On a generalization from ruin to default in a L´evyinsurance risk model

Runhuan Feng ∗ and Yasutaka Shimizu †‡

http://dx.doi.org/10.1007/s11009-012-9282-y

Abstract In a variety of insurance risk models, ruin-related quantities in the class of ex- pected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative L´evyrisk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current litera- ture. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution.

Key words: expected discounted penalty function; costs up to default; defective renewal equation; compound geometric distribution; L´evyrisk model; scale function; potential measure; operator calculus. MSC2010: 91B30; 60J45; 60G51.

1 Introduction

1.1 Generalization of ruin-related quantities In the recent literature of , a class of functions, known as expected discounted penalty function (EDPF) or the Gerber-Shiu function proposed by Gerber and Shiu [17], has gained enormous research interests due to its generality in representing a variety of ruin-related quantities, including the probability of ultimate ruin, the tri-variate Laplace transform of the time to ruin, surplus prior to ruin and deficit at ruin, etc. To show its precise definition, we introduce the mathematical structure of risk models. Consider a probability space (Ω, F, (Ft)t≥0, P) in which the evolution of an insurance company’s

∗Department of Mathematical Sciences, University of Wisconsin - Milwaukee, P.O. Box 413, Milwaukee, WI, USA 53202-0413; [email protected] †Graduate School of Engineering Science, Osaka University; 1-3, Machikaneyama-cho, Toyonaka-shi, Osaka 560-8531, Japan; [email protected] ‡Japan Science and Technology Agency, CREST; Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan.

1 2 Generalization from ruin to default in a L´evyrisk model

surplus is defined by a strong Markov process X = {Xt, t ≥ 0} together with a family of x x probability measures {P , x ≥ 0} such that P (X0 = x) = 1. The event of ruin is hence represented by τ0 = inf{t > 0 : Xt < 0}. The Gerber-Shiu function is defined by

[ ] − Ex δτ0 | | ∞ ≥ m(x) := e w(Xτ0−, Xτ0 )I(τ0 < ) , x 0, (1.1) where δ ≥ 0 is the discounting force of interest and the bivariate function w represents the penalty to be exercised, should ruin occur, depending on the surplus immediately prior to | | ruin Xτ0− as well as the deficit at ruin Xτ0 . A ground-breaking result in Gerber-Shiu [17] was to show that the EDPF in general satisfies a defective renewal equation (DRE) which yields analytical solutions once and for all ruin-related quantities in the same family. Gerber and Shiu [19], Li and Garrido [23] were among a sequence of papers leading to the discovery of DREs in renewal risk models. Gerber and Landry [16], Tsai and Willmot [35] proved DREs for the Gerber-Shiu function in compound Poisson model perturbed by a Brownian motion. Garrido and Morales [15], Morales [24] derived DREs when a risk process was driven by a L´evysubordinator. A further extension appeared in Biffis and Morales [4] for the analysis of the Gerber-Shiu function in a very general spectrally negative L´evy(SNL) risk model. Most remarkably, it is showed that solutions can also be represented as infinite series obtained through DREs (cf equation (43) in Biffis and Morales [4]). However, such a claim turns out to be overgeneralized as we shall demonstrate later. Another recent development in connecting ruin-related quantities with DRE was shown in Cai et al. [6] that a similar defective renewal equation holds for a generalized class of default-related quantities, referred to as the expected discounted cost up to default in the paper, with a piecewise-deterministic compound Poisson risk process. For comparison with the classic Gerber-Shiu function, this generalized function is defined by

[∫ ] τd x −δt H(x) := E e l(Xt)dt , x ≥ d, (1.2) 0 where the function l represents the operating cost depending on the current level of surplus and the time of default is given by τd = inf{t > 0 : Xt < d} with the convention that inf ∅ = ∞. While the generalized function can be shown to encompass all the cases of the Gerber-Shiu function, it also has a natural interpretation of running costs accumulated from the time of inception to the time of default event, which has long been studied as the objective of optimization in optimal control problems. (cf. Øksendal and Sulem [29] and references therein.) The major advantage of the formulation in ruin context is to provide a systematic approach to analyze many other ruin-related quantities such as total dividend payments to shareholders, aggregate claim costs, accumulated utilities up to default. A recent work by Cheung and Feng [8] has shown many further examples including higher moments of total dividends. In this paper, we intend to study the generalized function H in the context of an SNL risk model. In particular, we are interested in investigating to what extent the DREs exist for this general case. Runhuan Feng and Yasutaka Shimizu 3

1.2 Connection among various solution methods In the general setting of spectrally negative L´evyrisk process, another version of the EDPF is proposed in Biffis and Morales [4] and Biffis and Kyprianou [5], which is given by

x −δτ E 0 | | − ∞ ≥ ϕ(x) := [e w( Xτ0 ,Xτ0 ,Xτ0−)I(τ0 < )], x 0, (1.3) where Xt = inf0≤s≤t Xs is the running infimum of the surplus process. The paper em- ploys fluctuation theoretic techniques to show that ϕ also satisfies a DRE (cf. equation (23) of Biffis and Morales [4]). For interested readers, literature on fluctuation theoretic approaches which represent solutions to m and ϕ in terms of scale functions can be seen in Zhou [37], Biffis and Kyprianou [5] and references therein. As pointed out in Biffis and Kyprianou [5], a common feature of the classic ruin liter- ature is to reduce the analysis of Gerber-Shiu function to the study of integro-differential equations (IDEs), which in many cases can be converted to DREs. However, in the context of general L´evyprocess, the fluctuation theory appears to be the preferred approach in the recent literature. In comparison, the typical solutions from the DREs are represented as infinite series, whereas the solutions from the fluctuation theoretic approach are usu- ally expressed in terms of scale functions. Although in classical models the results are consistent, it is unclear in the current literature whether or not the DRE approach and fluctuation theoretic approach are completely interchangeable, as it seems to suggest by the above-mentioned papers in the context of SNL processes. Our work shows that the DRE approach works only when the corresponding L´evymeasure ν satisfies ∫ 1 zν(dz) < ∞. 0 Another novelty of the paper is to show that solutions from both methods are connected through an operator-based approach, which has been systematically exploited in recent literature. Since the introduction of operators in Shiu [34], Dickson and Hipp [10], Li and Garrido [23], operator-based approaches were developed in a number of papers including Gerber and Shiu [18], Albrecher et al. [1], Cai et al. [6]. An advantage of such arguments is that IDEs are often transformed into algebraic equations of operators, which are much easier to handle with rudimentary knowledge on operators and calculus, as opposed to the advanced machinery of fluctuation theory which requires one’s understanding of . This paper follows the same line of logic as in Feng [11, 12, 13] but differs in the model setup. The underlying risk process in the present paper is far more general than that analyzed in Feng [13] and requires non-trivial technical treatment of unbounded L´evy measure. We believe this line of development also contributes to the emerging application of operator calculus in ruin theory and can potentially facilitate symbolic calculation in computing systems such as Maple. The organization of the paper is summarized as follows. We first show in Section 2 that for any risk process driven by L´evy-type stochastic differential equation the ruin- related quantities in the family of H can be solved through an “all-in-one” formula of IDE under certain conditions. In applications, one could carry out the analysis of a quantity of interest in three steps. (1) Identify the cost function l associated with the quantity; (2) Find the specific form of the infinitesimal generator A of the underlying risk process; (3) Solve the IDE analytically or numerically for the function H. When the underlying process 4 Generalization from ruin to default in a L´evyrisk model

is a spectrally negative L´evyprocess, we can even afford the luxury of producing general explicit solution through a DRE, as shown in Section 3. This paper later establishes a connection between the IDE and the DRE by an operator based approach which has been proven to be useful for a variety of risk models in the aforementioned literature. Some preliminary lemmas for operator-based calculations are gathered in the Appendix. Section 4 is devoted to the demonstration of consistency between the results from the operator- based approach and fluctuation theoretic approach to the analysis of H under an SNL risk model, whereas Section 5 deals with various issues in applications of H. In Sections 5.1 and 5.2, we illustrate two examples that are only known previously to be in the family of H in other risk models. Section 5.3 presents an asymptotic result which could be an important tool for analyzing risk metrics at large surplus level. In Section 5.4, a statistical methodology to estimate H based on a certain surplus data is introduced, since many quantities in the DRE for H may be unknown in practice. We complete the paper with concluding remarks in Section 6.

2 A general integro-differential equation

For a wide variety of applications in ruin theory, we consider a risk process satisfying the general Markovian SDE defined on a filtered probability space (Ω, F, (Ft)t≥0, P): X0 = x, ∑p ∑q ∫ (k) dXt = µ(Xt−)dt + σk(Xt−)dWt + Fj(Xt−, z) N j(dt, dz), (2.1) k=1 j=1 E where E = R\{0}, µ, σ1, ··· , σp are measurable functions on R, F1, ··· ,Fq are measurable functions on R × E, W (1), ··· ,W (p) are mutually independent Brownian motions and N j(dt, dz) := Nj(dt, dz) − νj(dz)dt for j = 1, ··· , q are Ft-martingales with Nj being Poisson random measure on R+ × E independent of the Brownian motions and νj being the corresponding intensity measure which satisfies ∫ 2 z νj(dz) < ∞ (j = 1, . . . , q). |z|≤1 ∑ p We introduce the notation a(x, y) := k=1 σk(x)σk(y). It is known that if there exist constants K1 and K2 such that for all x1, x2 ∈ R,

2 |µ(x1) − µ(x2)| + |a(x1, x1) − 2a(x1, x2) + a(x2, x2)| ∑q ∫ 2 2 + |Fj(x1, z) − Fj(x2, z)| νj(dz) ≤ K1|x1 − x2| , (2.2) j=1 E and for all x ∈ R, ∑q ∫ 2 2 2 |µ(x)| + |a(x, x)| + |Fj(x, z)| νj(dz) ≤ K2(1 + |x| ), (2.3) j=1 E then the X exists and is pathwise unique; see, e.g., Section 6.2, Ap- plebaum [2]. In this case, the infinitesimal generator A of X is given for any f ∈ C2(R) Runhuan Feng and Yasutaka Shimizu 5

by

q ∫ 1 ∑ Af(x) = µ(x)f ′(x)+ a(x, x)f ′′(x)+ {f(x+F (x, z))−f(x)−F (x, z)f ′(x)}ν (dz), 2 j j j j=1 E where f ′ and f ′′ are the first and second derivatives of f.

Lemma 2.1. If a function f has bounded second and third derivatives, then there exists a function C such that, for |y − x| < 1,

|Af(y) − Af(x)| ≤ C(x)|y − x|. (2.4)

Proof. Without loss of generality, we shall prove the result for one dimensional case in which q = 1 and we simply write F1 = F , ν1 = ν in the sequel. For brevity, we denote

Q(x, y, z) = f(y + F (y, z)) − f(y) − F (y, z)f ′(y) − [f(x + F (x, z)) − f(x) − F (x, z)f ′(x)].

The proof ends if we can show that ∫ |Q(x, y, z)|ν(dz) < C(x)|y − x|, E because µ(x)f ′(x) + (1/2)a(x, x)f ′′(x) is clearly Lipschitz continuous due to (2.2) and that f has bounded first and second derivatives. Using Taylor’s formula with the integral expression of the remainder term, we have ∫ ∫ F (y,z) F (x,z) Q(x, y, z) = {F (y, z) − s}f ′′(s + y)ds − {F (x, z) − s}f ′′(s + x)ds 0 [∫ ∫ 0 ] F (y,z) F (x,z) = F (y, z) f ′′(s + y)ds + {f ′′(s + y) − f ′′(s + x)}ds (2.5) F (x,z) 0 ∫ F (x,z) +[F (y, z) − F (x, z)] f ′′(s + x)ds (2.6) ∫ 0 ∫ F (y,z) F (y,z) − s[f ′′(s + y) − f ′′(s + x)]ds − sf ′′(s + x)ds. (2.7) 0 F (x,z)

In the sequel, we denote the integrals in (2.5), (2.6) and (2.7) by I1,I2,I3, respectively, and (k) Mk = sup |f (x)| < ∞, for k = 2, 3. x∈E First, we note that

|I | ≤ M |F (y, z)||F (y, z) − F (x, z)| + M |F (y, z)F (x, z)||y − x| 1 2( )3 ≤ M |F (x, z)| + |F (y, z) − F (x, z)| |F (y, z) − F (x, z)| 2 ( ) 2 +M3 |F (x, z)| + |F (x, z)||F (y, z) − F (x, z)| |y − x| 2 ≤ M2|F (x, z)||F (y, z) − F (x, z)| + M2|F (y, z) − F (x, z)| 2 +M3|F (x, z)| |y − x| + M3|F (x, z)||F (y, z) − F (x, z)| 6 Generalization from ruin to default in a L´evyrisk model

using the fact that |y − x| < 1 in the last inequality. Hence, using Cauchy-Schwartz inequality, (2.2) and (2.3), we have ∫ 1/2 1/2 2 |I1|ν(dz) ≤ (M2 + M3)(K1K2) (1 + |x|) |x − y| + M2K1|x − y| + M3K2(1 + |x| )|x − y|. E Second, we have ∫ ∫ 1/2 2 1/2 |I2|ν(dz) ≤ M2 |F (y, z) − F (x, z)||F (x, z)|ν(dz) ≤ M2(K1K2) (1 + |x| ) |x − y|. E E Last, we estimate

∫ ∫ |F (y,z)| F (y,z) | | ≤ | − | 2 I3 2M3 y x s ds + 2M2 s ds 0 F (x,z) ∫ ∫ |F (x,z)| F (y,z) ≤ | − | 2M3 y x s ds + 2(M2 + M3) s ds 0 F (x,z) 2 2 2 = M3|F (x, z)| |y − x| + (M2 + M3)|F (y, z) − F (x, z)| = M |F (x, z)|2|y − x| + (M + M )|F (y, z) + F (x, z)||F (y, z) − F (x, z)| 3 2 3 ( ) 2 ≤ M3|F (x, z)| |y − x| + (M2 + M3) |2F (x, z)| + |F (y, z) − F (x, z)| |F (y, z) − F (x, z)|.

Hence, ∫ 2 1/2 2 1/2 2 |I3|ν(dz) ≤ M3K2(1 + |x| )|y − x| + 2(M2 + M3)(K1K2) (1 + |x| ) |x − y| E 2 +(M2 + M3)K1|x − y| .

Therefore, we obtain ∫ ∫ ∫ ∫

|Q(x, y, z)|ν(z) ≤ |I1|ν(dz) + |I2|ν(dz) + |I3|ν(dz) = C(x)|y − x|. E E E E

Remark 2.1. If νj(E) < ∞ for j = 1, ··· , p, then (2.4) also holds true even if f does not have bounded third derivative; Proof of the assertion can be found in Feng [13].

We are particularly interested in searching solutions to the following function, for which many applications in risk theory can be found in Section 5.1.

Definition 2.1. Let τd := inf{t > 0 : Xt < d} for a constant d ∈ R, and l be a measurable, or generalized function on R. The quantity of interest is H given by [∫ ] τd x −δt H(x) := E e l(Xt) dt , (2.8) 0

x x x where E is an expectation with respect to the probability P for which P {X0 = x} = 1. Runhuan Feng and Yasutaka Shimizu 7

x If x < d, then P {τd = 0} = 1 by definition and hence H(x) = 0. In the context of risk theory, τd and H can be interpreted as the time of default and the expected discounted total cost up to default, respectively. It is often useful to take l to be a generalized function such as Dirac delta function, which captures the exact moment of the underlying process reaching a certain level as shown in Section 5.1. We shall now present a solution method to H(x) when x ≥ d via the following IDE, the proof of which is nearly the same as that of Theorem 2.1 in Feng [13] and hence omitted.

Theorem 2.1. Suppose that {Xt, t ≥ 0} is a risk process given by (2.1), and that l is continuous on (d, ∞) except for a countable set of discontinuities D such that [∫ ] τd x −δt E |e l(Xt)|dt < ∞, for all x > d, (2.9) 0 and H has bounded second and third derivatives except on the set D, then H is the solution to the following integro-differential equation

AH(x) − δH(x) + l(x) = 0, x > d, x ̸∈ D. (2.10)

Remark 2.2. According to Remark 2.1, we can remove the boundedness of the third derivative of H in Theorem 2.1 if νj(E) < ∞ for all j = 1, ··· , p. Moreover, in those cases, H has the bounded second derivative if σ2 > 0 and l is bounded; see Feng [13].

3 Operator-based solutions to IDE

3.1 A spectrally negative L´evyprocess Even though Theorem 2.1 provides a shortcut to yield an integro-differential equation (IDE), it is generally difficult to find analytical solutions. However, there is indeed a fairly general class of risk processes for which explicit solutions to classic Gerber-Shiu functions are known to exist. Consider the L´evyrisk process introduced in Biffis and Morales [4] with slightly different notation, given by

Xt = x + ct − St − Zt, (3.1) where S is a subordinator with E[St] < ∞, Z is a spectrally positive L´evyprocess of unbounded variation with E[Zt] = 0 and E|Zt| < ∞ for each t ≥ 0 independent of S, or being trivially zero. The processes S and Z are often interpreted as the aggregate claim amount and an additional (martingale) perturbation respectively. According to the Ito-L´evydecomposition, we may rewrite the subordinator as ∫ ∫ t ∞ St = zN1(ds, dz), 0 0 where N1 is a Poisson random measure. The subordinator S has the Laplace exponent ∫ ∞ −sS1 −sz ψS(s) = log E[e ] = (e − 1)νS(dz), s ≥ 0. (3.2) 0 8 Generalization from ruin to default in a L´evyrisk model

There are at least two types of definitions of the Laplace exponent, one of which is pre- sented in (3.2). Interested readers may refer to Shimizu [33] for a more detailed account of various definitions. Similarly, the spectrally positive L´evyprocess can be written as a sum of independent components ∫ ∫ ∫ ∫ t ∞ t 1 Zt = at + σWt + zN2(ds, dz) + zN 2(ds, dz). 0 1 0 0 ∫ ∫ E E| | ∞ − ∞ ∞ ∞ Since [Z1] = 0 and Zt < , we must have a = 1 zνZ (dz) and 1 zνZ (dz) < (cf. Theorem 3.8 of Kyprianou [27]). Thus, ∫ ∫ t ∞ Zt = σWt + Lt where Lt = zN 2(ds, dz). 0 0

We denote the process L = {Lt, t ≥ 0}. Their Laplace exponents are given by ∫ 2 ∞ σ 2 −sz ψZ (s) = s + ψL(s); ψL(s) = (e − 1 + sz)νZ (dz), s ≥ 0. 2 0

We write ψS+L = ψS + ψL, which is the Laplace exponent of S + L. Hence the Laplace exponent of X − x denoted by ψX−x(s) satisfies

s(X1−x) ψX−x(−s) = log E[e ] = cs + ψS(s) + ψZ (s), s ≥ 0. Putting all pieces together, we see that ∫ ∫ ∫ ∫ t ∞ t ∞ Xt = x + ct − zN1(ds, dz) − σWt − zN 2(ds, dz) 0 0 ∫ ∫ 0 0 t ∞ ( ) = x +ct ˆ − σWt − z N 1(ds, dz) + N 2(ds, dz) , (3.3) 0 0 ∫ − ∞ wherec ˆ = c 0 zνS(dz). It shows that X is indeed in the class of L´evy-type stochastic integral given in (3.1). Hence for a function f in its domain, the infinitesimal generator of X is

∫ 2 ∞ σ ′′ ′ Af(x) = f (x) + cf (x) + {f(x − z) − f(x)} νS(dz) 2 0 ∫ ∞ ′ + {f(x − z) − f(x) + zf (x)} νZ (dz). 0 Under some regularities that we have discussed so far, F (x) = H(x + d) satisfies the following IDE: σ2 F ′′(x) + cF ′(x) − δF (x) + F (x) + l(x + d) = 0, 2 ν where ∫ ∞ ∫ ∞ ′ Fν(x) = {F (x − z) − F (x)} νS(dz) + {F (x − z) − F (x) + zF (x)} νZ (dz). (3.4) 0 0 Runhuan Feng and Yasutaka Shimizu 9

3.2 Operator-based derivation To facilitate the discussion, we introduce some operators for a suitable function f,

• (Identity) If(x) := f(x); • D d (Differentiation) f(x) := dx f(x);

• (Shift) Sdf(x) := f(x + d) for d ∈ R; ∫ • L ∞ −su ≥ (Laplace transform) f(s) := 0 e f(u) du, s 0. ∫ • T sx ∞ −sy ≥ (Dickson-Hipp) sf(x) := e x e f(y) dy, s 0. ∫ • E −sx x sy ≥ sf(x) := e 0 e f(y) dy, s 0. Some properties of these operators are given in the Appendix. We use the symbol ∗ to represent the convolution ∫ ∞ A ∗ B(x) = A(x − y)B(y) dy. 0 According to Theorem 2.1, the function H must satisfy the IDE ∫ 2 ∞ σ ′′ ′ ′ H (x) + cH (x) + {H(x − z) − H(x) + zH (x)}νZ (dz) − δH(x) + l(x) = 0, 2 0 for x > d. Note in this case we can no longer split the integral. Letting F := SdH and u = x − d, we obtain

σ2 F ′′(u) + cF ′(u) + F (u) − δF (u) + l(u + d) = 0, u > 0, (3.5) 2 ν where Fν is given by (3.4) and F (u) = 0 for all u < 0. We rewrite (3.5) in terms of operators: 2 (bI − D)(aI + D)F = [F + S l] , σ2 ν d where √ √ c + c2 + 2δσ2 −c + c2 + 2δσ2 a := ; b := . σ2 σ2 Then Lemma A.1 (1) and (2) yield that

2 F = E R, (3.6) σ2 a where σ2 R := T [F + S l] + U; U := F (0)△ . b ν d 2 0 10 Generalization from ruin to default in a L´evyrisk model

Choosing a nonnegative constant ρ ≠ b, and applying Lemma A.1 (4), we see that

R = Tρ [Fν + Sdl] − (b − ρ)Tρ(R − U) + U. Applying Lemma A.1 (5) yields that ( ) σ2 σ2 R = T [F + S l] − (b − ρ)(a + ρ) T F + (b − ρ) F + T U + U. (3.7) ρ ν d 2 ρ 2 ρ

Depending on whether or not the Levy measure νZ satisfies ∫ 1 zνZ (dz) < ∞, 0 the rest of the proof bifurcates into two routes, one of which leads to a DRE and the other leads to a solution via the scale function.

3.3 Defective renewal equation In this subsection, we obtain a defective renewal equation for H. Although the result is similar to the DRE for ϕ in Biffis and Morales [4] obtained through a fluctuation theoretic method, our approach is entirely different, more elementary, and can be consistently ex- tended to other risk processes (see Section 6). It is assumed in Biffis and Morales [4] that there exists a nonnegative function Jρ (denoted by Gρ in [4]) such that ψ (s) − ψ (ρ) LJ (s) = L L , s, ρ ≥ 0, s ≠ ρ. (3.8) ρ ρ − s However, as shown in Lemma 3.1, the assumption (3.8) does not hold in general for (3.1). ∫ 1 ∞ Lemma 3.1. The function Jρ exists if and only if 0 z νZ ( dz) < . Proof. For notational convenience, we define ∫ ∞ ρx −ρy Tρ ◦ νZ (x) := e e νZ (dy). x ∫ ∫ ∞ −sy − ∞ 1 ∞ The “if” part is easy to see: since 0 (e 1)νZ (dz) < if 0 zνZ (dz) < , we see by a direct computation that ∫ (∫ ) ∞ y −(s−ρ)x −ρy L{Tρ ◦ νZ }(s) = e dx e νZ (dy) 0 0 ∫ ∞ 1 −ρy −sy = [(e − 1) − (e − 1)]νZ (dy) s − ρ 0 ψ (s) − ψ (ρ) = L L , ρ − s which means that Jρ = Tρ ◦ νZ . Let us prove the “only if” part by contradiction. Suppose that Jρ exists and that ∫ 1 zνZ (dz) = ∞. (3.9) 0 Runhuan Feng and Yasutaka Shimizu 11

We can prove the following identity

sψ (ρ) − ρψ (s) L {T {T ◦ ν }} (s) = L L . (3.10) ρ 0 Z sρ(ρ − s)

Note that νZ is a finite measure on [ϵ, ∞) for any ϵ > 0. We define ∫ ∞ Lϵf(s) = e−sxf(x)dx. ϵ

ϵ ϵ L {T0 ◦ νZ }(ρ) − L {T0 ◦ νZ }(s) [ ∫ ∞ ∫ ∞ ] 1 −ρϵ −ρz −sϵ −sz = s (e − e )νZ (dz) − ρ (e − e )νZ (dz) ρs ϵ ϵ [ ∫ ∫ 1 ∞ ∞ = s(e−ρϵ + ρϵ − 1) ν (dz) − ρ(e−sϵ + sϵ − 1)e−sϵ ν (dz) ρs Z Z ϵ ϵ ] ∫ ∞ ∫ ∞ −sz −ρz +ρ (e + sz − 1)νZ (dz) − s (e + ρz − 1)νZ (dz) . (3.11) ϵ ϵ ∫ 2 ≥ ≤ 2 1 2 ∞ In view of the fact that ϵ I(z ϵ) z and 0 z νZ (dz) < , we observe that ∫ ∫ 1 s2 1 (e−sϵ + sϵ − 1) ν (dz) = [ ϵ2 + O(ϵ3)] ν (dz) Z 2 Z ϵ ∫ ϵ 2 1 s 2 ≤ ϵ I(z ≥ ϵ)νZ (dz) + O(ϵ) → 0, as ϵ → 0, 2 0 with the limit justified by the dominated convergence theorem. Letting ϵ → 0 in (3.11), we have ρψ (s) − sψ (ρ) L{T ◦ ν }(ρ) − L{T ◦ ν }(s) = L L , 0 Z 0 Z ρs and Lemma A.2 (2) leads to the identity (3.10). By the definition of Jρ, we also have

LJ (0) − LJ (s) sψ (ρ) − ρψ (s) L{T J }(s) = ρ ρ = L L . (3.12) 0 ρ s sρ(ρ − s)

Comparing (3.10) and (3.12), we must have by the uniqueness of Laplace transform that

T0Jρ(x) = TρT0 ◦ νZ (x) = T0Tρ ◦ νZ (x), x > 0 which implies that Jρ(x) = Tρ ◦ νZ (x), x > 0. However, in light of the condition (3.9),

L{Jρ}(s) = L{Tρ ◦ νZ }(s) ∫ ∞ 1 −(s−ρ)z −ρz = (1 − e )e νZ (dz) 0 s − ρ 12 Generalization from ruin to default in a L´evyrisk model

∫ 1 1 −(s−ρ)z ≥ (1 − e )(1 − ρz)νZ (dz) = ∞, 0 s − ρ ∫ 1 ∞ which causes a contradiction with (3.8). Therefore, it follows that 0 zνZ (dz) < .

The assumption (3.8) implies that L is a spectrally negative L´evyprocess of bounded variation and hence can be decomposed as − ∗ ≥ Lt = µt + St , t 0, where µ > 0 and S∗ is a surbordinator. In other words, the process L can be combined with −ct + St. Hence, without loss of generality, we only consider the case L ≡ 0 in this subsection. Theorem 3.1. Suppose that X is a L´evyrisk process given in (3.1) with Z being a ∫ ∞ Brownian motion or trivially zero, and the net profit condition c > 0 z νS(dz), and all the assumptions of Theorem 2.1 are satisfied. Then there exists a unique nonnegative solution ρ to the Lundberg fundamental equation:

ψX−x(−s) = δ. (3.13)

Furthermore, if the functions Tρl(x) and TρH(x) exist, then H satisfies the renewal equa- tion: ∫ x−d H(x) = hl(x) + H(x − y)g(y) dy, x ≥ d, (3.14) 0 where g and hl are defined as follows: { 2σ−2E {T ◦ ν }(y), σ2 > 0; g(y) := β ρ S (3.15) c−1T ◦ ν (y), σ2 = 0, { ρ S [ ] −2 −2 −β(x−d) 2 2σ EβTρl(x) + H(d) − 2σ EβTρl(d) e , σ > 0; hl(x) := −1 2 (3.16) c Tρl(x), σ = 0. with the constant β = 2c/σ2 + ρ and ∫ ∞ ρx −ρy Tρ ◦ νS(x) = e e νS(dy). (3.17) x

Proof. Let ϕ(s) = ψX−x(−s)−δ. We first prove the existence and uniqueness of a nonneg- ative root to the Lundberg equation (3.13) by showing the following. (1) ϕ(0) = −δ ≤ 0; (2) ϕ′(s) > 0 for all s ≥ 0; (3) ϕ(∞) = ∞. The first one is clear. Secondly, we find from the net profit condition that ∫ ∞ ∫ ∞ ′ −sz ϕ (s) = c − ze νS(dz) ≥ c − z νS(dz) > 0, s ≥ 0. 0 0 Finally, since e−x − 1 ≥ −x for any x ≥ 0, it follows that ∫ ∞ 2 σ 2 ψS(s) ≥ −s z νS(dz); ψZ (s) ≥ s . 0 2 Runhuan Feng and Yasutaka Shimizu 13

Therefore, it holds under the net profit condition that ( ∫ ) ∞ 2 σ 2 ϕ(s) = cs + ψS(s) + ψZ (s) − δ ≥ s c − z νS(dz) + s − δ → ∞, s → ∞. 0 2 Hence the proof for the existence and uniqueness of the nonnegative root is completed. Applying Lemma A.3 on the first term of (3.7) yields that ( ) σ2 R = F ∗ J + T S l + (ψ − (−ρ) − δ) T F + (b − ρ) F + T U + U. ρ ρ d X x ρ 2 ρ

Here, we choose ρ as the unique nonnegative root to (3.13) to obtain that ( ) σ2 R = F ∗ J + T S l + (b − ρ) F + T U + U. (3.18) ρ ρ d 2 ρ

We first consider the case where σ > 0. Substituting (3.18) for R in (3.6), and using Lemma A.1 (3) with a positive constant β ≠ a; relations EaEβR = EβEaR and EaTρU = 0, we see that 2 F = E (F ∗ J + T S l + U) + (b − ρ + β − a)E F. σ2 β ρ ρ d β

Since L ≡ 0, we identify that Jρ = Tρ ◦ νS according to Lemma A.2. We choose β = a − b + ρ = 2c/σ2 + ρ > 0 to obtain that 2 F = E (F ∗ J + T S l + U) σ2 β ρ ρ d 2 2 = F ∗ E J + E T S l + F (0)E △ (3.19) σ2 β ρ σ2 β ρ d β 0 The last equality is due to Remark A.1 since F is defined only on (0, ∞). Substituting with H(x) = F (x − d) in the last equality, and using Lemma A.1(6, 7), we can obtain the renewal-type equation (3.14). Now we consider the case where σ2 = 0. In this case, equation (3.5) is written as

′ ′ c(b − D)F = Fν + Sdl, b = δ/c.

Lemma A.1 (2) yields that

′ ′ cF = Tb′ R ,R = Fν + Sdl

Applying Lemma A.1 (4), we see that

′ ′ cF = TρR + c(ρ − b )TρF, and Lemma A.3 with L ≡ 0 yields that

cF = F ∗ Jρ + [ψX−x(−ρ) − δ]TρF + TρSdl.

Hence, choosing ρ as the nonnegative solution to (3.13), we have proved the result. 14 Generalization from ruin to default in a L´evyrisk model

Remark 3.1. We present alternative arguments for the derivation of the formulas (3.15) and (3.16) for σ2 = 0 which seems to be limiting case of σ2 → 0. For any function f, it follows that ∫ ∫ x ∞ 2 E 2 −βx βy e 2 βf(x) = 2 e e f(y) dy = fσ(x, z) dz, σ σ 0 0

e −(c+σ2ρ/2)z 2 2 where fσ(x, z) = e f(x − σ z/2)I(z ≤ 2x/σ ). Suppose that f is continuous. Then, for any x > 0, there exists a constant M > 0 such that

e −cz fσ(x, z) ≤ Me ∈ L1(0, ∞).

Therefore, by the dominated convergence theorem, we get

2 1 E f(x) → f(x), σ2 → 0, σ2 β c for each x > 0. Since it is clear that functions Tρ ◦ νS and Tρl are continuous and locally bounded, we also obtain the results for σ2 = 0 from those for σ2 > 0 by letting σ2 → 0.

Lemma 3.2. The renewal equation (3.14) is defective, i.e. when σ2 > 0,

∫ ∞ 2 E {T ◦ } 2 β ρ νS (y)dy < 1; σ 0 and when σ2 = 0, ∫ 1 ∞ Tρ ◦ νS(y)dy < 1. c 0 Proof. We only prove the case where σ2 > 0 and the arguments for σ2 = 0 can be carried out in a similar manner. It follows from Fubini’s theorem that ∫ ∫ ∞ 1 ∞ ψ (0) − ψ (ρ) E (T ◦ ν )(y)dy = (T ◦ ν )(y)dy = S S β ρ S β ρ S βρ 0 [∫0 ∫ ] 1 ∞ ∞ σ2 = e−ρ(z−y)ν (dz)dy + ρ β S 2 ∫ 0 ∫ y 1 ∞ ∞ σ2ρ ≤ ν (dz)dy + β S 2β ∫0 y 1 ∞ σ2ρ = zνS(dz) + . β 0 2β ∫ ∞ Hence the net profit condition: c > 0 zνS(dz), yields that ∫ ( ∫ ) ∞ ∞ 2 2 2 E T ◦ ≤ 2 1 σ ρ 2c/σ + ρ 2 β( ρ νS)(y)dy 2 zνS(dz) + < = 1. σ 0 σ β 0 2β β Runhuan Feng and Yasutaka Shimizu 15

Corollary 3.1. The explicit solution to H is hence given by ∫ 1 H(x) = hl(x − y) dGδ(y), x ≥ d, (3.20) 1 − p [0,x−d) ∫ ∞ −1 where p = 0 g(y) dy, qδ(x) = p g(x), and Gδ is the associated compound geometric distribution function: ∫ ∑∞ x − − k ∗k ≥ Gδ(x) = (1 p) + (1 p)p qδ (y) dy, x 0. k=1 0 Readers are referred to Theorem 9.1.1 by Willmot and Lin [31] for its derivation. Corollary 3.2. Suppose that the assumptions of Theorem 3.1 are satisfied. Then [∫ ] ∫ τ0 ∞ x −δt (δ) E e l(Xt)dt = r (x, y)l(y)dy, (3.21) 0 0 where the resolvent density r(δ)(x, y) for the process X killed on exiting [0, ∞) is given by ∫ (δ) 1 r (x, y) = Bδ(x − z, y)dGδ(z). (1 − p) [0,x)

2 When σ > 0, the function Bδ is given by  2  e−βx(eβy − e−ρy), 0 < y < x; σ2(β + ρ) Bδ(x, y) =  2  (eρx − e−βx)e−ρy, x ≤ y. σ2(β + ρ)

When σ2 = 0, 1 B (x, y) = eρ(x−y)I(x ≤ y). δ c The proof is almost the same as that of Corollary 3.1 of Feng [13] and hence omitted.

3.4 A solution via the scale function In this subsection, we consider the case where Z is a general spectrally negative L´evy process for which ∫ 1 zνZ (dz) ≤ ∞. 0 The solution solution to (3.5) relies on the scale function, denoted by W (δ). ∫ ∞ 1 e−sxW (δ)(x)dx = , s > ρ. (3.22) 0 ψX−x(−s) − δ Since W (δ)(0) > 0 when Z has bounded variation (cf. Kyprianou [27, Lemma 8.6]), we may introduce the measure with slight abuse of notation ∫ x ′ W (δ)[0, x) = W (δ)(0) + W (δ) (y)dy. 0 16 Generalization from ruin to default in a L´evyrisk model

Theorem 3.2. Suppose that X is a L´evyrisk process given in (3.1) with the net profit ∫ ∞ condition c > 0 z νS(dz), and the assumptions of Theorem 2.1 are satisfied. Moreover, suppose that the scale function W (δ) is differentiable. Then ∫ x−d σ2 H(x) = Tρl(x − y)K(dy) + H(d)K[0, x − d), x > d, (3.23) 0 2 where the measure K is defined by K(dx) = W (δ)(dx) − ρW (δ)(x)dx. Proof. We use the same notation as in the previous section. Taking the Laplace transform in the both sides of the equation (3.7), and applying Lemma A.3, we have that ( ) ψ (s) − ψ (ρ) σ2 LR = LF S+L S+L + LT S l + (b − ρ)L F + T U + LU ρ − s ρ d 2 ρ

+ (ψX−x(−ρ) − δ) LTρF, Here, we choose ρ as the unique nonnegative root to (3.13) to obtain that ( ) ψ (s) − ψ (ρ) σ2 LR = LF S+L S+L + LT S l + (b − ρ)L F + T U + LU, ρ − s ρ d 2 ρ Taking the Laplace transform in both sides of (3.6) and applying Lemma A.2 (1) gives ( ) σ2 ψ (s) − ψ (ρ) σ2 (s + a)LF = LF S+L S+L + LT S l + (b − ρ)L F + T U + LU, 2 ρ − s ρ d 2 ρ which simplifies to ψ − (−s) − δ X x LF (s) = M(s), s ≥ 0. s − ρ where it follows from Lemma A.2 (2) that (b − s)LU(s) − (b − ρ)LU(ρ) σ2 M(s) = LT S l(s) + = LT S l(s) + F (0). ρ d ρ − s ρ d 2 Note from the definition of W (δ) and integration by parts that ∫ s ∞ = e−sxW (δ)(dx). ψX−x(−s) − δ 0 Thus, ∫ s − ρ ∞ = e−sxK(dx). ψX−x(−s) − δ 0 Therefore, using Lemma A.1 (6), ∫ ∫ ∫ ∞ x 2 ∞ −sx σ −sx LF (s) = e SdTρl(x − y)K(dy) dx + F (0) e K(dx). 0 0 2 0 Inverting the Laplace transform, we obtain the result together with F (x) = H(x + d).

(δ) 1 Remark 3.2. W ∈ C (0, ∞) if Z ̸≡ 0, or νS has no atom when Z ≡ 0. In particular, W (δ) ∈ C2(0, ∞) if σ2 > 0. See, e.g., Kyprianou [22] for details. Runhuan Feng and Yasutaka Shimizu 17

4 Connecting classic and fluctuation-theoretic solutions

We confirm in this subsection that the results obtained with operator-based arguments are consistent with those by fluctuation theoretic approach in the SNL risk model.

Proposition 4.1. Suppose that Z is a Brownian motion or trivially zero. When σ2 > 0, the scale function for the risk process (3.1) is given by ∫ (δ) 2 ρ(x−z) − −β(x−z) W (x) = 2 (e e )dGδ(z), (4.1) σ (1 − p)(β + ρ) [0,x)

When σ2 = 0, ∫ (δ) 1 ρ(x−z) W (x) = e dGδ(z). (4.2) c(1 − p) [0,x)

Proof. We show that the expression in (4.1) matches the definition of the scale function. ∫ [ ] ∞ − L −sx (δ) 1 − (1 p)p qδ(s) e W (x)dx = 2 (1 p) + 0 (σ /2)(1 − p)(s − ρ)(s + β) 1 − pLqδ(s) 1 1 = , for s > ρ. (σ2/2)(s − ρ)(s + β) 1 − Lg(s)

Note that 2(ψ (s) − ψ (ρ)) 1 − Lg(s) = 1 − S S σ2(β + s)(ρ − s) (σ2/2)(β + s)(ρ − s) − (ψ (s) − ψ (ρ)) = S S (σ2/2)(β + s)(ρ − s) (σ2/2)ρ2 + cρ + ψ (ρ) − (σ2/2)s2 − cs − ψ (s) = S S (σ2/2)(β + s)(ρ − s) δ − ψ − (−s) = X x . (σ2/2)(β + s)(ρ − s)

Hence we arrive at the anticipated expression in (3.22). The formula (4.2) for σ2 = 0 can be obtained by passing to limit in (4.1).

Corollary 4.1. Suppose that X is a L´evyrisk process given in (3.1) with the net profit ∫ ∞ condition c > 0 z νS(dz), and the assumptions of Theorem 2.1 are satisfied. For any measurable function l, we have [∫ ] ∫ τ0 ∞ x −δt (δ) E e l(Xt)dt = r (x, y)l(y)dy, 0 0 where the resolvent density r(δ)(x, y) for the process X killed on exiting [0, ∞) is given by

r(δ)(x, y) = e−ρyW (δ)(x) − W (δ)(x − y), for x, y ≥ 0. (4.3) 18 Generalization from ruin to default in a L´evyrisk model

Proof. Let d = 0 and H(d) = 0. In the case that Z is a Brownian motion, substituting (4.1) and (4.2) into (4.3) respectively and using the fact that W (δ)(x) = 0, for all x < 0 produces precisely the expressions in (3.21). Similarly, in the case that Z is a general spectrally negative L´evyprocess, exchanging the order of integration in (3.23) and using the fact that W (δ)(x) = 0, for all x < 0 yields the desired expression (4.3).

Note that the expression (4.3) agrees with that in Corollary 8.8 of Kyprianou [27], although our approach differs significantly from that of Kyprianou [27]. To the best knowledge of the authors, the explicit expression of W (δ)(x) in terms of compound geometric distribution for the SNL risk model (3.1) appears to be new. We believe this simple result of Proposition 4.1 bridges a gap in the literature between ruin theoretic results and those of fluctuation theory. The series representation of the scale function can serve two purposes. (1) The repre- sentation explicitly shows the existence of the scale function, which may not be obvious due to its definition via Laplace transform. (2) It can be used for asymptotic and numer- ical analysis of the scale function when analytical inversion of the Laplace transform is impossible. Those approximations may be compared with numerical inversion of Laplace transforms.

Remark 4.1. We can also quickly retrieve many well-known results on the probability of ultimate survival from (4.1) by letting δ → 0. Since the unique non-negative solution to the Lundberg equation (3.13) in the case of δ = 0 is ρ = 0, we obtain ∫ (0) 1 −(2c/σ2)(x−y) W (x) = (1 − e )dG0(y), c(1 − p) [0,x) ∫ −1 ∞ where p = c 0 zνS(dz), which is clear from the proof of Lemma 3.2. It follows from a known identity (cf. Kyprianou [27, (8.15)]) that ∫ P{ ∞} ′ (0) − −(2c/σ2)(x−y) φ(x) := τ0 = = ψX−x(0)W (x) = (1 e )dG0(y). (4.4) [0,x)

It is easy to show using Laplace transforms that (4.4) is a generalization of the Pollaczek- Khinchin formula in disguise. Interested readers may read Huzak et al. [21, Theorem 3.1], Yang and Zhang [36, Theorem 1 & 2] for various forms of the generalized Pollaczek- Khinchin formula. However, one should be aware of the distinction between the definition of the convolution in their papers and that of ours.

Remark 4.2. Although the representation shown in Corollary 3.2 could have been repro- duced using known results on the scale function and the identities in Proposition 4.1, it is essentially required that H(0) = 0 and the function H for the rest of the paper is by definition more general to include cases where H(0) ≠ 0. For example, the results in Corollary 5.1 requires the cost function l to be the Dirac delta function, which turns out to be convenient in handling the issue of creeping. Runhuan Feng and Yasutaka Shimizu 19

5 Applications

5.1 Relationship with the classical Gerber-Shiu function The classical Gerber-Shiu expected discounted penalty function is defined by

− Ex δτ0 | | ∞ m(x) := [e w(Xτ0−, Xτ0 )I(τ0 < )], (5.1) where w is a bounded measurable function with w0 = w(0, 0). In the case of jump diffusion model, it is known to be composed of a jump component and a creeping component

m(x) = mj(x) + mc(x) − − Ex δτ0 | | ∞ Ex δτ0 ∞ := [e w(Xτ0−, Xτ0 )I(τ0 < ,Xτ0 < 0)] + w0 [e I(τ0 < ,Xτ0 = 0)].

Proposition 5.1. Suppose that X is a L´evyrisk process given in (3.1). Then the function m defined in (5.1) is a special case of H defined in (2.8) with d = 0 and

∫ ∞ l(x) = w0△0(x) + w(x, z − x) νS+L(dz), (5.2) x where △0 the Dirac delta function concentrated on the single point {0}.

Proof. Denote all jump contributions by ∆St := St − St− and ∆Lt := Lt − Lt−. Hence we can rewrite mj as [ ∑ x −δt mj(x) = E e ϖ(Xt−,Xt− + ∆St)I(∆St > 0) 0 0) ≤ 0[

Remark 5.1. Substituting (5.2) into (3.14) yields precisely the same DRE as the one in Corollary 4.1 by Biffis and Morales [4].

Remark 5.2. If w is bounded and continuous, then l given by (5.2) is continuous on (0, ∞), and l satisfies Condition (2.9) in Theorem 2.1.

Corollary 5.1. The probability of ultimate ruin due to jumps is given by ∫ P{ ∞ } − − p −(2c/σ2)(x−y) φj(x) := τ0 < ,Xτ0 < 0 = 1 G0(x) e dG0(y). (5.3) 1 − p [0,x)

The probability of ultimate ruin due to creeping is given by ∫ P{ ∞ } 1 −(2c/σ2)(x−y) φc(x) := τ0 < ,Xτ0 = 0 = e dG0(y). (5.4) 1 − p [0,x)

Proof. We focus on the following version of the Gerber-Shiu function

− − Ex δτ0 | | ∞ Ex δτ0 ∞ m(x) = [e w( Xτ0 )I(τ0 < ,Xτ0 < 0)] + w0 [e I(τ0 < ,Xτ0 = 0)]. ∫ x Let Qδ(x) = 0 qδ(y)dy where qδ is defined in Corollary 3.1. Using nearly identical argu- ments to those shown in Feng [13, Corollary 3.3], we can prove that ∫ ∫ p ∞ m(x) = w(z) qδ(z + x − y)dGδ(y)dz 1 − p 0 [0,x) [ ∫ ∞ ] ∫ 1 −β(x−y) + w0 − p w(−y)dQδ(y) e dGδ(y). 1 − p 0 [0,x)

The result (5.3) follows by letting δ = 0, w(x) = I(x > 0) and w0 = 0. Similarly the result (5.4) follows by letting δ = 0, w(x) = I(x = 0) and w0 = 1.

Remark 5.3. The decomposition of the ruin probabilities appears to be new for the spec- trally negative L´evyprocess (3.1). It is easy to check that the sum of the two probabilities (5.3) and (5.4) gives the probability of ultimate ruin which is consistent with the result on the probability of ultimate survival (4.4), i.e.

1 − φ(x) = φj(x) + φc(x).

The easy calculation may be viewed as another advantage of the operator-based analysis.

5.2 Relationship with total dividends paid up to ruin The significance of the generalization from the Gerber-Shiu function (5.1) to the function H (2.8) is to include many other default/ruin related quantities such as the expected present value of dividends paid up to ruin defined by [∫ ] τ0 x −δt V (x) = E e lb(Xt)dt , (5.5) 0 Runhuan Feng and Yasutaka Shimizu 21

where in the dividend threshold model lb is given by lb(x) = αI(x ≥ b). The infinitesimal generator of the risk model (3.1) with a dividend threshold is ∫ 2 ∞ σ ′′ ′ Af(x) = f (x) + [c − αI(x ≥ b)]f (x) + {f(x − z) − f(x)} νS(dz) 2 0 ∫ ∞ ′ + {f(x − z) − f(x) + zf (x)} νZ (dz). 0 Corollary 5.2. Suppose that X is a L´evyrisk process given in (3.1), the function V defined in (5.5) is given by ∫ 2α x V (x) = (1 − e−β(x−b)) + V (b)e−β(x−b) + V (x − y)g (y) dy σ2ρβ 1 ∫ 0 b −β(x−b) −e V (b − y)g1(y) dy, x > b; (5.6) 0 and ∫ x 1 −β(x−y) V (x) = C e dGδ(y), 0 ≤ x < b, (5.7) 1 − p 0 where the constants C and V (b) are determined by

V (b−) = V (b+); (5.8) σ2 σ2 V ′′(b+) + (c − α)V ′(b+) + α = V ′′(b−) + cV ′(b−). (5.9) 2 2

Proof. Since lb is bounded, the condition (2.9) is satisfied. Hence according to Theorem 3.1 the function V must satisfy the system of equations, ∫ 2 ∞ σ ′′ ′ V (x) + (c − α)V (x) + {V (x − z) − V (x)} νS(dz) 2 0 ∫ ∞ ′ + {V (x − z) − V (x) + zV (x)} νZ (dz) − δV (x) + α = 0, x ≥ b; (5.10) 0 ∫ 2 ∞ σ ′′ ′ V (x) + cV (x) + {V (x − z) − V (x)} νS(dz) 2 0 ∫ ∞ ′ + {V (x − z) − V (x) + zV (x)} νZ (dz) − δV (x) = 0, 0 ≤ x < b. (5.11) 0 Introducing F (x − b) = V (x) for all x ≥ 0 and letting y = x − b in (5.10), we obtain

σ2 F ′′(y) + (c − α)F ′(y) + F (y) − δF (y) + α = 0, y > 0, 2 ν which is in fact eq. (3.5) with l(x) = α and d = 0. However, one should note that F (y) = 0 for y < −b due to the shift in the argument. Therefore, it follows from (3.19) and Lemma A.4 that the function F must satisfy the renewal equation ∫ ∫ x+b 0 2α − −βx −βx − − −βx − F (x) = 2 (1 e )+F (0)e + F (x y)g1(y) dy e F (z)g1( z) dz, x > 0, σ ρβ 0 −b 22 Generalization from ruin to default in a L´evyrisk model

2 where β = 2(c−α)/σ +ρ1 and ρ1 is the unique non-negative solution to (c−α)s+ψS(s)+ ψZ (s) = δ and g1 is defined in (3.15) with ρ replaced by ρ1. Substituting F (x − b) by V (x) yields (5.10). Using a similar technique as in Lin and Pavlova [28], we see that the solution to V (x) for 0 ≤ x < b is a solution of the homogeneous integro-differential equation (5.11) and hence must be a multiple of the solution F to the same equation with an extended domain

σ2 F ′′(x) + cF ′(x) + F (x) − δF (x) = 0, x > 0, 2 ν which again by Theorem 3.1 has a solution with the boundary condition F (0) = 1, ∫ x 1 −β(x−y) F (x) = e dGδ(y), x ≥ 0. 1 − p 0 Therefore, V (x) for 0 ≤ x < b must be given by (5.11). In the end, we determine the two unknown coefficients C and V (b) by two conditions (5.8) and (5.9).

5.3 An asymptotic result Suppose that Z is a Brownian motion throughout this subsection. Then the DRE obtained in (3.14) also enables us to study the asymptotic behavior of H as in the classical theory with light tailed claim distribution. If H is chosen as a proper metric for quantifying and assessing insolvency risks, then the asymptotic result may indicate the magnitude of riskiness as the surplus level varies.

Theorem 5.1. Suppose that there exists a constnat R ∈ (0, β) such that

ψX−x(R) = δ, where β > 0 is given in Theorem 3.1, and that ∫ ∞ eRul(u)du < ∞. (5.12) 0 Then we must have the asymptotic result ∫ ∞ 0 A(u)du −R(u−d) H(u) ∼ ∫ ∞ e , u → ∞, 0 udB(u) where { [ ] } Ru 2 E T 2 E T −βu A(u) = e 2 β ρl(u) + F (0) + 2 β ρl(0) e ; ∫ σ σ u B(u) = eRyg(y) dy. 0 Proof. First, let us observe that ∫ ∞ eRyg(y) dy = 1. 0 Runhuan Feng and Yasutaka Shimizu 23

Recall that constants R and ρ satisfy the equalities

2 σ 2 ψ − (R) = R − cR + ψ (−R) = δ; X x 2 S 2 σ 2 ψ − (−ρ) = ρ + cρ + ψ (ρ) = δ, X x 2 S since Z is now a Brownian motion. Hence it follows from Lemma A.2 that ∫ ∞ Ru 2 LE T ◦ − e g(u)du = 2 β( ρ νS)( R) 0 σ 2 ψ (−R) − ψ (ρ) = S S σ2 (β − R)(R + ρ) cR − σ2R2/2 + σ2ρ2/2 + cρ = = 1. σ2ρ2/2 + cρ + cR − σ2R2/2

Define L(u) = eRuF (u). We multiply through (3.19) by eRu to obtain ∫ u L(u) = A(u) + L(u − x)dB(u). 0 Since it follows from the assumption and Lemma A.2 that Ll(−R) − Ll(ρ) LE T l(−R) = < ∞, β ρ (β − R)(R + ρ) ∫ ∞ ∞ we see that 0 A(u) du < . Moreover, it is also easy to see that A is directly Rieman integrable since R < β. Then the key renewal theorem (e.g., Lin and Willmot [26], Theorem 9.1.3) implies that ∫ ∞ 0 A(u) du lim L(u) = ∫ ∞ =: C, u→∞ 0 u dB(u) which means F (u) ∼ Ce−Ru ⇔ H(u) ∼ Ce−R(u−d).

5.4 Statistical inference Estimation of H from a set of data of risk process is one of the important issues in practice since the expression of H given in Theorem 3.2 includes many unknown quantities: ρ, σ2 and W (δ). From a practical point of view, these values are to be estimated from past data of the surplus process. Earlier work on this issue is found in e.g., Croux and Veraverbeke [9], Bening and Kolorev [3], Politis [30], Mnatsakanov et al. [25] for estimation of ruin probability, Shimizu [32, 33] for the classical Gerber-Shiu function for perturbation risk models. In particular, Shimizu [33] proposed a functional estimator for the classical Gerber-Shiu function in the same model as ours, spectrally negative L´evyrisk model, from discrete data: n ∪ n ∪ n Dn = X JS (ϵn) JZ (ϵn), 24 Generalization from ruin to default in a L´evyrisk model

n where X is a discrete sample from the path of the surplus X: for some ∆n > 0,

n { | } X = Xi∆n i = 0, 1, . . . , n ;

n JU (ϵn)(U = S, Z) is a set of jumps of a process U which are larger than a level ϵn > 0: n { | ∈ } JU (ϵn) = ∆Ut t [0, n∆n], ∆Ut > ϵn , where ∆Ut := Ut − Ut− for any t > 0. Since the Laplace transform of H is obtained, the same procedures as shown in Shimizu [33] are applied to the current case. Without loss of generality, we can assume that d = 0; see Remark 5.5. From the proof of Theorem 3.2, we see that [ ] L − L 2 − L l(s) l(ρ) σ ρ s H(s) = + H(0) 2 2 s − ρ 2 (σ /2)s + cs + ψS(s) + ψL(s) − δ

c2 c Using corresponding estimators σ , ρb, and ψU ’s (U = S, L) proposed in Lemmas 3.1– 3.3 by Shimizu [33], we can construct an estimator of LH, say LdH, by replacing the unknowns with the corresponding estimators, such that √ ( ) d n∆n LH(s) − LH(s) = OP(1), n → ∞, for each s ≥ 0 under a suitable asymptotics with n∆n → ∞ and ϵn ↓ 0. Thanks to the same device as in Shimizu [33], Section 4.1, the following Hb is a candidate of a “consistent” estimator of H:

b ϑxL−1 Ld · H(x) = e mn [ H(ϑ + )](x), (5.13)

L−1 where m is the regularized Laplace inversion given by Chauveau et al. [7], (3.5); mn and ϑ are positive numbers such that mn → ∞ as n → ∞, and that (c − E[S1])ϑ > δ. To show a “consistency” in rigorous sense, we need to check some regularity conditions as in Lemma 4.1 in Shimizu [33].

0 Remark 5.4. Estimation of H(0) depends on the function l. If Z ̸≡ 0, then P (τ0 = 0) = 1 by the regularity of the point x = 0 for (−∞, 0), and H(0) is given by one of the two cases: (1) If H is the classical Gerber-Shiu function as in (5.1) then H(0) = w0; (2) If l(0) is finite then H(0) = 0 by definition. However, if Z ≡ 0 then H(0) is not always clear except for some special cases: (1) If w ≡ 1 and δ = 0 (H is the probability of ruin) P0 ∞ −1 ′ ′ then H(0) = (τ0 < ) = c ψS(0); see Huzak et al. [21], and we can estimate ψS(0) c ′ by ψS (0); (2) Considering the classical Gerber-Shiu function for the Cram`er-Lundberg −1 risk model∫ where S is a , we see that H(0) = c Lω(ρ), where ∞ − ω(x) := x w(x, z x) νS(dz) which is also easy to be estimated; see Shimizu [33].

Remark 5.5. When d > 0, we can apply the same argument to the shifted function F (x) = H(x − d), with the cost function l∗(x) = l(x + d), and we have an estimator Fb(x) as in (5.13). Then we obtain Hb(x) = Fb(x + d). Runhuan Feng and Yasutaka Shimizu 25

Finally, we also remark that the statistical estimation of the adjustment coefficient R > 0 given in the previous section, is possible in a manner similar to the way of estimating ρ; see Section 3.3 in Shimizu [33]. The estimator is given as a positive solution to the following random equation in r, which is a estimated version of the Lundberg equation (3.13): σb2 r2 − cr + ψc(−r) + ψc(−r) = δ. 2 S L We note that the above equation has a unique positive root asymptotically as n → ∞ under the net profit condition. Under some further regularities, we can√ show that the b solution R is consistent for R, and asymptotically normal with the rate n∆n:, √ b d n∆n(R − R) → ΣZ, where Z is a standard normal random variable, Σ is a constant written explicitly in terms of c, σ, νS, νZ and R. This is an extension of the result by Grandell [20], Theorem 1.24.

6 Concluding remarks

It is worthwhile pointing out that there has been a wide variety of analytical tools used in ruin theory ranging from the Laplace transform techniques commonly seen in Cram`er- Lundberg risk model to the more recent fluctuation theoretic methods in L´evyrisk models. While all these tools have their own merits, we want to highlight the fact that the operator- based approach taken in this paper to deduce a renewal equation has been consistently developed across the board for a whole range of well-known risk models. See, for exam- ple, piecewise-deterministic compound Poisson model in Cai et al. [6], compound Poisson model perturbed by diffusion in Feng [13]. The matrix operator version of this approach has been shown to be concise and straightforward in phase-type renewal risk model in Feng [11] and in latest work by Feng and Shimizu [14] for a more general Markov additive risk process. Albrecher et al. [1] also used a similar operator-based approach with the same set of operators as ours in a fairly general renewal risk model. In addition to its advantage of accessibility to a broader audience, the operator-based approach appears to have the benefit of consolidating a wide variety of risk models with a consistent technical treatment.

Acknowledgement

The authors appreciate the helpful comments and suggestions from anonymous referees that have improved the presentation of this paper. They are also grateful to Dr. Xiaowen Zhou for making them aware of the development of fluctuation theory in the ruin context. This research has been supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 2011, no.21740073, and Japan Science and Technology Agency, CREST, and in part by the 2010 Research Grant of the Actuarial Foundation. 26 Generalization from ruin to default in a L´evyrisk model

A Appendix: properties of operators

The following lemmas give useful relations among various operators. The proofs are ele- mentary and hence omitted.

Lemma A.1. Let △0 be the Dirac delta function, s, t ≥ 0 and d ∈ R. Then the following properties hold for any continuously differentiable function f:

(1) Es(sI + D)f = f − f(0)Es△0;

(2) Ts(sI − D)f = f;

(3) (s − t)EsEtf = Esf − Etf, where s ≠ t;

(4) (s − t)TsTtf = Ttf − Tsf, where s ≠ t;

(5) (s + t)TtEsf = Esf + Ttf;

(6) TsSdf = SdTsf;

(7) EsSdf = SdEsf − Esf(d)Es△0.

Proof. See Feng [13].

Lemma A.2. The following identities are true for s ≠ t 1 (1) LE f(s) = Lf(s); t s + t Lf(s) − Lf(t) (2) LT f(s) = ; t t − s ψ (s) − ψ (t) (3) L{T ◦ ν }(s) = S S , t S t − s provided that all integrals exist.

To accommodate functions with support on negative real line, we use the following definition of bilateral Laplace transform only in the next lemma.

∫ ∞ Lf(s) = e−suf(u) du. −∞

Lemma A.3. For any constant ρ > 0 and a function F defined on (−∞, ∞) such that TρF and LF are well-defined, we have

ψ (s) − ψ (ρ) LT F (s) = LF (s) S+L S+L + ψ (ρ)LT F (s), s ≠ ρ, ρ ν ρ − s S+L ρ where Fν is defined in the same form as (3.4). Runhuan Feng and Yasutaka Shimizu 27

Proof. Note that ∫ [∫ ] ∞ ∞ { } ρu −ρy −ρz −ρz TρFν(u) = e e F (y − z) − (1 − e + e )F (y) νS(dz) dy u 0 ∫ ∞ [∫ ∞ ] ρu −ρy −ρz −ρz ′ +e e {F (y − z) − (1 − e − ρz + e + ρz)F (y) + zF (y)}νZ (dz) dy u 0 = I1(u) + I2(u) + ψS+L(ρ)TρF (u) where I1 and I2 are defined as ∫ [∫ ] ∞ ∞ { } ρu −ρy −ρz I1(u) := e e F (y − z) − e F (y) νS(dz) dy, u 0 ∫ ∞ [∫ ∞ ] ρu −ρy −ρz ′ I2(u) := e e {F (y − z) − (e + ρz)F (y) + zF (y)}νZ (dz) dy. u 0

Taking the bilateral Laplace transform of I1 and I2, we obtain that ∫ (∫ ∫ ) ∞ ∞ ∞ { } −(s−ρ)u −ρy −ρy LI1(s) = e e F (y − z) − e F (y) dy du νS(dz) 0 −∞ u ∫ ∞ (∫ ∞ {∫ ∞ ∫ ∞ } ) −(s−ρ)u −ρw −ρy −ρz = e e F (w) dw − e F (y) dy du e νS(dz) ∫0 (∫−∞ ∫ u−z ) u ∞ ∞ u −(s−ρ)u −ρy −ρz = e e F (y) dy du e νS(dz) ∫0 (∫−∞ ∫u−z ) ∞ ∞ y+z −ρy −(s−ρ)u −ρz = e F (y) e du dy e νS(dz) 0 −∞ y ∫ ∞ ( ∫ ∞ ∫ ∞ ) 1 −sz −sy −ρz −sy = e e F (y) dy − e e F (y) dy νS(dz) ρ − s 0 −∞ −∞ (∫ ∞ ∫ ∞ ) ∫ ∞ 1 −sz −ρz −sy = {e − 1}νS(dz) − {e − 1}νS(dz) e F (y) dy ρ − s 0 0 −∞ ψ (s) − ψ (ρ) = S S LF (s), ρ − s and that ∫ [∫ ∫ ∫ ] ∞ ∞ u+z ∞ −(s−ρ)u −ρy −su LI2(s) = e e F (y − z)dydu − e F (u)zdu νZ (dz) 0 −∞ u −∞ ∫ ∞ [ ∫ ∞ ∫ ∞ ] 1 −sz −su −ρz −su = {e e F (u) du − e e F (u) du} − LF (s)z νZ (dz) 0 ρ − s −∞ −∞ ∫ ∞ [ ] 1 −sz −ρz = {e LF (s) − e LF (s)} − LF (s)z νZ (dz) 0 ρ − s [∫ ∞ ∫ ∞ ] 1 −sz −ρz = LF (s) (e + sz − 1)νZ (dz) − (e + ρz − 1)νZ (dz) ρ − s 0 0 ψ (s) − ψ (ρ) = L L LF (s). ρ − s Hence we have ψ (s) − ψ (ρ) L (I + I ) = LF S+L S+L , 1 2 ρ − s 28 Generalization from ruin to default in a L´evyrisk model

This completes the proof.

Lemma A.4. For constants β, ρ > 0 and a function F defined on (−∞, ∞), we must have ∫ 0 −βx Eβ (F ∗ {Tρ ◦ νS}) = F ∗ Eβ {Tρ ◦ νS} − e F (z)Eβ {Tρ ◦ νS} (−z) dz. −∞

Proof. Denote Kρ = Tρ ◦ νS. We have

E (F ∗ K )(x) β ρ∫ [∫ ] x ∞ −βx βy = e e F (y − z)Kρ(z) dz dy ∫ 0 [∫0 ] x y −β(x−y) = e Kρ(y − z)F (z) dz dy ∫0 ∫ −∞ ∫ ∫ 0 x x x −β(x−y) −β(x−y) = F (z) e Kρ(y − z) dy dz + F (z) e Kρ(y − z) dy dz ∫−∞ ∫0 ∫0 ∫z 0 x−z x x−z −β(x−z−u) −β(x−z−u) = F (z) e Kρ(u) du dz + F (z) e Kρ(u) du dz −∞ −z ∫ 0 0 0 −βx = F ∗ EβKρ − e F (z)EβKρ(−z) dz. −∞

Note that EβKρ(x) = 0 for x ≤ 0 since Kρ only has support on (0, ∞).

Remark A.1. If F only has support on positive half-line, then

Eβ (F ∗ {Tρ ◦ νS}) = F ∗ (Eβ {Tρ ◦ νS}) .

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