Risk Modelling on Liquidations with Lévy Processes
aa r X i v : . [ q -f i n . R M ] J u l Risk Modelling on Liquidations with L´evy Processes
Aili Zhang ∗ , Ping Chen † , Shuanming Li ‡ and Wenyuan Wang § July 6, 2020
Abstract
It has been decades since the academic world of ruin theory defined the insolvency of an insurancecompany as the time when its surplus falls below zero. This simplification, however, needs carefuladaptions to imitate the real-world liquidation process. Inspired by Broadie et al. (2007) and Li et al.(2020), this paper uses a three-barrier model to describe the financial stress towards bankruptcy of aninsurance company. The financial status of the insurer is divided into solvent, insolvent and liquidatedthree states, where the insurer’s surplus process at the state of solvent and insolvent is modelled bytwo spectrally negative L´evy processes, which have been taken as good candidates to model insurancerisks. We provide a rigorous definition of the time of liquidation ruin in this three-barrier model. Byadopting the techniques of excursions in the fluctuation theory, we study the joint distribution of thetime of liquidation, the surplus at liquidation and the historical high of the surplus until liquidation,which generalizes the known results on the classical expected discounted penalty function in Gerberand Shiu (1998). The results have semi-explicit expressions in terms of the scale functions and the L´evytriplets associated with the two underlying L´evy processes. The special case when the two underlyingL´evy processes coincide with each other is also studied, where our results are expressed compactly viaonly the scale functions. The corresponding results have good consistency with the existing literatureson Parisian ruin with (or without) a lower barrier in Landriault et al. (2014), Baurdoux et al. (2016)and Frostig and Keren-Pinhasik (2019). Besides, numerical examples are provided to illustrate theunderlying features of liquidation ruin.
Keywords:
Spectrally negative L´evy process, Liquidation time, Expected discounted penalty func-tion, Discounted joint probability density, Liquidation probability.2000 Mathematical Subject Classification: 60G51, 91B30, 60G40
In the classical risk theory, the ultimate ruin occurs if an insurance company has no sufficient assets tomeet its liabilities. That is, when the surplus ever falls to zero or below, the insurance company is defined ∗ School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China. E-mail: [email protected]. † Department of Economics, The University of Melbourne, Parkville, Victoria 3010, Australia. E-mail:[email protected] ‡ Department of Economics, The University of Melbourne, Parkville, Victoria 3010, Australia. E-mail:[email protected] § Corresponding author. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, China. E-mail:[email protected]
1s insolvent. In this somewhat vague statement, the state of solvency is described by a single barrierzero of the surplus process. Under this definition, ruin theory has been an area of study for actuariesand mathematicians for many decades. However, the practical issue of solvency is far more complicatedthan this approximation.In the United States, bankruptcy is governed by the “US Bankruptcy Code”. To resolve the financialdistress of a business whose liability is greater than its assets, two options can be taken when filing forbankruptcy: Chapter 7 and Chapter 11. A business can seek Chapter 7 bankruptcy only if it immediatelyceases all operations and hence Chapter 7 is usually referred to as “liquidation”, this type of bankruptcyis a popular option for small business owners. Chapter 11 bankruptcy, on the other hand, is oftenreferred to as “reorganization” or “rehabilitation” bankruptcy because the process provides the businesswith an opportunity to reorganize its financial structure such as debts and to try to re-emerge as ahealthy organization while continuing to operate. Chapter 11, which is more expensive than Chapter 7,is typically intended for medium to large-sized businesses. According to Corbae and D’Erasmo (2017),80% of bankruptcies of publicly traded companies are filed under Chapter 11, while only 20% are Chapter7 liquidations.From academic point of view, it is natural and straightforward to describe Chapter 7 bankruptcyby a single barrier model of the surplus process, see for example Leland (1994), where bankruptcy istriggered when the firm’s asset value falls to the debt’s principal value. While Chapter 11 is a long andcostly process, the grant of a chance to reorganize debts has been an attractive feature to many largecorporations. Therefore, more and more researchers in finance try to imitate the process of Chapter11 bankruptcy in their models. Broadie et al. (2007) adopted a three-barrier model to characterizethe financial states of a bank: distress, Chapter 11 bankruptcy and Chapter 7 liquidation. Antill andGrenadier (2019) formulated a model in which shareholders can choose both their timing of default andthe chapter of bankruptcy. They also examined how this flexibility alters the capital structure decisionsof the firm.Given the significance of Chapter 11 filing, the regulators of insurance companies also keep up withthe trend aiming to provide crucial safeguards for policyholders and for the economy. A review on thechanges in the U.S. insurance regulatory system can be found in Li et al. (2020). We only give a briefintroduction on the current regulatory system in the U.S. and E.U. for self-completeness.In the United States, insurance companies are not subject to the federal bankruptcy code, and theirfinancial health is monitored by the National Association of Insurance Commissioners (NAIC) and varyby state. The primary goal is to make sure there is sufficient capital for insurers to operate and meettheir obligations to policyholders and other claimants. The U.S. method of measuring whether capital isadequate is called Risk Based Capital (RBC). Based on the amount of insurance the company writes, thelines of business it writes, the assets it invests in and other measures, an absolute least amount of capitalan insurer needs is calculated. If an insurer’s capital dwindles, the regulators have the opportunityto intercede. The closer the actual capital gets to the risk-based minimum, the more powerful theintervention can be.Analogous to the regulatory spirit in the U.S., the current supervisory regime for insurance companiesin the European Union is also a risk-based capital regime. Since January 2016, the E.U. insurers aregoverned by the Solvency II regulatory regime. A three-level capital requirement system is applied:technical provision is to fulfil the obligations towards policyholders and other beneficiaries; minimum2apital requirement (MCR) is a safety net and reflects a level of capital below which ultimate supervisoryaction would be triggered; solvency capital requirement (SCR) enables an insurance company to absorbsignificant unforeseen losses and that gives reasonable assurance to policyholders and beneficiaries.To imitate this real-world process of bankruptcy, the classical definition of ruin describes Chapter7 liquidation reasonably well by using a single barrier, but fails to describe the complex Chapter 11reorganization. In view of the recent regulatory development described in the above, and following thepioneering work by Li et al. (2020) in the insurance sector, we use a three-barrier model to describe thefinancial stress towards bankruptcy of an insurance company. This paper distinguishes between insolvent and bankruptcy . A company can be insolvent without being bankrupt, but it cannot be bankrupt withoutbeing insolvent. Insolvency is the inability to pay debts when they are due, while bankruptcy is usuallya final alternative with the failures of all the other attempts to clear debt.To better reflect the meaning of the three barriers, this paper adopts a different naming system ofthe three barriers from Li et al. (2020). The highest barrier is called “safety”barrier (denoted by c ).If the surplus process stays above this barrier, this indicates the insurer has enough buffer to settle allclaims in extreme situations which means the insurer is a healthy financial institution. Otherwise, aninvestigation of the insurer’s business is carried out by the regulator on the insurer’s financial capacity tomeet both its short-term and long-term liabilities. If the surplus process keeps shrinking, the intermediarybarrier is called “reorganization”or “rehabilitation”barrier (denoted by b ), which triggers the regulator’sintervention in the insurer’s business operation. The regulator may give a broad range of directions tothe insurer with respect to the carrying on of its business, including prohibiting the insurer from issuingfurther policies, prohibiting it from disposing of assets, requiring it to make provisions in its accounts andrequiring it to increase its paid-up capital. The regulator’s rehabilitation intervention continues until thesurplus climbs up to the safety barrier, or transfers to the wind-up procedure when the lowest barrier isbreached. We call the lowest barrier “liquidation”barrier (denoted by a ). Once an insurance companyhas been liquidated, it is completely dissolved and permanently ceases operations. The three-barriersystem with a < b < c divides the state of an insurer into solvent , insolvent and liquidated ( bankrupt )three states. The detailed transition between states is presented in Section 2.Another feature of this paper is the insurers surplus process is modelled by spectrally negative L´evyprocesses, which are stochastic processes with stationary independent increments and with sample pathsof no positive jumps. L´evy processes have been taken as good candidates to model insurance risk, seeShimizu and Zhang (2019) for a review for these features of L´evy processes. The application of spectrallynegative L´evy processes in risk theory can be seen in Yang and Zhang (2001), Huzak et al. (2004), Chiuand Yin (2005), Garrido and Morales (2006), Biffis and Morales (2010), Cheung et al. (2010), Yin et al.(2014), Albrecher et al. (2016), Loeffen et al. (2018), Wang et al. (2020) and Wang and Zhou (2020).Based on the time of liquidation, this paper studies an extended definition of the expected discountedpenalty function expresses in terms of the q-scale functions and the L´evy triplet associated with the L´evyprocess. The joint distribution of the time of liquidation, the surplus at liquidation and the historicalhigh of the surplus until liquidation is derived.From technical point of view, our mathematical argument is based on a heuristic idea presented inLi et al. (2020) which consists of distinguishing the excursions away from the three barriers (i.e., a , b and c ) of the underlying diffusion surplus process. In the context of L´evy processes, we provide arigorous definition of the time of liquidation ruin. With the help of the fluctuation theory for spectrallynegative L´evy processes (particularly in Loeffen et al. (2014), Wang et al. (2020), etc.) and delicate path3nalysis arguments, this paper derives a semi-explicit and compact expression of the extended Gerber-Shiu function at liquidation ruin, in terms of the so-called scale functions and the L´evy triplet associatedwith the underlying L´evy process. From our results, we can easily deduce the Gerber-Shiu distributionand the two-sided exit identities at Parisian ruin which was originally obtained by Landriault et al.(2014), Baurdoux et al. (2016), and Frostig and Keren-Pinhasik (2019). In addition, compared with thefluctuation identities obtained in Landriault et al. (2014) and Frostig and Keren-Pinhasik (2019) wherethe spectrally negative L´evy process is assumed to have bounded path variation, this paper unifies theresults for bounded and unbounded variation.The remaining part of this paper is organized as follows. In Section 2 we review the basics of thespectrally negative L´evy processes, the associated scale functions, and the existing results of the exitproblems. Section 3 presents the semi-explicit expression of the extended Gerber-Shiu function at thetime of liquidation ruin. The application of our main results in the case of Parisian ruin is provided inSection 4. In Section 5, several numerical examples are studied to illustrate the features of our results. In this section we review some preliminaries and fundamental results for fluctuation problems of thespectrally negative L´evy processes. Let X = { X t ; t ≥ } be a spectrally negative L´evy process definedon a filtered probability space (Ω , {F t ; t ≥ } , P ) with the natural filtration {F t ; t ≥ } , that is, X is a stochastic process with stationary and independent increments and no positive jumps. To avoidtrivialities, we exclude the case where X has monotone paths. Denote by P x the conditional probabilitygiven X = x , and by E x the associated conditional expectation. For notational convenience, we write P and E in place of P and E , respectively. The Laplace transform of spectrally negative L´evy process X is given by E (cid:16) e θX t (cid:17) = e tψ ( θ ) , for all θ ≥
0, where ψ ( θ ) = γθ + 12 σ θ + Z ∞ (e θz − θz (0 , ( z )) υ (d z ) , for γ ∈ ( −∞ , ∞ ) and σ ≥
0, where υ is a σ -finite measure on (0 , ∞ ) such that Z ∞ (1 ∧ z ) υ (d z ) < ∞ . The measure υ is called the L´evy measure of X , and ( γ, σ, υ ) is called the L´evy triplet of X . Since theLaplace exponent ψ is strictly convex and lim θ →∞ ψ ( θ ) = ∞ , there exists the right inverse of ψ definedby Φ q = sup { θ ≥ ψ ( θ ) = q } . We present the definition of the scale functions W q and Z q of X . For q ≥ W q : [0 , ∞ ) → [0 , ∞ ) is defined as the unique strictly increasing and continuous function on [0 , ∞ )with Laplace transform Z ∞ e − θx W q ( x )d x = 1 ψ ( θ ) − q , θ > Φ q . For convenience, we extend the domain of W q ( x ) to the whole real line by setting W q ( x ) = 0 for x < W q is the second scale function Z q defined by Z q ( x ) = 1 + q Z x W q ( z )d z, x ∈ [0 , ∞ ) , Z q ( x ) ≡ −∞ , W = W and Z = Z . In addition, for p, q ≥ x > x + w ≥
0, we define ω ( q,p ) ( w, x ) := W p ( w + x ) − ( p − q ) Z x W p ( z + w ) W q ( x − z )d z = W q ( x + w ) + ( p − q ) Z w W q ( x + w − z ) W p ( z )d z, (2.1)and ℓ ( q,p ) ( w, x ) := Z p ( w + x ) − ( p − q ) Z x Z p ( z + w ) W q ( x − z )d z = Z q ( x + w ) + ( p − q ) Z w W q ( x + w − z ) Z p ( z )d z. (2.2)It is well known that lim x →∞ W ( q ) ′ ( x ) W q ( x ) = Φ q , lim y →∞ W q ( x + y ) W q ( y ) = e Φ q x . (2.3)The first passage times of level w ∈ ( −∞ , ∞ ) for the process X is defined as τ + w = inf { t ≥ X t > w } and τ − w = inf { t ≥ X t < w } , with the convention that inf ∅ = ∞ . For w > x ≤ w , the two-sided exit problem is given by E x (cid:16) e − qτ + w { τ + w <τ − } (cid:17) = W q ( x ) W q ( w ) , x ∈ ( −∞ , w ] , (2.4) E x (cid:16) e − qτ − { τ − <τ + w } (cid:17) = Z q ( x ) − Z q ( w ) W q ( x ) W q ( w ) , x ∈ ( −∞ , w ] . (2.5)We recall the resolvent measure as follows. For q ≥ , w >
0, and x, y ∈ [0 , w ], Z ∞ e − qt P x ( X t ∈ d y, t < τ + w ∧ τ − )d t = (cid:18) W q ( x ) W q ( w − y ) W q ( w ) − W q ( x − y ) (cid:19) d y. (2.6)It is seen from Corollary 3.2 and Remark 3.3 in Wang et al. (2020) that E x (cid:16) e − qτ − w ; X ( τ − w − ) ∈ d y, − X ( τ − w ) ∈ d θ, τ − w < τ + z (cid:17) = W q ( x − w ) υ ( y + d θ ) (cid:20) W q ( z − y ) W q ( z − w ) − W q ( x − y ) W q ( x − w ) (cid:21) { w
0. The state of liquidation is an absorbingstate, that is, once an insurer’s surplus process hit a , it is completely dissolved and permanently ceasesoperations. When the surplus stays above c , the insurer is a healthy financial institution and is able tomeet all its liabilities. Once the surplus process falls below c , the insurer is under regulator’s supervision,which does not lead to the intervention of the insurer’s daily operations but an investigation on theinsurer’s financial status will be conducted. As the situation is getting worse, once b is down crossed, theregulators intervene the daily operation and hence the surplus process will follow a different spectrallynegative L´evy process as given in (2.9) in below.The three-barrier system with a < b < c divides the state of an insurer into solvent , insolvent and liquidated ( bankrupt ) three states. Before the final liquidation, the insurer remains at the state of solvent or insolvent , where different spectrally negative L´evy processes are adopted, indicating that the regulatorscan intervene the everyday operations of the insurer and hence can alter the dynamics of the surplusprocess. To measure the maximum tolerable time period for the regulator to allow the insurer as stayingin the state of insolvent , a “grace period”is granted. Then the state of insolvent refers to a period of timein which the surplus starts from a value equals to or below b until the lowest barrier a is hit and transfersdirectly to the state of liquidated , or until the up-crossing time of barrier c within the grace period andtransfers to the state of solvent , or until the staying period between a and c exceeds the granted graceperiod and transfers to the state of liquidated . In other words, a surplus value below b indicates a poorfinancial position of the insurer, the state of which can only be transferred to solvent when the safetybarrier c can be achieved, within the tolerable grace period, otherwise, the insurer will be liquidated.In this paper, the surplus process U will be given byd U t = ( d X t , when U t is in solvent state , d e X t , when U t is in insolvent state , (2.9)with e X being another L´evy process with L´evy triplet ( e γ, e σ, e υ ) and Laplace exponent e ψ . Let the twoscale functions W q and Z q associated with e X be defined in the same manner as W q and Z q while with ψ replaced by e ψ , and write W = W and Z = Z . In addition, denote by e P x the conditional probabilitygiven e X = x , and by e E x the associated conditional expectation. And, with a little abuse of notations,under e P x , τ + w ( τ − w ) is taken as the first up-crossing (down-crossing ) time of w for the process e X .Define ζ − b, = ζ − b := inf { t ≥ U t < b } and ζ + c, = ζ + c := inf { t ≥ ζ − b, ; U t > c } , respectively to be the first time that the process U down-crosses the level b and the first time U up-crossesthe level c after ζ − b, . We recursively define two sequences of stopping times { ζ − b,k } k ≥ and { ζ + c,k } k ≥ asfollows. For k ≥
2, let ζ − b,k := inf { t ≥ ζ + c,k − ; U t < b } and ζ + c,k := inf { t ≥ ζ − b,k ; U t > c } . Then, we define κ := inf { k ≥ ζ − b,k + e kλ < ζ + c,k } and let ζ b,c,e λ := ( ζ − b,κ + e κλ when κ < ∞ , ∞ , when κ = ∞ , { e kλ } k ≥ is a sequence of independent and exponentially distributed random variables with param-eter λ . We use { e kλ } k ≥ to model the duration of grace period granted by the regulator. We also assumethat { e kλ } k ≥ is independent of X and e X . Remark 1
A period of grace is the maximal amount of time the company is allowed to stay in the stateof insolvent. It is offered to enable the company to better shape its financial condition. In practice it couldrange from several months to several years, see Broadie et al. (2007). The grace period could be thoughtof as a multiple renegotiated period, according to the creditors’ interest and the regulators’ policies. Andhence is usually not a deterministic constant. As adopted by Li et al. (2020), we use a sequence ofindependent and exponentially distributed random variables to model the grace periods when the insureris in the state of insolvent.
Based on the passage times defined in the above, we introduce the time of liquidation as T := ζ − a ∧ ζ b,c,e λ . (2.10)To avoid the trivial case that liquidation occurs with probability one, we assume throughout the paperthat ψ ′ (0+) >
0, that is, the positive safety loading condition holds.
Time S u r p l u s acb Path IVPath IIIPath IIPath I -b,1 -b,2+c,1 -a Grace period +c,2 Figure 1: The bankruptcy scenarios.Figure 1 gives four possible paths of an insurer’s surplus process. In the setting of spectrally negativeL´evy processes, jumps are expected to occur. To avoid bewildering array of nodes in the coordinate axes,we omit the jumps to focus more on the idea of solvent and insolvent states. As sketched in Figure 1,7rom time 0 to time ζ − b, , the surplus process stays above barrier b and hence the state of solvent applies.As the financial situation gets worse, the insurer transferred to insolvent state from ζ − b, until the surplusrecovers to the safety barrier c at time ζ + c, , where the state of solvent resumes. At ζ − b, , the insurer turnsto be insolvent again. We depict four possible scenarios from ζ − b, of the surplus process. Path I representsthe case of direct liquidation during the insolvent period with the breaching of barrier a . Path II andPath III never touch the liquidation barrier a but have been staying below the safety barrier c for so longthat is unacceptable from the regulator, hence the insurer will still be concluded to liquidation. Path IVclimbs up to the safety barrier c successfully within the grace period, hence from ζ + c, onward, the insurerstays at the state of solvent in Path IV. Remark 2
Note that the way we define states using three barriers is different from Broadie et al. (2007),where a three-barrier model was also adopted but defined four states: liquid, equity dilution, default andliquidation. They used the three barriers directly as the division boundaries. In this paper, we only definethree states of the surplus process as the possible outputs of an insurer: solvent, insolvent and liquidation.From technical point of view, four states of the surplus process is also tractable in mathematics but maylead to tedious expressions. The track of defining states in this paper is consistent with Li et al. (2020),which is practically workable in the context of insurance industry, and also produces neat mathematicalresults.
Remark 3
As pointed out by Li et al. (2020), the definition of liquidation time in (2.10) includes severalexisting stopping times of insolvency as special cases. For example, as the liquidation barrier a goes to −∞ and let the rehabilitation barrier b equal to the safety barrier c , then the liquidation time T retrievesthe so-called Parisian ruin time, which is a popular topic of interest in recent years. We refer to Remark2.1 in Li et al. (2020) for more examples and their applications. This section aims to solve the Gerber-Shiu function at the time of liquidation in terms of the scalefunctions and the L´evy triplet associated with X and e X . As consequences, expressions of the discountedjoint probability density function of the liquidation time, the surplus at liquidation and the historicalhigh of the surplus until liquidation, the Laplace transform of the liquidation time conditional on thatliquidation occurs prior to the first up-crossing time of some fixed level, as well as the probability ofliquidation are also derived.By (2.7) and (2.8), we define for q ≥ λ > a < b < c , b ≤ x ≤ z and a ≤ w ≤ c , an auxiliaryfunction as followsΩ ( q,q + λ ) φ ( w, b, x, z ) := E x h e − qτ − b φ q + λ ( X τ − b − w ) { τ − b <τ + z } i = σ φ q + λ ( b − w ) (cid:20) W ′ q ( x − b ) − W q ( x − b ) W ′ q ( z − b ) W q ( z − b ) (cid:21) + Z z − b d y Z ∞ y φ q + λ ( y − θ + b − w ) υ (d θ ) (cid:20) W q ( z − b − y ) W q ( z − b ) W q ( x − b ) − W q ( x − b − y ) (cid:21) , (3.1)8ence by (2.3) one hasΩ ( q,q + λ ) φ ( w, b, x ) := lim z →∞ Ω ( q,q + λ ) φ ( w, b, x, z )= σ φ q + λ ( b − w ) (cid:0) W ′ q ( x − b ) − Φ q W q ( x − b ) (cid:1) + Z ∞ d y Z ∞ y φ q + λ ( y − θ + b − w ) υ (d θ ) (cid:0) e − Φ q y W q ( x − b ) − W q ( x − b − y ) (cid:1) . (3.2)The notations Ω ( q,q + λ ) φ ( w, b, x, z ) and Ω ( q,q + λ ) φ ( w, b, x ) will come up with φ = W , W ′ , Z . In particular,when W = W or Z = Z , by Lemma 2.2 and (19) of Loeffen et al. (2014), one has the following conciseexpression via the scale functionsΩ ( q,q + λ ) φ ( w, b, x, z ) = φ q + λ ( x − w ) − λ Z xb W q ( x − y ) φ q + λ ( y − w )d y − W q ( x − b ) W q ( z − b ) (cid:18) φ q + λ ( z − w ) − λ Z zb W q ( z − y ) φ q + λ ( y − w )d y (cid:19) , φ = W, Z. (3.3)The following Theorem 1 expresses the Gerber-Shiu function at the liquidation ruin time with expo-nentially distributed grace periods in terms of the scale functions and the L´evy triplet associated withthe L´evy processes X and e X . Theorem 1
For q, λ ∈ [0 , ∞ ) , a < b < c and a measurable function f , we have E x h e − qT f (cid:0) U T (cid:1) { T <ζ + z } i = e σ f ( a ) Ω ( q,q + λ ) W ′ ( a, b, x, z ) − Ω ( q,q + λ ) W ( a, b, x, z ) W ′ q + λ ( c ∧ z − a ) W q + λ ( c ∧ z − a ) ! + Z c ∧ za d y Z ∞ y − a f ( y − θ ) e υ (d θ ) (cid:18) Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c ∧ z − y ) W q + λ ( c ∧ z − a ) − Ω ( q,q + λ ) W ( y, b, x, z ) (cid:19) + λ Z c ∧ za f ( y ) (cid:18) Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c ∧ z − y ) W q + λ ( c ∧ z − a ) − Ω ( q,q + λ ) W ( y, b, x, z ) (cid:19) d y + Z zb d y Z ∞ y − a f ( y − θ ) υ (d θ ) (cid:18) W q ( z − y ) W q ( z − b ) W q ( x − b ) − W q ( x − y ) (cid:19) + Ω ( q,q + λ ) W ( a, b, x, z ) { z>c } W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) " e σ f ( a ) Ω ( q,q + λ ) W ′ ( a, b, c, z ) − Ω ( q,q + λ ) W ( a, b, c, z ) W ′ q + λ ( c − a ) W q + λ ( c − a ) ! + Z ca d y Z ∞ y − a f ( y − θ ) e υ (d θ ) (cid:18) Ω ( q,q + λ ) W ( a, b, c, z ) W q + λ ( c − y ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( y, b, c, z ) (cid:19) + λ Z ca f ( y ) (cid:18) Ω ( q,q + λ ) W ( a, b, c, z ) W q + λ ( c − y ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( y, b, c, z ) (cid:19) d y + Z zb d y Z ∞ y − a f ( y − θ ) υ (d θ ) (cid:18) W q ( z − y ) W q ( z − b ) W q ( c − b ) − W q ( c − y ) (cid:19)(cid:21) , b < x ≤ z. (3.4) Proof.
Denote by ~ ( x ) the left hand side of (3.4). Note that, if liquidation occurs, then the surplusprocess U has to down-cross the level b beforehand, otherwise liquidation would never happen. In9ddition, one should bear in mind that, at the moment U down-crosses b , U may or may not down-cross a , in the former case liquidation occurs. Using the piecewise strong Markov property of the surplusprocess U , one obtains ~ ( x ) = E x (cid:20) e − qτ − b { τ − b <τ + z } { τ − b <τ − a } e E X τ − b (cid:16) e − q ( ζ − a ∧ ζ b,c,eλ ) f ( U ζ − a ∧ ζ b,c,eλ ) { ζ − a ∧ ζ b,c,eλ <ζ + z } (cid:17)(cid:21) + E x h e − qτ − b { τ − b <τ + z } { τ − b = τ − a } f ( X τ − b ) i = E x (cid:20) e − qτ − b { τ − b <τ + z } { X ( τ − b ) ≥ a } e E X τ − b (cid:16) e − q ( τ − a ∧ e λ ) f ( U τ − a ∧ e λ ) { τ − a ∧ e λ <τ + c } (cid:17)(cid:21) + E x (cid:20) e − qτ − b { τ − b <τ + z } { X ( τ − b ) ≥ a } e E X τ − b (cid:16) e − qτ + c { τ + c <τ − a ∧ e λ } (cid:17) ~ ( c ) (cid:21) + E x h e − qτ − b { τ − b <τ + z } { X ( τ − b ) a } e E X τ − b (cid:16) e − ( q + λ ) τ − a f ( e X τ − a ) { τ − a <τ + z } (cid:17)(cid:21) + E x (cid:20) e − qτ − b { τ − b <τ + z } { X ( τ − b ) >a } e E X τ − b (cid:16) e − qe λ f ( e X e λ ) { e λ <τ − a ∧ τ + z } (cid:17)(cid:21) + E x h e − qτ − b { τ − b <τ + z } { X ( τ − b ) ≤ a } f ( X τ − b ) i , b < x ≤ z ≤ c, which together with (2.6), (2.7) and (2.8) yields (3.4). The proof of Theorem 1 is complete.Define U t := sup ≤ s ≤ t U s , then T < ζ + z is equivalent to U T < z . Given a sequence of exponentiallydistributed grace periods, the following Corollary 1 characterizes the joint discounted probability density11unction of the liquidation time, the surplus at and the running supreme of the surplus until the liquidationtime. Corollary 1
For q ≥ , λ > , a < b < c and b < x ≤ z , we have E x (cid:0) e − qT , U T ∈ d u, U T ∈ d z (cid:1) = Z ca ∂ Ω ( q,q + λ ) W ( a, b, x, z ) ∂z W q + λ ( c − y ) W q + λ ( c − a ) − ∂ Ω ( q,q + λ ) W ( y, b, x, z ) ∂z ! e υ ( y − d u ) { z>c } d y d z + Z za ∂∂z (cid:18) Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( z − y ) W q + λ ( z − a ) − Ω ( q,q + λ ) W ( y, b, x, z ) (cid:19) e υ ( y − d u ) { z ≤ c } d y d z +Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ (0+) W q + λ ( z − a ) e υ ( z − d u )d z { z ≤ c } + W q (0+) W q ( z − b ) W q ( x − b ) υ ( z − d u )d z + Z zb ∂∂z (cid:20) W q ( z − y ) W q ( z − b ) (cid:21) W q ( x − b ) υ ( y − d u )d y d z + Ω ( q,q + λ ) W ( a, b, x, z ) { z>c } W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) " Z ca ∂ Ω ( q,q + λ ) W ( a, b, c, z ) ∂z W q + λ ( c − y ) W q + λ ( c − a ) − ∂ Ω ( q,q + λ ) W ( y, b, c, z ) ∂z ! e υ ( y − d u )d y d z + Z zb ∂∂z (cid:20) W q ( z − y ) W q ( z − b ) (cid:21) W q ( c − b ) υ ( y − d u )d y d z +d zυ ( z − d u ) W q (0+) W q ( z − b ) W q ( c − b ) + ∂∂z " Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) × " Z ca (cid:18) Ω ( q,q + λ ) W ( a, b, c, z ) W q + λ ( c − y ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( y, b, c, z ) (cid:19) e υ ( y − d u )d y d z + Z zb (cid:18) W q ( z − y ) W q ( z − b ) W q ( c − b ) − W q ( c − y ) (cid:19) υ ( y − d u )d y d z { z>c } , for u ∈ ( −∞ , a ) , and E x (cid:2) e − qT , U T ∈ d u, U T ∈ d z (cid:3) = λ d u d z (cid:20) ∂ Ω ( q,q + λ ) W ( a,b,x,z ) ∂z W q + λ ( c − u ) { z>c } W q + λ ( c − a ) + ∂ Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( z − u ) W q + λ ( z − a ) ∂z { z ≤ c } + Ω ( q,q + λ ) W ( a, b, x, z ) { z>c } W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) (cid:20) ∂ Ω ( q,q + λ ) W ( a, b, c, z ) ∂z W q + λ ( c − u ) W q + λ ( c − a ) − ∂ Ω ( q,q + λ ) W ( u, b, c, z ) ∂z (cid:21) + ∂∂z (cid:20) Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) (cid:21)(cid:20) Ω ( q,q + λ ) W ( a, b, c, z ) W q + λ ( c − u ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( u, b, c, z ) (cid:21) { z>c } (cid:21) − λ d u d z ∂ Ω ( q,q + λ ) W ( u, b, x, z ) ∂z − e σ δ a (d u ) d z ∂∂z Ω ( q,q + λ ) W ( a, b, x, z ) W ′ q + λ ( z − a ) W q + λ ( z − a ) ! { z ≤ c } + e σ δ a (d u ) d z " ∂ Ω ( q,q + λ ) W ′ ( a, b, x, z ) ∂z − ∂ Ω ( q,q + λ ) W ( a, b, x, z ) ∂z W ′ q + λ ( c − a ) W q + λ ( c − a ) { z>c }
12 Ω ( q,q + λ ) W ( a, b, x, z ) { z>c } W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) (cid:20) ∂ Ω ( q,q + λ ) W ′ ( a, b, c, z ) ∂z − ∂ Ω ( q,q + λ ) W ( a, b, c, z ) ∂z W ′ q + λ ( c − a ) W q + λ ( c − a ) (cid:21) + ∂∂z (cid:20) Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) (cid:21)(cid:20) Ω ( q,q + λ ) W ′ ( a, b, c, z ) − Ω ( q,q + λ ) W ( a, b, c, z ) W ′ q + λ ( c − a ) W q + λ ( c − a ) (cid:21) { z>c } (cid:21) , for u ∈ [ a, c ∧ z ] . Here δ a (d u ) denotes the Dirac measure which assigns unit mass to the singleton { a } . Proof.
Let f ( w ) = { w ≤ u } in (3.4) one may have E x h e − qT { U T ≤ u } { U T
For q, λ ∈ (0 , ∞ ) , a < b < c < z and b < x ≤ z , we have E x (cid:20) e − qT { T <ζ + z } (cid:21) = qq + λ (cid:18) Ω ( q,q + λ ) Z ( a, b, x, z ) − Z q + λ ( c − a ) W q + λ ( c − a ) Ω ( q,q + λ ) W ( a, b, x, z ) (cid:19) + λq + λ Z q ( x − b ) − Z q ( z − b ) W q ( z − b ) W q ( x − b ) − Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c − a ) ! + Ω ( q,q + λ ) W ( a, b, x, z ) W q + λ ( c − a ) − Ω ( q,q + λ ) W ( a, b, c, z ) (cid:20) qq + λ (cid:18) Ω ( q,q + λ ) Z ( a, b, c, z ) − Z q + λ ( c − a ) W q + λ ( c − a ) Ω ( q,q + λ ) W ( a, b, c, z ) (cid:19) + λq + λ Z q ( c − b ) − Z q ( z − b ) W q ( z − b ) W q ( c − b ) − Ω ( q,q + λ ) W ( a, b, c, z ) W q + λ ( c − a ) ! . (3.14) Proof.