Eric J. Baurdoux, Juan Carlos Pardo, Jose Lius Perez,
Jean-François Renaud
Gerber–Shiu distribution at Parisian ruin for
Lévy insurance risk processes
Article (Accepted version)
(Refereed)
Original citation:
Baurdoux, Erik J., Pardo, Juan Carlos, Perez, Jose Luis and Renaud, Jean-Francois (2016)
Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes. Journal of Applied
Probability. ISSN 0021-9002
© 2016 Applied Probability Trust
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Applied Probability Trust (22 June 2015)
GERBER–SHIU DISTRIBUTION AT PARISIAN RUIN
FOR L
´
EVY INSURANCE RISK PROCESSES
ERIK J. BAURDOUX,
∗
London School of Economics
JUAN CARLOS PARDO,
∗∗
Centro de Investigaci´on en Matem´aticas
JOS
´
E LUIS P
´
EREZ,
∗∗∗
Universidad Nacional Aut´onoma de M´exico
JEAN-FRANC¸ OIS RENAUD,
∗∗∗∗
Universit´e du Qu´ebec `a Montr´eal (UQAM)
Abstract
Inspired by works of Landriault et al. [11, 12], we study the Gerber–Shiu
distribution at Parisian ruin with exponential implementation delays for a
spectrally negative L´evy insurance risk process. To be more specific, we study
the so-called Gerber–Shiu distribution for a ruin model where at each time the
surplus process goes negative, an independent exponential clock is started. If
the clock rings before the surplus becomes positive again then the insurance
company is ruined. Our methodology uses excursion theory for spectrally
negative L´evy processes and relies on the theory of so-called scale functions.
In particular, we extend recent results of Landriault et al. [11, 12].
Keywords: Scale functions; Parisian ruin; L´evy processes; excursion theory;
fluctuation theory; Gerber–Shiu function; Laplace transform
2010 Mathematics Subject Classification: Primary 60G51
Secondary 60J99
∗
Postal address: Department of Statistics, London School of Economics. Houghton street, London,
WC2A 2AE, United Kingdom. Email: e.j.baurdoux@lse.ac.uk
∗∗
Postal address: Centro de Investigaci´on en Matem´aticas A.C. Calle Jalisco s/n. C.P. 36240,
Guanajuato, Mexico. Email: jcpardo@cimat.mx
∗∗∗
Postal address: Department of Probability and Statistics, IIMAS, UNAM. , C.P. 04510, Mexico,
D.F., Mexico. Email: garmendia@sigma.iimas.unam.mx
∗∗∗∗
Postal address: D´epartement de math´ematiques, Universit´e du Qu´ebec `a Montr´eal (UQAM). 201
av. Pr´esident-Kennedy, Montr´eal (Qu´ebec) H2X 3Y7, Canada. Email: renaud.jf@uqam.ca
1
2 E.J. Baurdoux et al.
1. Introduction and main results
Originally motivated by pricing American claims, Gerber and Shiu [8, 9] introduced
in risk theory a function that jointly penalizes the present value of the time of ruin,
the surplus before ruin and the deficit after ruin for Cram´er–Lundberg-type processes.
Since then this expected discounted penalty function, by now known as the Gerber–
Shiu function, has been deeply studied. Recently, Biffis and Kyprianou [3] characterized
a generalized version of this function in the setting of processes with stationary and
independent increments with no positive jumps, also known as spectrally negative L´evy
processes, using scale functions. In the current actuarial setting, we refer to the latter
class of processes as L´evy insurance risk processes.
In the traditional ruin theory literature, if the surplus becomes negative, the com-
pany is ruined and has to go out of business. Here, we distinguish between being
ruined and going out of business, where the probability of going out of business is
a function of the level of negative surplus. The idea of this notion of going out of
business comes from the observation that in some industries, companies can continue
doing business even though they are technically ruined (see [11] for more motivation).
In this paper, our definition of going out of business is related to so-called Parisian ruin.
The idea of this type of actuarial ruin has been introduced by A. Dassios and S. Wu
[7], where they consider the application of an implementation delay in the recognition
of an insurer’s capital insufficiency. More precisely, they assume that ruin occurs if the
excursion below the critical threshold level is longer than a deterministic time. It is
worth pointing out that this definition of ruin is referred to as Parisian ruin due to its
ties with Parisian options (see Chesney et al. [4]).
In [7], the analysis of the probability of Parisian ruin is done in the context of the
classical Cram´er–Lundberg model. More recently, Landriault et al. [11, 12] and Loeffen
et al. [13] considered the idea of Parisian ruin with respectively a stochastic implemen-
tation delay and a deterministic implementation delay, but in the more general setup
of L´evy insurance risk models. In [11], the authors assume that the deterministic
delay is replaced by a stochastic grace period with a pre-specified distribution, but
they restrict themselves to the study of a L´evy insurance risk process with paths of
bounded variation; explicit results are obtained in the case the delay is exponentially
Gerber–Shiu distribution at Parisian ruin for L´evy insurance risk processes 3
distributed. The model with a deterministic delay has also been studied in the L´evy
setup by Czarna and Palmowski [6] and by Czarna [5].
In this paper, we study the Gerber–Shiu distribution at Parisian ruin for general
L´evy insurance risk processes, when the implementation delay is exponentially dis-
tributed. Since the L´evy insurance risk process does not jump at the time when
Parisian ruin occurs, the Gerber–Shiu function that we present here only considers the
discounted value of the surplus at ruin. Our results extend those of Landriault et al.
[11], in the exponential case, by simultaneously considering more general ruin-related
quantities and L´evy insurance risk processes of unbounded and bounded variation. Our
approach is based on a heuristic idea presented in [12] and which consists in marking
the excursions away from zero of the underlying surplus process. We will fill this gap
and provide a rigorous definition of the time of Parisian ruin. Our main contribution
is an explicit and compact expression, expressed in terms of the scale functions of the
process, for the Gerber–Shiu distribution at Parisian ruin. From our results, we easily
deduce the probability of Parisian ruin originally obtained by Landriault et al. [11, 12].
The rest of the paper is organized as follows. In the remainder of Section 1, we
introduce L´evy insurance risk processes and their associated scale functions and we
state some well-known fluctuation identities that will be useful for the sequel. We also
introduce, formally speaking, the notion of Parisian ruin in terms of the excursions
away from 0 of the L´evy insurance risk process and we provide the main results of
this paper. As a consequence, we recover the results that appear in Landriault et al.
[11, 12] and remark on an interesting link with recent findings in [1] on exit identities
of spectrally negative L´evy processes observed at Poisson arrival times. Section 2 is
devoted to the proofs of the main results.
1.1. L´evy insurance risk processes
In what follows, we assume that X = (X
t
, t ≥ 0) is a spectrally negative L´evy process
with no monotone paths (i.e. we exclude the case of the negative of a subordinator)
defined on a probability space (Ω, F, P). For x ∈ R denote by P
x
the law of X when it
is started at x and write for convenience P in place of P
0
. Accordingly, we shall write
E
x
and E for the associated expectation operators. It is well known that the Laplace
4 E.J. Baurdoux et al.
exponent ψ : [0, ∞) → R of X, defined by
ψ(λ) := log E
h
e
λX
1
i
, λ ≥ 0,
is given by the so-called L´evy-Khintchine formula
ψ(λ) = γλ +
σ
2
2
λ
2
−
Z
(0,∞)
1 − e
−λx
− λx1
{x<1}
Π(dx),
where γ ∈ R, σ ≥ 0 and Π is a measure on (0, ∞) satisfying
Z
(0,∞)
(1 ∧ x
2
)Π(dx) < ∞,
which is called the L´evy measure of X. Even though X only has negative jumps, for
convenience we choose the L´evy measure to have mass only on the positive instead of
the negative half line.
It is also known that X has paths of bounded variation if and only if
σ = 0 and
Z
(0,1)
x Π(dx) < ∞.
In this case X can be written as X
t
= ct − S
t
, t ≥ 0, where c = γ +
R
(0,1)
xΠ(dx) and
(S
t
, t ≥ 0) is a driftless subordinator. Note that necessarily c > 0, since we have ruled
out the case that X has monotone paths. In this case its Laplace exponent is given by
ψ(λ) = log E
e
λX
1
= cλ −
Z
(1,∞)
1 − e
−λx
Π(dx).
The reader is referred to Bertoin [2] and Kyprianou [10] for a complete introduction
to the theory of L´evy processes.
A key element of the forthcoming analysis relies on the theory of so-called scale
functions for spectrally negative L´evy processes. We therefore devote some time in
this section reminding the reader of some fundamental properties of scale functions.
For each q ≥ 0, define W
(q)
: R → [0, ∞), such that W
(q)
(x) = 0 for all x < 0 and on
[0, ∞) is the unique continuous function whose Laplace transform satisfies
Z
∞
0
e
−λx
W
(q)
(x)dx =
1
ψ(λ) − q
, λ > Φ(q),
where Φ(q) = sup{λ ≥ 0 : ψ(λ) = q} which is well defined and finite for all q ≥ 0, since
ψ is a strictly convex function satisfying ψ(0) = 0 and ψ(∞) = ∞. The initial value