[TABLE]

Since $X$ is of bounded variation paths (i.e., $\mathbb{W}_{\lambda}(0)>0$), we have

[TABLE]

Also, we have

[TABLE]

where

[TABLE]

We then obtain the following expression for the distribution of the occupation time $\mathcal{O}^{U}_{t}$ for $t>0$ :

[TABLE]

Acknowledgements

Support from grants from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by David Landriault and Bin Li (grant numbers 341316 and 05828, respectively). Support from the Canada Research Chair Program is gratefully acknowledged by David Landriault.