Generalized Expected Discounted Penalty Function at General Drawdown for Lévy Risk Processes
aa r X i v : . [ q -f i n . P R ] J un Generalized Expected Discounted Penalty Function at GeneralDrawdown for L ´evy Risk Processes ✩ Wenyuan WANG a , Ping CHEN b ∗ , Shuanming LI b a School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China. b Department of Economics, The University of Melbourne, Parkville, Victoria 3010, Australia.
Abstract
This paper considers an insurance surplus process modeled by a spectrally negative L´evy pro-cess. Instead of the time of ruin in the traditional setting, we apply the time of drawdown as therisk indicator in this paper. We study the joint distribution of the time of drawdown, the runningmaximum at drawdown, the last minimum before drawdown, the surplus before drawdown andthe surplus at drawdown (may not be deficit in this case), which generalizes the known results onthe classical expected discounted penalty function in Gerber and Shiu (1998). The results havesemi-explicit expressions in terms of the q -scale functions and the L´evy measure associated withthe L´evy process. As applications, the obtained result is applied to recover results in the literatureand to obtain new results for the Gerber-Shiu function at ruin for risk processes embedded with aloss-carry-forward taxation system or a barrier dividend strategy. Moreover, numerical examplesare provided to illustrate the results. Keywords:
Spectrally negative L´evy process; general drawdown time; generalized expecteddiscounted penalty function; scale function; excursion theory.
1. Introduction
In the classical model of risk theory, the behavior of the insurer’s risk process is analysedthrough the expected discounted penalty function, which is commonly referred to as Gerber-Shiufunction in the ruin literature, see Gerber and Shiu (1998). Based on the Cram´er-Lundberg modelfor the surplus process, they studied the joint distribution of three key quantities of interest: theprobability of ruin, the distribution of surplus immediately prior to ruin and deficit at time of ruin.Thereafter the Gerber-Shiu function has been studied extensively in the literature. The diversityof techniques and perspectives under which this function was studied initiated a special issue of
Insurance: Mathematics and Economics on the topic of Gerber-Shiu functions in 2010. ✩ This work was partially supported by the National Natural Science Foundation of China (Nos. 11661074 and11701436), the Program for New Century Excellent Talents in Fujian Province University (No. Z0210103) and theFundamental Research Funds for the Central Universities (Nos. 20720170096 and 2018IB019). ∗ Corresponding author. Tel: +
61 90358053
Email addresses: [email protected] (Wenyuan WANG a ), [email protected] (Ping CHEN b ), [email protected] (Shuanming LI b ) Preprint submitted to Elsevier June 5, 2019
INTRODUCTION 2Motivated by the recent development in risk measurements, this paper studies an extended defi-nition of the classical Gerber-Shiu function. The main idea is to replace the time of ruin by a moregeneral risk indicator: the time of drawdown, which is widely used in industry. Generally, a draw-down time refers to a moment when the surplus process declines from the peak to the subsequenttrough during a specific recorded period of an investment, fund or commodity security. We adopta general drawdown definition which includes not only the ruin time as a special case, but alsomany other forms (linear or non-linear), see Remark 1 and Remark 2 in Avram et al. (2017) forexamples and their explanations. In terms of the surplus process of an insurer, the sooner the draw-down time occurs, the more risk the insurer is bearing. Accordingly, the surplus before drawdownand the surplus at drawdown also play an important role in determining an insurer’s financial risk.The level of those surpluses can help manage the financial decision of the insurer. For example, aninsurer may try to minimize the probability of drawdowns of 20% or greater before increasing itspremium rate to avoid even worse situitions. The mathematical formulation was first introducedby Taylor (1975) who studied the maximum drawdown of a drifted Brownian motion. This resultwas later extended to other situations, see Avram et al. (2004), Landriault et al. (2015), Landriaultet al. (2017), Li et al. (2017), Wang and Zhou (2018) and the references therein.In the context of finance and actuarial studies, the application of drawdown risks has been flour-ishing in recent years. Shepp and Shiryaev (1993) proposed a new put option where the optionbuyer receives the maximum price that the option has ever traded from the purchase time and theexercise time. More recent applications in option pricing can be found in Avram et al. (2004) andCarr (2014). In terms of portfolio selection, Grossman and Zhou (1993) pioneered this researchtopic by adopting a strict drawdown constraint on the optimal investment strategy. Extended workalong this line is abundant, to name a few but not limited to, see Cvitanic and Karatzas (1995)for a multi-asset framework, Cherny and Obloj (2013) for a general semimartingale framework,Roche (2006) for an optimal consumption-investment problem, and Elie and Touzi (2008) for theoptimization over a general class of utility functions. Along another line in the portfolio selection,the probability of drawdown is minimized instead of imposing a constraint on drawdown. Variousscenarios have been considered, see Chen et al. (2015) for a pure investment formulation, An-goshtari et al. (2016) for a case with constant consumption constraint, and Han et al. (2018) foran optimal reinsurance case. In terms of dividend optimization problems, Wang and Zhou (2018)considered a general version of de Finetti’s optimal dividend problem in which the ruin time isreplaced with a general drawdown time.Another feature of this paper is the insurer’s surplus process is modelled by a spectrally nega-tive L´evy process, which is a stochastic process with stationary independent increments and withsample paths of no positive jumps. It often serves as a surplus process in risk theory, where thedownward jumps describe the outgoing payments of claims. The application of spectrally negativeL´evy processes in risk theory can be seen in Yang and Zhang (2001), Garrido and Morales (2006),Bi ffi s and Morales (2010), Avarm et al. (2017) and Loe ff en et al. (2018). Based on the time ofdrawdown, this paper studies an extended definition of the expected discounted penalty functionin terms of the q -scale functions and the L´evy measure associated with the L´evy process. The jointdistribution of the time of drawdown, the running maximum at drawdown, the last minimum be-fore drawdown, the surplus before drawdown and the surplus at drawdown is derived. Unlike thetime of ruin, the surplus process may not fall below zero at the time of drawdown, therefore, we INTRODUCTION 3use the surplus at drawdown instead of the deficit at ruin in the extended definition of Gerber-Shiufunction.From technical point of view, the classical ruin theory is mainly based on renewal equationtechniques to obtain some delicate results, see Gerber and Shiu (1998), or some specific meth-ods for some particular risk models such as a Gamma process, see Dufresne and Gerber (1993).When the risk process is extended to a spectrally negative L´evy process, the results on ruin prob-abilities follow from the fluctuation theory of L´evy processes, see Kyprianou (2006). In the caseof ruin, di ff erent mathematical subtleties from the fluctuation theory are used for specific modelformulation, such as the Laplace exponent when the classical risk process and the gamma processare perturbed by di ff usion, see Yang and Zhang (2001); the Laplace transform of the time to ruinwhen the aggregate claims process is a subordinator, see Garrido and Morales (2006); the over-shoots results when the Parisian ruin problem is investigated, see Loe ff en et al. (2018). The ruinprobability is usually expressed in terms of a multi-fold convolution of some distribution functions,or the q -scale functions associated to the L´evy process.The results of this paper (the case of drawdown) are derived based on the excursion-theoreticalapproach, which is also from the fluctuation theory of L´evy processes and has been proving itse ffi ciency in solving the related boundary crossing problems. Using this approach, Kyprianou andPistorius (2003) derived the Laplace transform of a crossing time which is the key quantity to theevaluation of the Russian option; Avram et al. (2004) determined the joint Laplace transform ofthe exit time and exit position from an interval containing the origin of the process reflected in itssupremum, which is then applied to an optimal stopping problem associated with the pricing ofRussian options and their Canadized versions; Pistorius (2004) derived the q -resolvent kernels forthe L´evy process reflected at its supremum killed upon leaving [0 , a ]; Pistorius (2007) solved theproblem of Lehoczky and the Skorokhod embedding problem for the L´evy process reflected at itssupremum; Baurdoux (2007) investigated the density of the resolvent measure of the killed L´evyprocess reflected at its infimum; Kyprianou and Zhou (2009) obtained the generalized version ofthe Gerber-Shiu function for a taxed L´evy risk process where a loss-carry-forward taxation systemis embedded.The existing literature has been witnessing the applications of the powerful excursion-theoreticalapproach in the fields of financial mathematics and stochastic process theory. However, as far asthe authors know, this approach is rarely used in the field of actuarial risk theory. This paper at-tempts to apply the approach to study the expression of the extended Gerber-Shiu function in thecase of drawdown. One merit of applying the excursion-theoretical approach is, we do not needto use specific features of the underlying L´evy process except for a generic path decomposition interms of excursions from the running maximum, which falls into the framework of Poisson pointprocess. Thence we are allowed to use the theory of Poisson point process, such as the compen-sation formula, in the whole manipulation of our target problem. We mention that all results inthis paper are expressed elegantly in terms of scale functions and the L´evy measure associatedwith the L´evy process. We also mention that, when the first passage over the non-constant gen-eral drawdown boundary is reduced to the case of constant boundary, we are able to recover thecorresponding results in the existing literature.We point out that Li et al. (2017) also applied the excursion approach to study the exit problemsinvolving a general drawdown time for spectrally negative L´evy processes. However their focus PRELIMINARIES OFSPECTRALLY NEGATIVEL´EVY PROCESS 4was on the joint Laplace transform for the process at the drawdown time, while this paper aims tostudy the joint distribution involving general drawdown times. It is true that the joint distributionof random quantities is uniquely determined by the corresponding Laplace transform, and canbe obtained by inverting the Laplace transform either analytically by the Bromwich integral ornumerically. Unfortunately, many problems of mathematical interest or physical interest lead toLaplace transforms whose inverses are not readily expressed in terms of tabulated functions. Andalso, to the best of our knowledge, all the current numerical inversion methods are unstable in thesense that small “input”errors, arising say from computer roundo ff or from parameter selectioninherent in the algorithm, can be disastrously magnified in the inversion, which should be avoidedin some practical problems such as the survival probability of a population in a finite capacityenvironment or in tumor growth models, see Albano and Giorno (2006) for an example. Evenwith the development of the modern technology, it is hard to find a universal algorithm that workswell to all the cases. A nice review of these methods can be found in Davies (2002). Thereforeit is still worthwhile to study the joint distribution even when the Laplace transform is readilyavailable.Besides, in the computation of the joint distribution we add a constraint on the surplus levelwhich is another di ff erence between our paper and Li et al (2017). The motivation of this con-straint comes from Bi ffi s and Morales (2010) where the last minimum of the surplus before ruinwas considered to better reflect the company’s financial condition. In our paper we extend thisterminology to the last minimum of the surplus before drawdown, which serves as a warning lineof the company’s financial activities. We refer to Remark 3.1 for more detailed interpretations.This paper is organized as follows. Section 2 presents some preliminary facts on spectrallynegative L´evy processes. The main results, proofs and discussions are provided in Section 3. InSection 4, the main results are applied to study the Gerber-Shiu function at ruin for L´evy riskprocesses with tax and dividends. Section 5 provides some numerical examples to illustrate ourresults. Section 6 concludes this paper.
2. Preliminaries of spectrally negative L´evy process
Write X = { X ( t ); t ≥ } , defined on a probability space with probability laws { P x ; x ∈ R } and natural filtration {F t ; t ≥ } , for a spectrally negative L´evy process.We denote its runningsupremum and running infimum process, respectively, as { X ( t ) = sup ≤ s ≤ t X ( s ); t ≥ } and { X ( t ) = inf ≤ s ≤ t X ( s ); t ≥ } .A function ξ defined on (0 , ∞ ) is called a general drawdown function if it is continuous and ξ ( x ) = x − ξ ( x ) > x >
0. The general drawdown time with respect to the general drawdownfunction ξ ( · ), also called the ξ -drawdown time for short, is defined as τ ξ = inf { t ≥ X ( t ) < ξ ( X ( t )) } . Remark 2.1.
Note that when ξ ( · ) ≡ , τ ξ reduces to the classical ruin time. Another example isa linear function of the running supreme, say, ξ ( X ( t )) = . X ( t ) − . . Then X ( t ) < ξ ( X ( t )) isequivalent to . X ( t ) − X ( t ) > . . Accordingly, τ ξ refers to the first time the surplus process drops . units below 80% of its maximum to date. In practice, the risk manager can use it as a turning PRELIMINARIES OFSPECTRALLY NEGATIVEL´EVY PROCESS 5 point of taking some actions, such as increasing the premium rate or negotiating by the companywith the capital providers to prevent even worse situations. For non-linear forms of ξ ( · ) , we referto Remark 2 in Avram et al. (2017) for the examples and their explanations. We also define the first down-crossing time of level a and up-crossing time of level b , respec-tively, as follows τ − a : = inf { t ≥ X ( t ) < a } and τ + b : = inf { t ≥ X ( t ) > b } . Let the Laplace exponent of X be given by ψ ( θ ) = ln E x (cid:16) e θ ( X (1) − x ) (cid:17) = γθ + σ θ − Z (0 , ∞ ) (cid:16) − e − θ x − θ x (0 , ( x ) (cid:17) υ (d x ) , where υ is the L´evy measure satisfying R (0 , ∞ ) (cid:16) ∧ x (cid:17) υ (d x ) < ∞ . It is known that ψ ( θ ) is finite for θ ∈ [0 , ∞ ) in which case it is strictly convex and infinitely di ff erentiable. As in Bertoin (1996), the q -scale functions { W q ; q ≥ } of X are defined as follows. For each q ≥ W q : [0 , ∞ ) → [0 , ∞ ) isthe unique strictly increasing and continuous function with Laplace transform Z ∞ e − θ x W q ( x )d x = ψ ( θ ) − q , for θ > Φ q , where Φ q is the largest solution of the equation ψ ( θ ) = q . Further define W q ( x ) = x <
0, andwrite W for short for the 0-scale function W . Remark 2.2.
Scale functions appear in the vast majority of known identities concerning boundarycrossing problems and related path decompositions. This in turn has consequences for their use ina number of classical applied probability models which rely heavily on such identities. We refer toKuznetsov et al. (2012) for intuitive examples and explanations. We have to point out that for allspectrally negative L´evy processes, q-scale functions exist for all q ≥ . However, since the scalefunction is defined via its Laplace transform, and in most cases it is not possible to find an explicitexpression for the inverse of a Laplace transform, hence the expression in terms of the q-scalefunctions can be called semi-explicit expression. For those who cannot find explicit expressions,we resort to numerical methods which allow one to compute scale functions easily and e ffi ciently,see also Kuznetsov et al. (2012). We also briefly recall concepts in excursion theory for the reflected process { X ( t ) − X ( t ); t ≥ } ,and we refer to Bertoin (1996) for more details. The process { L ( t ) : = X ( t ) − x , t ≥ } serves as alocal time at 0 for the Markov process { X ( t ) − X ( t ); t ≥ } under P x . Let the corresponding inverselocal time be defined as L − ( t ) : = inf { s ≥ | L ( s ) > t } = sup { s ≥ | L ( s ) ≤ t } . Let further L − ( t − ) = lim s ↑ t L − ( s ). The Poisson point process of excursions indexed by this localtime is denoted by { ( t , ε t ); t ≥ } , where ε t ( s ) : = X ( L − ( t )) − X ( L − ( t − ) + s ) , s ∈ (0 , L − ( t ) − L − ( t − )] , MAIN RESULTS 6whenever L − ( t ) − L − ( t − ) >
0. For the case of L − ( t ) − L − ( t − ) =
0, define ε t = Υ with Υ being an additional isolated point. Accordingly, we denote a generic excursion as ε ( · ) (or, ε forshort) belonging to the space E of canonical excursions. The intensity measure of the process { ( t , ε t ); t ≥ } is given by d t × d n where “ n ”is a measure on the space of excursions. The lifetimeof a canonical excursion ε is denoted by ζ , and its excursion height is denoted by ε = sup t ∈ [0 ,ζ ] ε ( t ).The first passage time of a canonical excursion ε will be defined by ρ + b : = inf { t ∈ [0 , ζ ]; ε ( t ) > b } , (1)with the convention inf ∅ = ζ . In addition, define α b : = sup t ∈ [0 ,ρ + b ) ε ( t ) , (2)which is the excursion height prior to ρ + b . Remark 2.3.
An excursion is a segment of the path that has zero value only at its two endpoints. Itrefers to a maximal open time interval such that the path is away from 0. Naturally, a L´evy processcan be decomposed into a sequence of excursions. Due to the stochastic nature, say, a Brownianmotion (the only continuous L´evy process) started from zero can hit zero infinitely often in anytime interval, then the excursions are labelled by local times rather than by the starting time of aparticular excursion. Considering there are only countably many excursions, hence there are onlycountably many local times which pertain to an excursion. This motivates the idea of taking theset of excursions as a Poisson Point Process on local times. Therefore, a measure can be definedto describe the intensity of the Poisson point process of excursions, which is the intuitive meaningof measure “n”in our setting. For more detailed explanations, we refer to Kyprianou (2006).
3. Main results
This section introduces the extended expected discounted penalty function at the general draw-down time for a L´evy risk process, then express it in terms of the q -scale functions and the L´evymeasure associated with the L´evy process.The following three technical lemmas turn out to be helpful when we derive the main results.Lemma 3.1 characterizes the atom at 0 of the discounted distribution law of the overshoot at firstup-crossing time of a canonical excursion ε with respect to the excursion measure. We present aproof here for self-completeness. Lemma 3.1.
For q > and s > x ≥ , we haven (cid:16) e − q ρ + s ; ε > s , ε ( ρ + s ) = s (cid:17) = σ (cid:16) W ′ q ( s ) (cid:17) W q ( s ) − W ′′ q ( s ) . (3) MAIN RESULTS 7 Proof : From (14) of Pistorius (2007), we read o ff that E x (cid:16) e − q τ − ; X ( τ − ) = (cid:17) = E (cid:16) e − q τ −− x ; X ( τ −− x ) = − x (cid:17) = σ (cid:16) W ′ q ( x ) − Φ q W q ( x ) (cid:17) . (4)By the definition of ruin, one knows that there must exist an excursion with a positive lifetime,such that τ − lies in between the left and right end points of this excursion. Denote this excursionby ε θ with θ ≥
0, then ε θ is the last excursion whose left-end point is less than τ − , and we have ε θ > x + θ and ε t ≤ x + t for all excursions ε t with t < θ . That is, ruin does not occur duringthe lifetime of ε t with t < θ , while it does occur during the lifetime of ε θ . Furthermore, one cantranslate the ruin time and the surplus at ruin time by the excursion ε θ , respectively, through τ − = L − ( θ − ) + ρ + x + θ ( θ ) , X ( τ − ) = x + θ − ε θ (cid:0) ρ + x + θ ( θ ) (cid:1) , where ρ + x + θ ( θ ) is defined via (1) with ε replaced by ε θ and b replaced by x + θ . Therefore, the lefthand side of (4) can be translated as E x (cid:16) e − q τ − ; X ( τ − ) = (cid:17) = E x X θ ≥ e − qL − ( θ − ) { L − ( θ − ) <τ − } e − q ( τ − − L − ( θ − ) ) { S ( L − ( θ − ) ) = X ( τ − ) = x + θ } { X ( τ − ) = } = E x X θ ≥ e − qL − ( θ − ) Y t <θ { ε t < x + t } e − q ρ + x + θ ( θ ) { ε θ > x + θ } { ε θ ( ρ + x + θ ( θ ) ) = x + θ } , (5)where ε θ represents the last excursion prior to τ − , and ρ + x + θ ( θ ) is given by (1) with ε replaced by ε θ and b replaced by x + θ . By the compensation formula (cf., Bertoin (1996)) in the excursion theorytogether with (5), we obtain E x (cid:16) e − q τ − ; X ( τ − ) = (cid:17) = Z ∞ E x e − qL − ( θ − ) Y t <θ { ε t < x + t } n (cid:16) e − q ρ + x + θ ( θ ) { ε θ > x + θ, ε θ ( ρ + x + θ ( θ ) ) = x + θ } (cid:17) d θ = Z ∞ E x (cid:16) e − q τ + x + θ { τ + x + θ <τ − } (cid:17) n (cid:16) e − q ρ + x + θ { ε> x + θ, ε ( ρ + x + θ ) = x + θ } (cid:17) d θ = Z ∞ x W ( q ) ( x ) W ( q ) ( s ) n (cid:16) e − q ρ + s { ε> s , ε ( ρ + s ) = s } (cid:17) d s . (6)Equating (4) and (6) and then di ff erentiating both side of the resulting equation with respect to x yields (3).We recall from (1) and (2) that, ρ + x and α x refer respectively to the first passage time of theexcursion ε over x and the excursion height prior to ρ + x . The following result gives the discountedjoint distribution involving ρ + x and α x under the excursion measure. It generalizes Lemma 2.2 ofKyprianou and Zhou (2009) by imposing a constraint on the excursion height α x ≤ x − v , where v ∈ (0 , x ] can be taken as the minimum capital requirement on the last minimum of the surplusbefore ruin. MAIN RESULTS 8 Lemma 3.2.
For x , y , z ∈ (0 , + ∞ ) and v ∈ (0 , x ] , we haven (cid:16) e − q ρ + x ; x − ε (cid:0) ρ + x − (cid:1) ∈ d y , ε (cid:0) ρ + x (cid:1) − x ∈ d z , α x ≤ x − v (cid:17) = W ′ q ( x − y ) − W q ( x − y ) W ′ q ( x − v ) W q ( x − v ) ! υ (d z + y ) { y < x } d y + W q (0) υ (d z + x ) δ x (d y ) , (7) where δ x (d y ) is the Dirac measure which assigns unit mass to the point x.Proof : Recall X ( t ) = inf ≤ s ≤ t X ( s ) for t ≥
0, then X ( τ − − ) refers to the last minimum of the surplusbefore ruin, see Bi ffi s and Morales (2010). By Theorem 1 in Bi ffi s and Kyprianou (2010) one has E x (cid:16) e − q τ − ; X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , X ( τ − − ) ∈ d v (cid:17) = e − Φ q ( y − v ) (cid:16) W ′ q ( x − v ) − Φ q W q ( x − v ) (cid:17) υ (d z + y ) d v d y , x , y , z ∈ (0 , + ∞ ) , v ∈ (0 , x ∧ y ] , which yields E x (cid:16) e − q τ − ; X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , X ( τ − − ) ≥ v (cid:17) = Z w ∈ [ v , x ∧ y ] e Φ q w (cid:16) W ′ q ( x − w ) − Φ q W q ( x − w ) (cid:17) d w e − Φ q y υ (d z + y ) d y = e − Φ q y υ (d z + y ) d y { y < x } Z w ∈ [ v , y ] d (cid:16) − e Φ q w W q ( x − w ) (cid:17) + { y ≥ x } Z w ∈ [ v , x ) d (cid:16) − e Φ q w W q ( x − w ) (cid:17) + Z w ∈{ x } d (cid:16) − e Φ q w W q ( x − w ) (cid:17)!! = e − Φ q y υ (d z + y ) d y (cid:16) { y < x } (cid:16) e Φ q v W q ( x − v ) − e Φ q y W q ( x − y ) (cid:17) + { y ≥ x } (cid:16) e Φ q x W q (0) + (cid:16) e Φ q v W q ( x − v ) − e Φ q x W q (0) (cid:17)(cid:17)(cid:17) , x , y , z ∈ (0 , ∞ ) , v ∈ (0 , x ∧ y ] . (8)By the same language of excursions as in (5), we rewrite (8) as E x (cid:16) e − q τ − ; X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , X ( τ − − ) ≥ v (cid:17) = E x X θ ≥ e − qL − ( θ − ) Y t <θ { ε t < x + t − v } e − q ρ + x + θ ( θ ) { ε θ > x + θ } × { x + θ − ε θ ( ρ + x + θ ( θ ) − ) ∈ d y , − ( x + θ − ε θ ( ρ + x + θ ( θ ) )) ∈ d z , α x + θ ( θ ) ≤ x + θ − v } (cid:17) , x , y , z ∈ (0 , ∞ ) , v ∈ (0 , x ∧ y ] , (9)where ε θ represents the last excursion prior to τ − , ρ + x + θ ( θ ) and α x + θ ( θ ) are defined by (1) and (2)with ε replaced by ε θ and b replaced by x + θ , and X ( τ − − ) = inf ≤ t <θ ( x + t − ε t ) ∧ ( x + θ − α x + θ ( θ )) . By (9), the well known two-sided exit identity (see, Kyprianou (2006)), E x (cid:16) e − q τ + s { τ + s <τ − v } (cid:17) = W q ( x − v ) W q ( s − v ) , < v < x < s < ∞ , MAIN RESULTS 9and the compensation formula (cf., Bertoin (1996)) in excursion theory, one can obtain E x (cid:16) e − q τ − ; X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , X ( τ − − ) ≥ v (cid:17) = Z ∞ E x e − qL − ( θ − ) Y t <θ { ε t < x + t − v } × n (cid:16) e − q ρ + x + θ ( θ ) { ε θ > x + θ } × { x + θ − ε θ ( ρ + x + θ ( θ ) − ) ∈ d y , − ( x + θ − ε θ ( ρ + x + θ ( θ ) )) ∈ d z , α x + θ ( θ ) ≤ x + θ − v } (cid:17) d θ = Z ∞ E x (cid:16) e − q τ + x + θ { τ + x + θ <τ − v } (cid:17) n (cid:16) e − q ρ + x + θ { ε> x + θ } { x + θ − ε ( ρ + x + θ − ) ∈ d y , ε ( ρ + x + θ ) − ( x + θ ) ∈ d z , α x + θ ≤ x + θ − v } (cid:17) d θ = Z ∞ x E x (cid:16) e − q τ + s { τ + s <τ − v } (cid:17) n (cid:16) e − q ρ + s ; ε > s , s − ε (cid:0) ρ + s − (cid:1) ∈ d y , ε (cid:0) ρ + s (cid:1) − s ∈ d z , α s ≤ s − v (cid:17) d s = Z ∞ x W q ( x − v ) W q ( s − v ) n (cid:16) e − q ρ + s ; s − ε (cid:0) ρ + s − (cid:1) ∈ d y , ε (cid:0) ρ + s (cid:1) − s ∈ d z , α s ≤ s − v (cid:17) d s , which combined with (8) yields1 W q ( x − v ) n (cid:16) e − q ρ + x ; x − ε (cid:0) ρ + x − (cid:1) ∈ d y , ε (cid:0) ρ + x (cid:1) − x ∈ d z , α x ≤ x − v (cid:17) = W ′ q ( x − y ) W q ( x − v ) − W q ( x − y ) W ′ q ( x − v ) (cid:16) W q ( x − v ) (cid:17) { x > y } υ (d z + y ) d y + W q (0) W q ( x − v ) υ (d z + x ) δ x (d y ) , x , y , z ∈ (0 , ∞ ) , v ∈ (0 , x ∧ y ] , which is (7). The proof is complete.The following Lemma 3.3 solves the general drawdown based two-sided exit problem, and canbe found in Proposition 3.1 of Li et al. (2017) and Lemma 3.2 of Wang and Zhou (2018). Lemma 3.3.
For q > and s > x and general drawdown function ξ , we have E x (cid:16) e − q τ + s { τ + s <τ ξ } (cid:17) = exp − Z sx W ′ q ( ξ ( z )) W q ( ξ ( z )) d z , where ξ ( z ) = z − ξ ( z ) . Motivated by the last minimum of the surplus before ruin proposed by Bi ffi s and Morales (2010),this paper includes a constraint on the last minimum surplus before drawdown intending to betterreflect the company’s financial condition. Let ϑ : [ x , ∞ ) → [0 , ∞ ) be a measurable functionsatisfying 0 ≤ ϑ ( z ) < ξ ( z ). In our main result in below, we use ϑ ( X ( t )) to describe the minimumcapital requirement of the surplus above the drawdown level. Note that when ϑ ( X ( t )) equals toa constant v and ξ ( z ) ≡ ς ( z ) = ϑ ( z ) + ξ ( z ) and ς ( z ) = z − ( ϑ ( z ) + ξ ( z )) = ξ ( z ) − ϑ ( z ), hence ς is also a general drawdown function. MAIN RESULTS 10Theorem 3.1 derives the extended expected discounted penalty function at the general drawdowntime for L´evy risk processes, where a similar excursion approach as in Pistorius (2007), Kyprianouand Zhou (2009) and Li et al (2017) was adopted. Theorem 3.1.
Denote by υ the L´evy measure of − X, by ℓ : = L − ( L ( τ ξ ) − ) the first time when theL´evy process X hits the running maximum prior to the general drawdown time.(a) For s ∈ ( x ∨ y , ∞ ) , y ∈ [ ξ ( s ) , ∞ ) , z ∈ ( − ξ ( s ) , ∞ ) and q , λ ≥ , we have E x e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ d s , X ( τ ξ − ) ∈ d y , − X ( τ ξ ) ∈ d z , inf t ∈ [ 0 ,τ ξ ) (cid:16) X ( t ) − ς ( X ( t )) (cid:17) ≥ ! = exp − Z sx W ′ q ( ς ( w )) W q ( ς ( w )) d w ! W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( ς ( s )) W λ ( ς ( s )) W λ ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s . (10) (b) For s ∈ ( x , ∞ ) and q , λ ≥ , we have E x (cid:16) e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ d s , X ( τ ξ ) = ξ ( s ) (cid:17) = σ − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w (cid:16) W ′ λ (cid:16) ξ ( s ) (cid:17)(cid:17) W λ (cid:16) ξ ( s ) (cid:17) − W ′′ λ (cid:16) ξ ( s ) (cid:17) d s , (11) where the expression is understood to be equal to if σ = . At the time of drawdown,the surplus stays at the general drawdown level with a positive probability if and only if theGaussian part of the L´evy process is nontrivial. Remark 3.1.
Note that on the left hand side of equation (10) we include a constraint inf t ∈ [ 0 ,τ ξ ) (cid:16) X ( t ) − ς ( X ( t )) (cid:17) ≥ ⇔ X ( t ) − ξ ( X ( t )) ≥ ϑ ( X ( t )) , ≤ t < τ ξ , which in fact is a constraint on X ( t ) − ξ ( X ( t )) , that is, the level of surplus that is above the draw-down level ξ ( X ( t )) . We require this level to be at least ϑ ( X ( t )) ≥ , which could be linked with theconfidence level of the company and hence to be dependent with the historical running maximumX ( t ) . The lower ϑ ( X ( t )) , the worse the financial conditions that need to be negotiated with thecompany’s capital providers, and the more urgent for the company to examine its financial activi-ties. The level of ϑ ( X ( t )) provides a warning line of the company’s inadvisable financial decisionssuch as a low premium rate, also serves as a bu ff er towards future’s capital injections. MAIN RESULTS 11
Proof of Theorem 3.1 : Similar to the idea in Lemma 3.1, the definition of general drawdown leadsto the existence of an excursion with a positive lifetime, such that τ ξ lies in between the left andright end points of this excursion. Denote this excursion by ε θ with θ ≥
0, then ε θ is the lastexcursion whose left-end point is less than τ ξ , and we have ε θ > ξ ( x + θ ) and ε t ≤ ξ ( x + t ) for allexcursions ε t with t < θ . That is, a general drawdown does not occur during the lifetime of ε t with t < θ , while it does occur during the lifetime of ε θ . Furthermore, one can translate immediatelythe surplus before and at the general drawdown time by the excursion ε θ through X ( τ ξ − ) = x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) − (cid:19) , X ( τ ξ ) = x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) (cid:19) , where ρ + ξ ( x + θ ) ( θ ) is defined via (1) with ε replaced by ε θ and b replaced by ξ ( x + θ ). Meanwhile, thegeneral drawdown time can be rewritten as τ ξ = L − ( θ − ) + ρ + ξ ( x + θ ) ( θ ) . Therefore we have that, for y , z ∈ ( −∞ , + ∞ ), q , λ ≥ B ⊆ ( x , ∞ ), E x e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ B , X ( τ ξ − ) ∈ d y , − X ( τ ξ ) ∈ d z , inf t ∈ [0 ,τ ξ ) (cid:16) X ( t ) − ς ( X ( t )) (cid:17) ≥ ! = E x X θ ∈ B − x e − qL − ( θ − ) Y t <θ { ε t <ς ( x + t ) } e − λρ + ξ ( x + θ ) ( θ ) { ε θ >ξ ( x + θ ) } × { x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) − (cid:19) ∈ d y , − (cid:18) x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) (cid:19)(cid:19) ∈ d z , α ξ ( x + θ ) ≤ ς ( x + θ ) } ! , (12)where ε θ represents the last excursion prior to τ ξ , ρ + ξ ( x + θ ) ( θ ) is given by (1) with ε replaced by ε θ and b replaced by ξ ( x + θ ), and B − x : = { y − x | y ∈ B } .Using the compensation formula (see, Corollary 11 in Chapter IV.4 of Bertoin (1996)) in theexcursion expression in (12), we obtain E x e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ B , X ( τ ξ − ) ∈ d y , − X ( τ ξ ) ∈ d z , inf t ∈ [0 ,τ ξ ) (cid:16) X ( t ) − ς ( X ( t )) (cid:17) ≥ ! = Z B − x E x e − qL − ( θ − ) Y t <θ { ε t <ς ( x + t ) } × n (cid:18) e − λρ + ξ ( x + θ ) ( θ ) { ε θ >ξ ( x + θ ) } × { x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) − (cid:19) ∈ d y , − (cid:18) x + θ − ε θ (cid:18) ρ + ξ ( x + θ ) ( θ ) (cid:19)(cid:19) ∈ d z , α ξ ( x + θ ) ≤ ς ( x + θ ) } ! d θ = Z B − x E x (cid:16) e − q τ + x + θ { τ + x + θ <τ ς } (cid:17) × n (cid:18) e − λρ + ξ ( x + θ ) { ε>ξ ( x + θ ) } × { x + θ − ε (cid:18) ρ + ξ ( x + θ ) − (cid:19) ∈ d y , ε (cid:18) ρ + ξ ( x + θ ) (cid:19) − ( x + θ ) ∈ d z , α ξ ( x + θ ) ≤ ς ( x + θ ) } ! d θ = Z B E x (cid:16) e − q τ + s { τ + s <τ ς } (cid:17) n (cid:18) e − λρ + ξ ( s ) ; ε > ξ ( s ) , s − ε (cid:18) ρ + ξ ( s ) − (cid:19) ∈ d y , ε (cid:18) ρ + ξ ( s ) (cid:19) − s ∈ d z , α ξ ( s ) ≤ ς ( s ) (cid:19) d s MAIN RESULTS 12 = Z B exp − Z sx W ′ q ( ς ( w )) W q ( ς ( w )) d w ! { y ≥ ξ ( s ) } { z > − ξ ( s ) } × n (cid:18) e − λρ + ξ ( s ) ; ξ ( s ) − ε (cid:18) ρ + ξ ( s ) − (cid:19) ∈ − ξ ( s ) + d y , ε (cid:18) ρ + ξ ( s ) (cid:19) − ξ ( s ) ∈ ξ ( s ) + d z , α ξ ( s ) ≤ ς ( s ) (cid:19) d s = Z B exp − Z sx W ′ q ( ς ( w )) W q ( ς ( w )) d w ! { y ≥ ξ ( s ) , z > − ξ ( s ) } W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( ς ( s )) W λ ( ς ( s )) W λ ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s , (13)where in the last equality we have used Lemma 3.2, and in the last but one equality we have usedLemma 3.3. The arbitrariness of the open interval B together with (13) yields (10).It remains to prove Case ( b ). In fact, by the compensation formula (see, Corollary 11 in ChapterIV.4 of Bertoin (1996)) we have E x (cid:16) e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ B , X ( τ ξ ) = ξ (cid:16) X ( τ ξ ) (cid:17)(cid:17) = E x X θ ∈ B − x e − qL − ( θ − ) Y t <θ { ε t <ξ ( x + t ) } e − λρ + ξ ( x + θ ) ( θ ) { ε θ >ξ ( x + θ ) } { ε θ ( ρ + ξ ( x + θ ) ( θ )) = ξ ( x + θ ) } = Z B − x E x e − qL − ( θ − ) Y t <θ { ε t <ξ ( x + t ) } × n (cid:18) e − λρ + ξ ( x + θ ) ( θ ) { ε θ >ξ ( x + θ ) } { ε θ ( ρ + ξ ( x + θ ) ( θ )) = ξ ( x + θ ) } (cid:19) d θ = Z B − x E x (cid:16) e − q τ + x + θ { τ + x + θ <τ ξ } (cid:17) n (cid:18) e − λρ + ξ ( x + θ ) { ε>ξ ( x + θ ) } { ε ( ρ + ξ ( x + θ ) ) = ξ ( x + θ ) } (cid:19) d θ = Z B exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w n (cid:18) e − λρ + ξ ( s ) ; ε > ξ ( s ) , ε (cid:18) ρ + ξ ( s ) (cid:19) = ξ ( s ) (cid:19) d s = Z B exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w σ (cid:16) W ′ λ (cid:16) ξ ( s ) (cid:17)(cid:17) W λ (cid:16) ξ ( s ) (cid:17) − W ′′ λ (cid:16) ξ ( s ) (cid:17) d s , (14)where in the last equality we have used Lemma 3.1, and in the last but one equality we have usedLemma 3.3. The arbitrariness of the open interval B together with (14) yields (11). This completesthe proof of Case ( b ) of Theorem 3.1. Remark 3.2.
It should be pointed out that the excursion methodology adopted here to study thegeneral drawdown is based on L´evy processes, and hence can not be easily applied to other un-derlying models, say for example, di ff usion processes or jump di ff usion processes. In Landriault etal. (2017), a methodology called “short-time pathwise analysis” was proposed to study the clas-sical drawdown involved fluctuation problems for general time-homogeneous Markov processes.Their method is supposed to be more general in the sense that can be adapted to study fluctuationproblems involving the general drawdown time, but still seems restrictive in our case where boththe time to reach the running maximum at τ ξ and the surplus level before τ ξ are involved. MAIN RESULTS 13Note that in Part ( a ) of Theorem 3.1, y ∈ [ ξ ( s ) , ∞ ) is equivalent to y − ξ ( s ) ∈ [0 , ∞ ), and z ∈ ( − ξ ( s ) , ∞ ) is equivalent to z + ξ ( s ) ∈ (0 , ∞ ), we readily have the following result. Corollary 3.1.
For z > , y ≥ , s ≥ y ∨ x, and q , λ ≥ , we have E x e − q ℓ − λ ( τ ξ − ℓ ); X ( τ ξ ) ∈ d s , X ( τ ξ − ) − ξ ( s ) ∈ d y , ξ ( s ) − X ( τ ξ ) ∈ d z , inf t ∈ [ 0 ,τ ξ ) (cid:16) X ( t ) − ς ( X ( t )) (cid:17) ≥ ! = exp − Z sx W ′ q ( ς ( w )) W q ( ς ( w )) d w ! W λ (0 + ) υ ( ξ ( s ) + d z ) δ ξ ( s ) (d y ) + W ′ λ ( ξ ( s ) − y ) − W ′ λ ( ς ( s )) W λ ( ς ( s )) W λ ( ξ ( s ) − y ) ! υ ( y + d z ) { y <ξ ( s ) } d y ! d s . Furthermore, when q = λ , ξ ≡ and ϑ ≡ v ∈ [0 , x ) we have E x (cid:16) e − q τ − ; X ( τ − ) ∈ d s , X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , X ( τ − ) ≥ v (cid:17) = W q ( x − v ) W q ( s − v ) W q (0 + ) υ ( s + d z ) δ s (d y ) + W ′ q ( s − y ) − W ′ q ( s − v ) W q ( s − v ) W q ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s , integrating which with respect to s over [ x , ∞ ) and recalling that lim x →∞ W q ( s − y ) W q ( s − v ) = e − Φ q ( y − v ) , one canrecover (8) . Hence, as a special case, the result of Theorem 3.1 coincides with that of Theorem 1in Bi ffi s and Kyprianou (2010). When ϑ ≡ ξ ≡
0, then the general drawdown time is reduced to the classical ruin time τ ξ = τ − , ℓ = L − ( L ( τ − ) − ), and the results of Theorem 3.1 are specialized to the following Corollary 3.2,which coincides well with Theorem 1.3 in Kyprianou and Zhou (2009) with γ ≡ Corollary 3.2.
The generalized expected discounted penalty function at the classical ruin timecan be characterized as follows.(a ′ ) For s ∈ ( x ∨ y , ∞ ) , y ∈ [0 , ∞ ) , z ∈ (0 , ∞ ) and q , λ ≥ , we have E x (cid:16) e − q ℓ − λ ( τ − − ℓ ); X ( τ − ) ∈ d s , X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z (cid:17) = W q ( x ) W q ( s ) W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( s ) W λ ( s ) W λ ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s . (b ′ ) For s ∈ ( x ∨ y , ∞ ) and q , λ ≥ , we have E x (cid:16) e − q ℓ − λ ( τ − − ℓ ); X ( τ − ) ∈ d s , X ( τ − ) = (cid:17) = σ W q ( x ) W q ( s ) (cid:16) W ′ λ ( s ) (cid:17) W λ ( s ) − W ′′ λ ( s ) d s , where the expression is understood to be equal to if σ = . MAIN RESULTS 14
Remark 3.3.
Furthermore, if there is no Brownian part in the L´evy-Itˆo decomposition of X (i.e., σ = ), then by ( a ′ ) one may find, for b > x, y ∈ [0 , ∞ ) , z ∈ (0 , ∞ ) and q ≥ E x (cid:16) e − q τ − ; X ( τ − − ) ∈ d y , − X ( τ − ) ∈ d z , τ − < τ + b (cid:17) = Z s ∈ ( x , b ) W q ( x ) W q ( s ) W q (0 + ) υ ( s + d z ) δ s (d y ) + W ′ q ( s − y ) − W ′ q ( s ) W q ( s ) W q ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s = W q ( x ) W q ( y ) W q (0 + ) υ ( y + d z ) { x < y } d y + Z s ∈ ( x , b ) W q ( x ) W q ( s ) W ′ q ( s − y ) − W ′ q ( s ) W q ( s ) W q ( s − y ) ! υ ( y + d z ) (cid:16) { y ≤ x } + { x < y < s } (cid:17) d y d s = W q ( x ) W q ( y ) W q (0 + ) υ ( y + d z ) { x < y } d y + Z s ∈ ( x , b ) W q ( x ) W q ( s ) W ′ q ( s − y ) − W ′ q ( s ) W q ( s ) W q ( s − y ) ! d s υ ( y + d z ) { y ≤ x } d y + Z s ∈ ( y , b ) W q ( x ) W q ( s ) W ′ q ( s − y ) − W ′ q ( s ) W q ( s ) W q ( s − y ) ! d s υ ( y + d z ) { x < y } d y = W q ( x ) W q ( y ) W q (0 + ) υ ( y + d z ) { x < y } d y + W q ( x ) υ ( y + d z ) W q ( s − y ) W q ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bs = x { y ≤ x } d y + W q ( s − y ) W q ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bs = y { x < y } d y = W q ( x ) υ ( y + d z ) W q ( b − y ) W q ( b ) − W q ( x − y ) W q ( x ) ! d y . (15) where we have used the fact that W ′ q ( s − y ) W q ( s ) − W ′ q ( s )[ W q ( s )] W q ( s − y ) = dd s (cid:16) W q ( s − y ) W q ( s ) (cid:17) in the fourth equation, andthe fact that W q ( x − y ) = for y > x in the last equation. One can find that (15) coincides wellwith Theorem 5.5 on page 41 of Kyprianou (2013), noting that we have υ ( y + d z ) = λ F ( y + d z ) inthe classical Cram´er-Lundberg risk process with claim distribution F. Remark 3.4.
We say that the results in Theorem 3.1 is the generalized version of the classicalexpected penalty function at the general drawdown time. In fact, the classical expected penaltyfunction at the general drawdown time can be written as φ ( x ) : = E x (cid:16) e − q τ ξ ω (cid:16) X ( τ ξ − ) , X ( τ ξ ) (cid:17) { τ ξ < ∞} (cid:17) , for some bounded measurable bivariate function ω ( · , · ) : ( −∞ , + ∞ ) → (0 , ∞ ) . In actuarialsciences, ω is called the penalty function (cf., Gerber and Shiu (1998)). Using Theorem 3.1, one APPLICATIONS 15 can solve the classical expected penalty function at the general drawdown as φ ( x ) = Z s ∈ ( x , ∞ ) Z y ∈ [ ξ ( s ) , s ] Z z ∈ ( − ξ ( s ) , ∞ ) ω ( y , − z ) exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w W q (0 + ) υ ( s + d z ) δ s (d y ) + W ′ q ( s − y ) − W ′ q ( ξ ( s )) W q ( ξ ( s )) W q ( s − y ) υ ( y + d z ) { y < s } d y d s + σ Z s ∈ ( x , ∞ ) ω ( ξ ( s ) , ξ ( s )) exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w (cid:16) W ′ q (cid:16) ξ ( s ) (cid:17)(cid:17) W q (cid:16) ξ ( s ) (cid:17) − W ′′ q (cid:16) ξ ( s ) (cid:17) d s . (16) The following interesting example of penalty function can be found in Gerber and Shiu (1998), ω ( x , y ) = max (cid:0) K − e a − y , (cid:1) . In this case, φ ( x ) is the payo ff of a perpetual American put option with K as the exercise price and e a be the value of an option-exercise boundary.
4. Applications
This section is focused on the applications of Theorem 3.1. By specifying the general drawdownfunction, the methodology in this paper can be adapted naturally to recover results in the literatureand to obtain new results for the Gerber-Shiu function at ruin for risk processes embedded with aloss-carry-forward taxation system or a barrier dividend strategy. In fact, it was Landriault et al.(2017) and Li (2015) who first pointed out that ruin problems in loss-carry-forward taxation (resp,De Finetti’s dividend) models can be transformed to general drawdown problems for the classicalmodels without taxation (resp, dividend). However, they proposed the idea without the implemen-tation of a particular ruin problem. In addition, the drawdown function studied in Landriault et al.(2017) and Li (2015) was the classical drawdown function in the form of ξ ( x ) = x − d with d > In Kyprianou and Zhou (2009), a L´evy risk model with a general loss-carry-forward tax struc-ture was first considered U γ ( t ) : = X ( t ) − Z t γ ( X ( s ))d X ( s ) = X ( t ) − Z X ( t ) x γ ( w )d w , (17)where γ : [0 , + ∞ ) → [0 ,
1) is measurable and R ∞ (1 − γ ( w )) d w = ∞ . In this formulation, taxesare paid whenever the company is in a profitable situation, defined as being at a running maximumof the surplus process. APPLICATIONS 16We claim that the version of Gerber-Shiu function at ruin obtained in Kyprianou and Zhou(2009) can be recovered by specifying a special drawdown function ξ in Theorem 3.1. To thispurpose, for x ∈ (0 , ∞ ), let ξ γ ( z ) : = Z zx γ ( w )d w , z ∈ [ x , ∞ ) , (18)which is indeed a drawdown function. One can make the following three observations.( i ) The ξ γ -drawdown time of X coincides with the ruin time of the taxed risk process (17) τ ξ γ = inf { t ≥ X ( t ) < ξ γ ( X ( t )) } = inf { t ≥ U γ ( t ) < } : = τ − ( γ ) , and hence ℓ = L − ( L ( τ − ( γ )) − ), i.e., the last moment that tax is paid before the ruin of (17).( ii ) The running supremum process { U γ ( t ) : = sup ≤ s ≤ t U γ ( s ); t ≥ } can be rewritten as U γ ( t ) = x + Z t (1 − γ ( X ( s )))d X ( s ) = ξ γ ( X ( t )) , and hence, U γ ( τ − ( γ )) = ξ γ ( X ( τ ξ γ )) and U γ ( τ − ( γ )) ∈ ( s , s + △ s ) ⇔ X ( τ ξ γ ) ∈ (cid:18)(cid:16) ξ γ (cid:17) − ( s ) , (cid:16) ξ γ (cid:17) − ( s + △ s ) (cid:19) , s ≥ x , △ s > , with (cid:16) ξ γ (cid:17) − being the well-defined inverse function of ξ γ .( iii ) The taxed surplus level at and immediately before the ruin time τ − ( γ ) are rewritten as U γ ( τ − ( γ )) = X ( τ ξ γ ) − ξ γ ( X ( τ ξ γ )) , U γ ( τ − ( γ ) − ) = X ( τ ξ γ − ) − ξ γ ( X ( τ ξ γ )) , and hence, we have, for z ≥ △ z > − U γ ( τ − ( γ )) ∈ ( z , z + △ z ) ⇔ − X ( τ ξ γ ) ∈ (cid:16) − ξ γ ( X ( τ ξ γ )) + z , − ξ γ ( X ( τ ξ γ )) + z + △ z (cid:17) , and for y ≥ △ y > U γ ( τ − ( γ ) − ) ∈ ( y , y + △ y ) ⇔ X ( τ ξ γ − ) ∈ (cid:16) ξ γ ( X ( τ ξ γ )) + y , ξ γ ( X ( τ ξ γ )) + y + △ y (cid:17) . The above three observations combined with Theorem 3.1 yields, for s ≥ x > y , z > △ s , △ y , △ z ∈ (0 , ∞ ) E x (cid:16) e − q ℓ − λ ( τ − ( γ ) − ℓ ); U γ ( τ − ( γ )) ∈ ( s , s + △ s ) , U γ ( τ − ( γ ) − ) ∈ ( y , y + △ y ) , − U γ ( τ − ( γ )) ∈ ( z , z + △ z ) (cid:17) = Z s ∈ (cid:18) ( ξ γ ) − ( s ) , ( ξ γ ) − ( s + △ s ) (cid:19) Z y ∈ ( ξ γ ( s ) + y , ξ γ ( s ) + y + △ y ) Z z ∈ ( − ξ γ ( s ) + z , − ξ γ ( s ) + z + △ z ) × E x (cid:16) e − q ℓ − λ ( τ ξγ − ℓ ); X ( τ ξ γ ) ∈ d s , X ( τ ξ γ − ) ∈ d y , − X ( τ ξ γ ) ∈ d z (cid:17) , APPLICATIONS 17which combined with (10) (with ϑ ≡
0) yields E x (cid:16) e − q ℓ − λ ( τ − ( γ ) − ℓ ); U γ ( τ − ( γ )) ∈ d s , U γ ( τ − ( γ ) − ) ∈ d y , − U γ ( τ − ( γ )) ∈ d z (cid:17) = − γ (cid:18)(cid:16) ξ γ (cid:17) − ( s ) (cid:19) exp − Z ( ξ γ ) − ( s ) x W ′ q ( ξ γ ( w )) W q ( ξ γ ( w )) d w W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( s ) W λ ( s ) W λ ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s , which recovers the first equation in Theorem 1.3 of Kyprianou and Zhou (2009). Similarly, for s ≥ x > E x (cid:16) e − q ℓ − λ ( τ − ( γ ) − ℓ ); U γ ( τ − ( γ )) ∈ ( s , s + △ s ) , U γ ( τ − ( γ )) = (cid:17) = Z s ∈ (cid:18) ( ξ γ ) − ( s ) , ( ξ γ ) − ( s + △ s ) (cid:19) Z z ∈{ ξ γ ( s ) } E x (cid:16) e − q ℓ − λ ( τ ξγ − ℓ ); X ( τ ξ γ ) ∈ d s , X ( τ ξ γ ) ∈ d z (cid:17) , which together with (11) yields E x (cid:16) e − q ℓ − λ ( τ − ( γ ) − ℓ ); U γ ( τ − ( γ )) ∈ ( s , s + △ s ) , U γ ( τ − ( γ )) = (cid:17) = − γ (cid:18)(cid:16) ξ γ (cid:17) − ( s ) (cid:19) exp − Z ( ξ γ ) − ( s ) x W ′ q ( ξ γ ( w )) W q ( ξ γ ( w )) d w σ (cid:16) W ′ λ ( s ) (cid:17) W λ ( s ) − W ′′ λ ( s ) d s , which recovers the second equation in Theorem 1.3 of Kyprianou and Zhou (2009). Consider the following L´evy risk process with a barrier dividend strategy R b ( t ) : = X ( t ) − (cid:16) X ( t ) − b (cid:17) ∨ , (19)where b ∈ ( x , ∞ ) is the dividend barrier level. The risk process (19) is well-known as the “DeFinetti’s dividend model”. For a variety of dividend risk models driven by compound Poissonprocesses or Brownian motions, the Gerber-Shiu function has been studied by many authors (seefor example, Lin et al. (2003)), typically involves an “infinitesimal time interval argument”orthe approach of conditioning on the time and amount of the first claim, which di ff ers from ourexcursion argument. However, to the best knowledge of the authors, the Gerber-Shiu function inthe context of general L´evy risk processes with a barrier dividend strategy, has not yet been studied.Here in this subsection, based on the surplus process (19), we attempt to express the correspondingGerber-Shiu function in terms of the scale functions and the L´evy measure associated with X .To fix (19) into our drawdown setup, let ξ b ( z ) : = ( z − b ) ∨ , z ∈ [ x , ∞ ) , which is indeed a drawdown function. One can make the following three observations. APPLICATIONS 18( i ′ ) The ξ b -drawdown time of X coincides with the ruin time of the risk process (19) τ ξ b = inf { t ≥ X ( t ) < ξ b ( X ( t )) } = inf { t ≥ R b ( t ) < } : = τ − ( b ) . ( ii ′ ) The running supremum process { R b ( t ) : = sup ≤ s ≤ t R b ( s ); t ≥ } can be rewritten as R b ( t ) = X ( t ) ∧ b = ξ b ( X ( t )) , and hence, for small △ s > R b ( τ − ( b )) ∈ [ s , s + △ s ) ⇔ X ( τ ξ b ) ∈ [ s , s + △ s ) , x ≤ s < b , X ( τ ξ b ) ∈ [ b , ∞ ) , x ≤ s = b , ∅ , otherwise . ( iii ′ ) The surplus level (with dividend deducted) at and immediately before τ − ( b ) are R b ( τ − ( b )) = X ( τ ξ b ) − ξ b ( X ( τ ξ b )) , R b ( τ − ( b ) − ) = X ( τ ξ b − ) − ξ b ( X ( τ ξ b )) , and hence, for z ≥ △ z > − R b ( τ − ( b )) ∈ [ z , z + △ z ) ⇔ − X ( τ ξ b ) ∈ h − ξ b ( X ( τ ξ b )) + z , − ξ b ( X ( τ ξ b )) + z + △ z (cid:17) , and for y ∈ [0 , b ] and small △ y > R b ( τ − ( b ) − ) ∈ [ y , y + △ y ) ⇔ X ( τ ξ b − ) ∈ h ξ b ( X ( τ ξ b )) + y , ξ b ( X ( τ ξ b )) + y + △ y (cid:17) . The above observations and Theorem 3.1 yield, for s ∈ [ x , b ), y ∈ [ x , b ), z > △ s , △ y , △ z ∈ (0 , ∞ ), E x (cid:16) e − q ℓ − λ ( τ − ( b ) − ℓ ); R b ( τ − ( b )) ∈ [ s , s + △ s ) , R b ( τ − ( b ) − ) ∈ [ y , y + △ y ) , − R b ( τ − ( b )) ∈ [ z , z + △ z ) (cid:17) = Z s ∈ [ s , s + △ s ) Z y ∈ [ ξ b ( s ) + y , ξ b ( s ) + y + △ y ) Z z ∈ [ − ξ b ( s ) + z , − ξ b ( s ) + z + △ z ) × E x (cid:16) e − q ℓ − λ ( τ ξ b − ℓ ); X ( τ ξ b ) ∈ d s , X ( τ ξ b − ) ∈ d y , − X ( τ ξ b ) ∈ d z (cid:17) , which together with (10) (with ϑ ≡
0) and the fact that ξ b ( s ) = s for s ∈ [ x , b ], yield E x (cid:16) e − q ℓ − λ ( τ − ( b ) − ℓ ); R b ( τ − ( b )) ∈ d s , R b ( τ − ( b ) − ) ∈ d y , − R b ( τ − ( b )) ∈ d z (cid:17) = W q ( x ) W q ( s ) W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( s ) W λ ( s ) W λ ( s − y ) ! υ ( y + d z ) { y < s } d y ! d s , s , y ∈ [ x , b ) , z > . NUMERICAL EXAMPLES 19For y ∈ [ x , b ], z > △ s , △ y , △ z ∈ (0 , ∞ ) we have E x (cid:16) e − q ℓ − λ ( τ − ( b ) − ℓ ); R b ( τ − ( b )) = b , R b ( τ − ( b ) − ) ∈ [ y , y + △ y ) , − R b ( τ − ( b )) ∈ [ z , z + △ z ) (cid:17) = Z s ∈ [ b , ∞ ) Z y ∈ [ ξ b ( s ) + y , ξ b ( s ) + y + △ y ) Z z ∈ [ − ξ b ( s ) + z , − ξ b ( s ) + z + △ z ) × E x (cid:16) e − q ℓ − λ ( τ ξ b − ℓ ); X ( τ ξ b ) ∈ d s , X ( τ ξ b − ) ∈ d y , − X ( τ ξ b ) ∈ d z (cid:17) , which together with (10) (with ϑ ≡
0) yields E x (cid:16) e − q ℓ − λ ( τ − ( b ) − ℓ ); R b ( τ − ( b )) = b , R b ( τ − ( b ) − ) ∈ d y , − R b ( τ − ( b )) ∈ d z (cid:17) = Z s ∈ [ b , ∞ ) exp − Z sx W ′ q ( ξ b ( w )) W q ( ξ b ( w )) d w W λ (0 + ) υ ( ξ b ( s ) + d z ) δ ξ b ( s ) (d y ) + W ′ λ ( ξ b ( s ) − y ) − W ′ λ ( ξ b ( s )) W λ ( ξ b ( s )) W λ ( ξ b ( s ) − y ) υ ( y + d z ) { y <ξ b ( s ) } d y d s , y ∈ [ x , b ] , z > . Similarly, we have E x (cid:16) e − q ℓ − λ ( τ − ( b ) − ℓ ); R b ( τ − ( b )) ∈ d s , R b ( τ − ( b )) = (cid:17) = σ W q ( x ) W q ( s ) (cid:16) W ′ λ ( s ) (cid:17) W λ ( s ) − W ′′ λ ( s ) [ x , b ) ( s ) d s + δ b (d s ) Z ∞ b exp − Z zx W ′ q ( ξ b ( w )) W q ( ξ b ( w )) d w (cid:16) W ′ λ (cid:16) ξ b ( z ) (cid:17)(cid:17) W λ (cid:16) ξ b ( z ) (cid:17) − W ′′ λ (cid:16) ξ b ( z ) (cid:17) d z .
5. Numerical examples
The results in Section 3 are illustrated with several examples in this section. One quantity ofinterest is the probability of drawdown, which includes the probability of ruin as a special case.The other item is the joint density of the drawdown time and the first time when the runningmaximum prior to the drawdown time is hit.
The numerical results in this subsection is based on equation (16) where we let the penalty func-tion be ω ( x , y ) ≡ q ≡
0. Then the expected penalty function at generaldrawdown is specialized to φ ( x ) = E x (cid:16) { τ ξ < ∞} (cid:17) = P x ( τ ξ < ∞ ), which refers to the probability ofgeneral drawdown. For simplicity, we consider the general drawdown function in a linear form ξ ( x ) = ax − b . According to the definition of general drawdown function in Section 2, we require a ∈ ( −∞ ,
1) and b >
0. Then X ( t ) < ξ ( X ( t )) = aX ( t ) − b is equivalent to aX ( t ) − X ( t ) > b , whichrefers to the surplus process drops b units below 100 a percent of its maximum to date. In thissection, we compare the evolution of the probability of drawdown according to the following foursets of parameters. NUMERICAL EXAMPLES 20It is obvious from the table that (I) refers to the ruin case. Starting from the value X (0) = x , say x =
1, suppose that at some time point t > X ( t ) <
0, then 0 . X ( t ) − X ( t ) ≥ . − X ( t ) > . > .
5. That is, the drawdown time for case (IV) must occur before the ruin time, which isdenoted by τ IV < τ I . Similarly, we have τ IV < τ III < τ II and τ IV < τ III < τ I . Accordingly, thesooner the general drawdown time occurs in theory, the higher occurring probability correspondingto it. We point out that the relation between case (I) and case (II) is not clear, which depends onthe underlying risk process and the parameter setting. We can see the di ff erence in the followingexamples. Example 5.1.
Cram´er-Lundberg model with exponential jumps.
Suppose that the L´evy processis given by the Cram´er-Lundberg model with exponential jumps. To be specific, when X ( t ) isreduced to a compound Poisson process with Poisson arrival rate λ >
0, premium rate c , andclaim sizes following an exponential distribution with mean 1 /µ > X ( t ) = x + ct − N ( t ) X i = Y i , t ≥ , then the expected discounted penalty function at the general drawdown time, that is the probabilityof general drawdown is given by (16) with υ (d z ) = λ F (d z ) = λ µ e − µ z d z and σ =
0. The explicitexpression for the scale function is available as, W q ( x ) = A ( q ) c e θ ( q ) x − A ( q ) c e θ ( q ) x , x ≥ , with A ( q ) = µ + θ ( q ) θ ( q ) − θ ( q ) and A ( q ) = µ + θ ( q ) θ ( q ) − θ ( q ) , where θ ( q ) = λ + q − c µ + √ ( c µ − λ − q ) + cq µ c and θ ( q ) = λ + q − c µ − √ ( c µ − λ − q ) + cq µ c . Due to σ =
0, we only need to compute the first two lines of integralsin (16), which represent the risk measurement brought by exponentially distributed claims. Weare interested in the impact of the initial capital x and the premium rate c on the probability ofdrawdown.We list the parameter values as follows: x = c = . λ = µ =
2. In the classical risktheory, the higher x or c , the lower probability of ruin. This can be verified by Figure 1. We canalso observe a similar trend for the probability of drawdown, which is decreasing along with x or c . Figure 1 also verify our prior analysis on the relations between the results of di ff erent drawdownfunctions. That is, case (IV) has the highest value, followed by case (III), then case (II) and (I).Unlike the unclear theoretical comparison between case (I) and case (II), we observe that case (II)stays on top of case (I) in Figure 1. This may due to the exponentially distributed claim size natureof the underlying risk process, which results in a similar relation for the jump-di ff usion model inFigure 3.Another observation from Figure 1 is, the probability of drawdown is more sensitive to thechange of premium rate c than the initial capital x . Similar slops are produced from part ( a ) as x increases from unit 1 to unit 10, and from part (b) as c increases from 1.1 to 2.1. This gives therisk manager a hint that, the adjustment on the premium rate has a more immediate e ff ect on therisk level than an injection of capital. NUMERICAL EXAMPLES 21 Table 1: The parameters for the linear drawdown function. a b (I) 0 0(II) 0.3 0.5(III) 0.5 0.5(IV) 0.6 0.5 x D r a w do w n P r obab ili t y (a) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 c D r a w do w n P r obab ili t y (b) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 Figure 1: (a) The drawdown probability as a function of x ∈ [1 ,
10] for c = . c ∈ [1 . , . x = NUMERICAL EXAMPLES 22
Example 5.2.
Brownian motion with drift.
Brownian motion (with or without drift) is the onlycontinuous L´evy process. When X is reduced to a Brownian motion with drift X t = x + µ t + σ B t , t ≥ , µ , , σ > . Then the explicit expression for the scale function is written as W q ( x ) = a ( e λ x − e λ x ) , x ≥ , (20)where a = (2 q σ + µ ) − , λ = (2 q σ + µ ) − µσ and λ = − (2 q σ + µ ) − µσ . Under this continuous riskprocess, the L´evy measure υ (d z ) =
0. Then the expected discounted penalty function at generaldrawdown time is given by the third line only in equation (16). x D r a w do w n P r obab ili t y (a) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 D r a w do w n P r obab ili t y (b) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 Figure 2: (a) The drawdown probability as a function of x ∈ [1 ,
10] for µ = . µ ∈ [1 . , . x = Taking σ = µ increas-ing from 1.1 to 2.1 as plotted in Figure 2. Accordingly, we expect a low probability of ruin aswell as the probability of general drawdown. This can be seen in Figure 2 that the levels are muchlower than those in Figure 1. Comparing to Figure 1, the overall trends in Figure 2 are similar.The only di ff erence is the blur relation between case (I) and case (II), which is also blur in theoryand may comes from the small fluctuations described by the Brownian motion. NUMERICAL EXAMPLES 23 Example 5.3.
Jump-di ff usion process. When X is reduced to a jump-di ff usion process, X ( t ) = x + ct + σ W t − N ( t ) X i = Y i , t ≥ , where σ> { N ( t ) , t ≥ } is a Poisson process with arrival rate λ , and Y i ’s are a sequence ofi.i.d. random variables distributed with Erlang (2 , α ) . The scale function associated with X can bederived as (cf., Loe ff en (2008)) W q ( x ) = X j = D j ( q ) e θ j ( q ) x , x ≥ , (21)where D j ( q ) = (cid:16) α + θ j ( q ) (cid:17) σ Q i = , i , j (cid:16) θ j ( q ) − θ i ( q ) (cid:17) , and θ j ( q ) ( j = , · · ·
4) are the (distinct) zeros of the polynomial (cid:16) c θ − λ + λ α ( α + θ ) + σ θ − q (cid:17) ( α + θ ) = σ θ + ( ασ + c ) θ + (cid:0) σ α − λ − q + c α (cid:1) θ + (cid:2) c α − λ + q ) α (cid:3) θ − q α . Then the expected discounted penalty function at the general drawdown is given by equation (16)with υ (d z ) = λ α z e − α z d z and W q given by (21).The parameters in this example are: σ = . λ = α =
2. Comparing to the aforementionedtwo examples, we have neither ν ( dz ) = σ = ff usion process, which resultsin the involvement of all the three lines of equation (16) in our computation. Then it is natural toexpect a much higher probability of drawdown than the previous two examples. Intuitively speak-ing, the drawdown of the jump-di ff usion process is composed of two parts: the small fluctuationsdescribed by the Brownian motion W t and the large jumps described by the compound Poissonprocess P N ( t ) i = Y i . This has been verified in our computation that we have to choose a much higherpremium rate c than the previous examples, otherwise the ruin probability would be 1. In Figure3, we let the initial capital increase from 1 to 10 in Part ( a ), and premium rate increase from 3 to7.5 in part (b), then we produce a similar trend as in Figure 1. τ ξ and ℓ . The first time when the running maximum prior to τ ξ is hit is denoted by ℓ in this paper. Thissubsection intends to study the joint distribution of the drawdown time τ ξ and ℓ , as well as thee ff ects of the model parameters. By Theorem 3.1, the joint Laplace transform of ℓ and τ ξ is E x (cid:16) e − q ℓ − λ ( τ ξ − ℓ ) { τ ξ < ∞} (cid:17) NUMERICAL EXAMPLES 24 x D r a w do w n P r obab ili t y (a) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 c D r a w do w n P r obab ili t y (b) a=b=0a=0.3,b=0.5a=0.5,b=0.5a=0.6,b=0.5 Figure 3: (a) The drawdown probability as a function of x ∈ [1 ,
10] for c = c ∈ [3 , .
5] for x = ff usion process. NUMERICAL EXAMPLES 25 = Z s ∈ ( x , ∞ ) Z y ∈ [ ξ ( s ) , s ] Z z ∈ ( − ξ ( s ) , ∞ ) exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w W λ (0 + ) υ ( s + d z ) δ s (d y ) + W ′ λ ( s − y ) − W ′ λ ( ξ ( s )) W λ ( ξ ( s )) W λ ( s − y ) υ ( y + d z ) { y < s } d y d s + σ Z s ∈ ( x , ∞ ) exp − Z sx W ′ q ( ξ ( w )) W q ( ξ ( w )) d w (cid:16) W ′ λ (cid:16) ξ ( s ) (cid:17)(cid:17) W λ (cid:16) ξ ( s ) (cid:17) − W ′′ λ (cid:16) ξ ( s ) (cid:17) d s . (22)For computational simplicity, our numerical results are based on Brownian motion with drift
Example 5.2 . The main parameters are x = µ = . σ =
1. In terms of τ ξ we stilluse the linear drawdown function defined in the previous subsection. And, we only consider twocases for the parameters: the ruin case with a = b = a = . b = .
5. In the following, we take equation (22) as the joint Laplace transform of τ ξ and ℓ , andthen take the inverse transformation to derive the joint density distribution. Our algorithm is basedon the method of Fourier series expansion proposed by Moorthy (1995), which is one of the mostworthwhile methods in the numerical inversion of Laplace transforms. There are also many otherbasic methods available in the literature, say for example, the Laguerre function expansion andCombination of Gaver functions. These basic methods breed over 100 algorithms on the subject.As we have explained in Section 1, it is hard to find a universal algorithm that works for all thecases and we are not intending to develop a perfect algorithm in this paper. Therefore, we justfollow the Fourier series expansion method to write our codes in Matlab, and accept the instabilitybehavior at the boundaries of the produced results.In Figure 4, f ( t , t ) refers to the joint density function of random variables τ ξ (correspondsto t ) and ℓ (corresponds to t ). We make the following observations. Firstly, the overall trendof f ( t , t ) goes to zero as t or t gets bigger and bigger indicating that the drawdown time orthe running maximum hitting time occurring at a later time has a smaller probability, which isconsistent with the path behavior of a Brownian motion with positive drift . In fact, due to µ > t →∞ X ( t ) = ∞ , which means that, if drawdown occurs, it occurs at a finite time, and less andless likely to occur at a later time until infinity. Secondly, since ℓ refers to a time point that is priorto τ ξ , then the value of the probability density f ( t , t ) = t < t . This can be seen clearlyin Figure 4 that all the positively valued f ( t , t ) are distributed on the side of t axis. Thirdly,the drawdown time is expected to occur before the ruin time, then the value of f ( t , t ) is moreconcentrated at smaller values of t and t in the drawdown case than in the ruin case. We can seeclearly that in Figure 4 that the drawdown case in (b) builds up a higher value than the ruin case in(a).Next we look at the e ff ects of parameters x , µ and σ on the value of f ( t , t ). The basic parame-ters are x = µ = . σ = a = . b = .
5, which lead to (a) of Figure 5. When we changethe initial capital x from 1 to 2 in (b), the drawdown time is expected to occur later intuitively,and correspondingly we observe a lower but fatter joint density function in (b). Similarly, a biggervalue of µ helps to build up the surplus value which in turn leads to a later drawdown time. Wealso observe a lower but fatter distribution in (c) comparing to (a). The e ff ect of σ goes to the otherdirection, the resulted joint density in (d) is higher and sharper comparing to (a), which explains CONCLUSION 26the higher uncertainty brought by a larger value of σ .
6. Conclusion
In this paper, the generalized Gerber-Shiu function at general drawdown time is considered fora spectrally negative L´evy process. It is shown that in the present model, the extended Gerber-Shiu function can be expressed in terms of the q -scale functions and the L´evy measure associatedwith the L´evy process. This expression makes it possible to study the joint distribution of thetime of drawdown, the running maximum at drawdown, the last minimum before drawdown, thesurplus before drawdown and the surplus at drawdown, which broaden the family of risk indicatorsand measurements. The motivation of such an extension from the time of ruin to the time ofdrawdown is two folds. First, thanks to the development of the excursion approach in solvingboundary crossing problems related with L´evy processes, such that the derivation of the extendedGerber-Shiu function is possible. Second, the time of drawdown has a clearer description ofthe company’s financial position than the time of ruin. Then the insurer can take actions morepromptly and e ff ectively, such as adjusting the premium rate or injecting more capital to preventeven worse situations. Acknowledgements
The authors are very grateful to Professor Xiaowen Zhou at Concordia University for his helpfulcomments on this paper.
References [1] Albano, G. and Giorno, V., 2006. A stochastic model in tumor growth.
Journal of Theoretical Biology , ,329-336.[2] Angoshtari, B., Bayraktar, E. and Young, V., 2016. Minimizing the probability of lifetime drawdown underconstant consumption. Insurance: Mathematics and Economics , , 210-223.[3] Avram, F., Kyprianou, A. and Pistorius, M., 2004. Exit problems for spectrally negative L´evy processes andapplications to (Canadized) Russian options. The Annals of Appllied Probability , , 215-238.[4] Avram, F., Vu, N. L. and Zhou, X., 2017. On taxed spectrally negative L´evy processes with drawdown stopping. Insurance: Mathematics and Economics , , 69-74.[5] Baurdoux, E., 2007. Fluctuation Theory and Stochastic Games for Spectrally Negative L´evy Processes. DoctoralThesis, de Universiteit Utrecht.[6] Bertoin, J., 1996. L´evy Processes. Cambridge University Press .[7] Bi ffi s E. and Kyprianou A., 2010. A note on scale functions and the time value of ruin for L´evy insurance riskprocesses. Insurance: Mathematics and Economics , , 85-91.[8] Bi ffi s, E. and Morales, M., 2010. On a generalization of the Gerber-Shiu function to path-dependent penalties. Insurance: Mathematics and Economics , , 92-97.[9] Carr, P., 2014. First-order calculus and option pricing. Journal of Financial Engineering , , 1450009.[10] Chen, X., Landriault, D., Li, B. and Li, D., 2015. On minimizing drawdown risks of lifetime investments. Insurance: Mathematics and Economics , , 46-54.[11] Cherny, V. and Obloj, J., 2013. Portfolio optimisation under non-linear drawdown constraints in a semimartin-gale financial model. Finance and Stochastics , , 771-800. EFERENCES 27 [12] Cvitanic, J. and Karatzas, I., 1995. On portfolio optimization under “drawdown”constraints.
IMA Volumes inMathematics and its Applications , , 35-46.[13] Davies, B., 2002. Integral Transforms and Their Applications. Springer, New York .[14] Dufresne, F. and Gerber, H., 1993. The probability of ruin for the inverse Gaussian and related processes.
Insurance: Mathematics and Economics , , 9-22.[15] Elie, R. and Touzi, N., 2008. Optimal lifetime consumption and investment under a drawdown constraint. Fi-nance and Stochastics , , 299-330.[16] Garrido, J. and Morales, M., 2006. On the expected discounted penalty function for L´evy risk processes. NorthAmerican Actuarial Journal ,
10 (4) , 196-216.[17] Gerber H. and Shiu, E., 1998. On the time value of ruin.
North American Actuarial Journal , , 48-72.[18] Grossman, S. and Zhou, Z., 1993. Optimal investment strategies for controlling drawdowns. Mathematical Fi-nance , , 241-276.[19] Han, X., Liang, Z. and Yuen, K. C., 2018. Optimal proportional reinsurance to minimize the probability ofdrawdown under thinning-dependence structure. Scandinavian Actuarial Journal , , 863-889.[20] Kyprianou, A., 2013. Gerber-Shiu risk theory (EAA Series), Springer-Verlag .[21] Kyprianou, A. and Pistorius, M., 2003. Perpetual options and Canadization through fluctuation theory.
TheAnnals of Appllied Probability , , 1077-1098.[22] Kyprianou, A., 2006. Introductory Lectures on Fluctuations of L´evy Processes with Applications, Springer .[23] Kyprianou, A. and Zhou, X., 2009. General tax structures and the L´evy insurance risk model.
Journal of AppliedProbability , , 1146-1156.[24] Kuznetsov, A., Kyprianou, A. and Rivero, V., 2012. The theory of scale functions for spectrally negative L´evyprocesses. In L´evy Matters II (Lecture Notes Math. 2061), Springer , Heidelberg, 97-186.[25] Landriault, D., Li, B. and Zhang, H., 2015. On the frequency of drawdowns for brownian motion processes.
Journal of Applied Probability , , 191-208.[26] Landriault, D., Li, B. and Zhang, H., 2017. On magnitude, asymptotics and duration of drawdowns for L´evymodels. Bernoulli , , 432-458.[27] Loe ff en, R., 2008. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negativeL´evy processes. Annals of Applied Probability , , 1669-1680.[28] Loe ff en, R., Palmowski, Z. and Surya, B., 2018. Discounted penalty function at Parisian ruin for L´evy insurancerisk process. Insurance: Mathematics and Economics , , 190-197.[29] Li, B., Vu, N. and Zhou, X. 2017. Exit problems for general drawdown times of spectrally negative L´evyprocesses. arXiv 1702.07259 .[30] Li, S., 2015. Adaptive policies and drawdown problems in insurance risk models. Doctor thesis .[31] Lin, X., Willmot, G. and Drekic, S., 2003. The classical risk model with a constant dividend barrier: analysis ofthe Gerber-Shiu discounted penalty function.
Insurance: Mathematics and Economics , , 551-566.[32] Moorthy, M., 1995. Numerical inversion of two-dimensional Laplace transforms-Fourier series representation. Applied Numerical Mathematics , , 119-127.[33] Pistorius, M., 2004. On exit and ergodicity of the spectrally one-sided L´evy process reflected at its infimum. Journal of Theoretical Probability , , 183-220.[34] Pistorius, M., 2007. An excursion-theoretical approach to some boundary crossing problems and the skorokhodembedding for refrected L´evy processes. In S´eminaire de Probabilit´es
XL, 287-307.
Springer .[35] Roche, H., 2006. Optimal consumption and investment strategies under wealth ratcheting. Preprint. Availableat: http: // ciep.itam.mx / ∼ hroche / Research / MDCRESFinal.pdf.[36] Shepp, L. and Shiryaev, A., 1993. The Russian option: reduced regret.
The Annals of Applied Probability , ,631-640.[37] Taylor, H., 1975. A stopped Brownian motion formula. The Annals of Applied Probability , , 234-246.[38] Wang, W. and Zhou, X., 2018. General drawdown based de Finetti optimization for spectrally negative L´evyrisk processes. Journal of Applied Probability , , 513-542.[39] Yang, H. and Zhang, L., 2001. Spectrally negative L´evy processes with applications in risk theory. Advances inApplied Probability , , 281-291. EFERENCES 28 J o i n t den s i t y f ( t ,t ) (a) Ruin case with a=b=0 t t J o i n t den s i t y f ( t ,t ) (b) Drawdown case with a=0.6, b=0.5 t t Figure 4: The joint density f ( t , t ) for τ ξ and ℓℓ
The Annals of Applied Probability , ,631-640.[37] Taylor, H., 1975. A stopped Brownian motion formula. The Annals of Applied Probability , , 234-246.[38] Wang, W. and Zhou, X., 2018. General drawdown based de Finetti optimization for spectrally negative L´evyrisk processes. Journal of Applied Probability , , 513-542.[39] Yang, H. and Zhang, L., 2001. Spectrally negative L´evy processes with applications in risk theory. Advances inApplied Probability , , 281-291. EFERENCES 28 J o i n t den s i t y f ( t ,t ) (a) Ruin case with a=b=0 t t J o i n t den s i t y f ( t ,t ) (b) Drawdown case with a=0.6, b=0.5 t t Figure 4: The joint density f ( t , t ) for τ ξ and ℓℓ . EFERENCES 29
040 402 204 200 0 040 402 204 200 0040 402 204 200 0 040 402 204 200 0
Figure 5: The joint density f ( t , t ) for τ ξ and ℓℓ