An ODE Approach for the Expected Discounted Penalty at

Ruin in Jump Diffusion Model

Yu-Ting Chen, Institute of FinanceNational Chiao Tung University, Hsinchu, Taiwan

Cheng-Few Lee, Department of FinanceRutgers University, New Brunswick, NJ, USA

Yuan-Chung Sheu∗, Department of Applied MathematicsNational Chiao Tung University, Hsinchu, Taiwan

Abstract

For a general penalty function, the expected discounted penalty at ruin was consid-ered by, for example, Gerber and Shiu(1998) and Gerber and Landry (1998) in insuranceliterature. On the other hand, many pricing functionals in mathematical finance(e.g.,options pricing, credit risk modelling) can be formulated in terms of expected discountedpenalties. Under the assumption that the asset value follows a jump diffusion, we showthe expected discounted penalty satisfies a homogeneous ODE. Based on ODE the-ory, we obtain a general form for the expected discounted penalty. In particular, ifonly downward phase-type jumps are allowed, we obtain an explicit formula in termsof the penalty function. On the other hand, if downward jump distribution is a mix-ture of exponential distributions (and upward jumps are determined by a general Levymeasure), we obtain closed form solutions for the expected discounted penalty. Forearlier and related results, see Gerber and Landry(1998), Hilberink and Rogers(2002),Mordecki(2002), Kou and Wang(2004), Asmussen et al.(2004) and others.

Key words: jump diffusion, integro-differential equation, expected discounted penalty,phase-type distribution

JEL Classification: D92, G32, G33Mathematics Subject Classification(2000): 60G40Running Title: Expected Discounted Penalty at Ruin

∗Corresponding author. Tel.: +886-3-5712121x56428; fax: +886-3-5724679. E-mail address:sheu@math.nctu.edu.tw

1

1 Introduction

In the classical model of ruin theory, the process

Xt = x+ ct− Zt, t ≥ 0, (1.1)

stands for the surplus process of an insurance company. Here, x > 0 is the initial surplus,c > 0 is the rate at which the premiums are received, and Z = (Zt; t ∈ R+) is a compoundPoisson process which represents the aggregate claims between time 0 and t. (Note thatin insurance Z has only downward jumps.) Ruin is the event that Xt ≤ 0 for some t > 0.Let τ be the time of ruin and Xτ the negative surplus when ruin occurs. Given a penaltyscheme g, Gerber and Shiu(1998) considered the expected discounted penalty

Φ(x) = E[e−rτg(Xτ )

]. (1.2)

Here, r is the risk-free rate and we use the convention that e−r·(+∞) = 0. By taking g ≡ 1and r = 0, the ruin probability is a special case of (1.2). Results about this and relatedproblems can be found in Asmussen(2000).

Gerber(1970) extended the classical model (1.1) by adding an independent diffusion.And the surplus process takes the form

Xt = x+ ct+ σWt − Zt, t ≥ 0. (1.3)

Here σ > 0 and W = (Wt; t ≥ 0) is a standard Brownian motion that is independent ofZ. In this case ruin may be caused by oscillation (that is, Xτ = 0) or by a claim (that is,Xτ < 0). Dufresne and Gerber(1991) studied the probability of ruin caused by oscillationand the probability of ruin caused by a claim. Moreover, as in Gerber and Shiu(1998), Ger-ber and Landry(1998) considered the discounted expected penalty (1.2). They heuristicallyderived the integro-differential equation for Φ and showed that it satisfies a renewal integralequation. Based on this integral equation, they investigated the asymptotic behavior of Φand, as an application of the integral equation, obtained the optimal exercise strategy ofa perpetual American option. For extension and related results, see Wang and Wu(2000),Tsai and Wilmott(2002), Chan(2005) and others. In another interesting paper, Kou andWang(2003) considered the model (1.3) with the so-called double exponential jump distri-bution. Instead of giving the solutions Φ for all bounded Borel function g, they consideredthree cases: g ≡ 1, g ≡ 10, and g ≡ 1(−∞,y), y ≤ 0. In these cases, they gave explicitformulae for Φ. (See Theorem 3.1 in Kou and Wang(2003) and Example 2 below.)

In addition to the study of ruin problem, the function (1.2) is also considered in math-ematical finance. For example, consider a perpetual American put option with strike priceK. Let V = (Vt; t ∈ R+) be the price process of the security such that log V is a Levyprocess and τ(L) is the first entry time of (−∞, L] by V . Mordecki(2002) showed that forsome L > 0, the price of the perpetual American put option is given by

EQ[e−rτ(L)(K − Vτ(L))

+], where Q is a risk-neutral probability measure. (1.4)

Set Xt = log VtL . Then τ(L) is the time of ruin for X, and (1.4) is equal to Φ(log(V0/L)),

where g(x) = (K−Lex)+. Mordecki(2002) also gave the price formula for this option underthe assumption that X is the difference of a spectrally nonnegative Levy process and a com-pound Poisson process whose jump distribution is a mixture of exponential distributions.

2

In addition to option theory, the expected discounted penalty also finds its importancein the structural form modelling of credit risk. For example, Hilberink and Rogers(2002)extended the model of Leland (1994) by adding downward jumps in the dynamics of thefirm’s asset. More precisely, they considered a firm which has a constant debt structure,has an exponential jump diffusion asset value and issues a coupon-paying debt with finitematurity. And bankruptcy cost and tax effect are both allowed. Denote by τ(B) the firsttime that the value of the firm’s asset falls below some level B. Then, by rolling the debtindefinitely and ”exponentially”, which has an analogy of sinking fund in real world, thevalue of the debt and the value of the firm are given by

D(V,B) = Value of the Debt = C1(1− EV[e−m1τ(B)

]) + C2EV

[Vτ(B)e

−m1τ(B)],

and

F (V,B) = Value of the Firm = V + C3(1− EV[e−m2τ(B)

]) + C4EV

[e−m2τ(B)Vτ(B)

]for some constants Ci andmj > 0.(See Eq.(2.7) and Eq.(2.8) in Hilberink and Rogers(2002).)Then Hilberink and Rogers(2002) gave explicit descriptions for the two values up to Fouriertransforms. However, the Fourier inversion is not an easy work. On the other hand, setXt = log(Vt/B) and formulate D(V,B) and F (V,B) in terms of Φ in (1.2). By using the for-mulae of Φ’s(see Theorem 4.3 below), we may obtain closed form solutions of the two valuesif upward jumps are allowed and downward jump distribution is a mixture of exponentialdistributions.

The next problem in Leland’s model is the choice of optimal B maximizing the equityvalue F (V,B)−D(V,B). By imposing smooth-pasting condition, Hilberink and Rogers(2002)surprisingly derived the closed form solution of optimal default boundary in terms of theWiener-Hopf factors of a Levy process. Finally, they showed numerically that the creditspreads do not tend to zero if time to maturity goes to zero. For recent works and relatedresults, see Chen and Kou(2005), Chen et al.(2006a) and Dao and Jeanblanc(2006).

The main purpose of this paper is to study the function Φ defined in (1.2). In Section2 we introduce the process X and derive an integro-differential equation for Φ(Theorem2.2). In Section 3 we assume that the jump distribution for X is a two-sided phase-typedistribution, and we transform the integro-differential equation into an ordinary differentialequation(Theorem 3.1). Based on the ODE theory, we get the general form for Φ undersome technical conditions. If only downward jumps are allowed, we have an explicit formulafor the first order derivative of Φ at 0 in terms of g(Theorem C.1), and hence, by a recursiveformula, we get all higher order derivatives of Φ at 0(Lemma 3.2). From these, we obtain anexplicit formula for Φ in terms of g(Theorem 3.2). In Section 4 we consider a general Levyprocess X which is a difference of a spectrally nonnegative Levy process and a compoundPoisson process with only upward jumps. We assume further that the jump distributionof the compound Poisson process is a mixture of exponential distributions. Based on theresults in Section 3, we conjecture the solution form of Φ. And by using the Feymann-Kac formula we verify our conjecture and show that the value of Φ can be computed inclosed form(Theorem 4.3). Section 5 concludes the paper. It is interesting to compareour results with those of Asmussen et al.(2004). By martingale stopping and Wiener-Hopffactorization, they obtained similar and general results as ours. But for special cases, ourresults are more straightforward and give more computable results. For more details, seethe remark after Theorem 3.2 and Asmussen et al.(2004).

3

2 Integro-Differential Equation

To start with, we specify a Levy process that we consider in this paper unless otherwisestated. We are given a probability space (Ω,F ,P) on which there are a standard Brownianmotion W = (Wt; t ∈ R+) and a compound Poisson process Z = (Zt =

∑Ntn=1 Yn; t ∈

R+). Here the Poisson process N = (Nt; t ∈ R+) has parameter λ > 0 and the randomvariables (Yn;n ∈ N) are independent and identically distributed. We assume further thatthe distribution F of Y1 has a bounded density f that is continuous on R\0. In addition,W,N, (Yn) are assumed to be independent. For every x ∈ R, let Px be the law of the process

Xt = X0 + ct+ σWt − Zt, t ≥ 0, (2.1)

where c ∈ R, σ > 0 and X0 = x. Write P0 for P and Ex[Z] =∫Z(ω)dPx(ω) for a random

variable Z. For every ξ ∈ iR, we have

E0

[eζX1

]= eψ(ζ), (2.2)

whereψ(ζ) = Dζ2 + cζ + λ

∫e−ζydF (y)− λ

and

D =σ2

2. (2.3)

(ψ is called the characteristic exponent of X in literature.) Moreover, the infinitesimalgenerator L of X has a domain containing C2

0(R) and for any h ∈ C20(R),

Lh(x) = Dh′′(x) + ch′(x) + λ

∫h(x− y)dF (y)− λh(x). (2.4)

(For details, see Bertoin(1996).) On the other hand, let (Ft) be the usual augmentation ofthe natural filtration of X. Then for every Borel set A, the entry time of A by X,

τA = inft ≥ 0;Xt ∈ A, (2.5)

is an (Ft)−stopping time. (See Chapter III Theorem 2.17 in Revuz and Yor(2005).)

As stated in Section 1, the main purpose of this paper is to study the function Φ givenby

Φ(x) = Ex[e−rτg(Xτ )

], x ∈ R, (2.6)

where g is a bounded Borel measurable function on (−∞, 0], r ≥ 0 and τ = τ(−∞,0]. Notethat Φ(x) = g(x) for x ≤ 0 and Φ is called expected discounted penalty for the penaltyfunction g in insurance literature. From now on, we will fix a bounded Borel penaltyfunction g on (−∞, 0] and let Φ(x) be given by (2.6). We also write τ for τ(−∞,0].

4

Notation. We write R+ = x ∈ R;x ≥ 0, R++ = x ∈ R;x > 0, and analogously R−and R−−.

Let I be an interval in R and n ∈ N. We introduce the following function spaces:

1. C(I) is the space of real-valued continuous functions f on I.

2. Cb(I) is the space of bounded continuous functions f on I.

3. C0(I) is the space of continuous functions f on I with limx→∞ f(x) = 0(resp. limx→−∞ f(x) =0) provided that I is not bounded above(resp. below). Also, C0,b(I) = C0(I) ∩ Cb(I).

4. Cc(I) is the space of continuous functions f on I with compact supports.

5. Cn(I) is the space consisting of f ∈ C(I) with f (n) ∈ C(I). Here on I, f (n) is theusual n times derivative. If x is the left(resp. right) hand boundary point of I and isin I, f (n)(x) is the n times right(resp. left) hand derivative at x. Cn0 (I), Cnb (I), Cnc (I)and Cn0,b(I) are similarly defined.

6. C∞c (I) =⋂k Ckc (I), C∞0,b(I) =

⋂k Ck0,b(I), and C∞0 (I) =

⋂k Ck0 (I).

In Gerber and Landry(1998) and Tsai and Wilmott(2002), the following regularitieswere used implicitly. For a rigorous proof and related works, see Wang and Wu(2000), Caiand Yang(2005) and Chen et al.(2006b).

Theorem 2.1 The function Φ in (2.6) has the following properties:

1. For all r ≥ 0, Φ ∈ C1b (R+) ∩ C2(R++).

2. If r > 0 or r = 0 and E[X1] > 0, then Φ ∈ C10(R+).

Remark. If r = 0, it is not always true that limx→∞ Φ(x) = 0. See Kou and Wang(2003)Corollary 3.1 for an example.

Lemma 2.1 The process e−rτ∧tΦ(Xτ∧t); t ∈ R+ is a (Px,Ft∧τ )−martingale and, for allt,

Ex[e−rτg(Xτ )|Fτ∧t

]= e−rt∧τΦ(Xτ∧t). (2.7)

Hence, for all (Ft∧τ )−stopping time η,

Ex[e−rt∧η∧τΦ(Xt∧τ∧η)

]= Φ(x). (2.8)

Proof. Fix t ≥ 0. On [t ≥ τ ], using local property of conditional expectation(See Kallen-berg(2002) Lemma 6.2) and the fact that Φ = g on R−,

Ex[e−rτg(Xτ )|Fτ∧t

]= Ex

[e−rτg(Xτ )|Fτ

]= e−rτΦ(Xτ ). (2.9)

Similarly, on [t < τ ],

Ex[e−rτg(Xτ )|Fτ∧t

]= Ex

[e−rτg(Xτ )|Ft

].

By Strong Markov Property of X, on [t < τ ],

Ex[e−rτg(Xτ )|Fτ∧t

]= e−rtEXt [e

−rτg(Xτ )] = e−rtΦ(Xt). (2.10)

Combining (2.9) and (2.10), we get (2.7).

5

Lemma 2.2 For x > 0, (L− r)Φ(y) → (L− r)Φ(x), as y → x.

Proof. Since Φ ∈ C2(R++), it suffices to show that, as y → x∫Φ(y − z)dF (z) →

∫Φ(x− z)dF (z).

For any y > 0, write∫Φ(y − z)dF (z) =

∫ ∞

yg(y − z)dF (z) +

∫ y

−∞Φ(y − z)dF (z).

Let ε > 0 and find M > x such that∫(−∞,x−M+1]∪[x+M−1,∞) dF (z) < ε. Then for all y > 0

such that |x− y| ≤ 1/2,∣∣∣∣∫ Φ(y − z)dF (z)−∫

Φ(x− z)dF (z)∣∣∣∣

≤2‖g‖∞ε+∣∣∣∣∫ y+M

yg(y − z)dF (z)−

∫ x+M

xg(x− z)dF (z)

∣∣∣∣+

∣∣∣∣∫ y

y−MΦ(y − z)dF (z)−

∫ x

x−MΦ(x− z)dF (z)

∣∣∣∣≤2‖g‖∞ε+

∣∣∣∣∫ 0

−Mg(z)[f(y − z)− f(x− z)]dz

∣∣∣∣ +∣∣∣∣∫ M

0Φ(z)[f(y − z)− f(x− z)]dz

∣∣∣∣ ,where f is the density function for F . Since g and Φ are bounded and f is continuous onR\0, by Dominated Convergence Theorem, we have

lim supy→x

∣∣∣∣∫ Φ(y − z)dF (z)−∫

Φ(x− z)dF (z)∣∣∣∣ ≤ 2‖g‖∞ε.

Since ε > 0 is arbitrary, the proof is completed.

Theorem 2.2 The function Φ is in C2b (R++) and satisfies

(L− r)Φ(x) = 0, x > 0, (2.11)

where L is defined in (2.4). Furthermore, if r > 0, we have Φ ∈ C20,b(R++).

Proof. That Φ′′ ∈ C0,b(R++) for r > 0 and ∈ Cb(R++) for r = 0 is clear once we haveestablished that (L− r)Φ(x) = 0 for all x > 0. For then we have

Φ′′(x) = − c

DΦ′(x)− λ

D

∫Φ(x− y)dF (y) +

λ+ r

DΦ(x) ∈

C0,b(R++), r > 0,Cb(R++), r = 0.

To establish (2.11), we fix x > 0 and let ε ∈ (0, x). By Theorem 2.1, we have Φ ∈C1b (R+) ∩ C2(R++). Since the behavior of Φ′′ near 0+ and +∞ are unclear, we stop the

6

process X at the time τ , where τ is the entry time of (−∞, ε]∪ [x+ ε). (See (2.5).) Then τis an (Ft)−stopping time. Moreover, since τ ≤ τ , we see that τ is a (Fτ∧t)−stopping time.

Next, we will apply Dynkin’s formula(see Sato(1999) Exercise 44.20) to verify the fol-lowing identity:

Ex[e−rt∧τΦ(Xτ∧t)

]= Ex

[∫ τ∧t

0e−ru(L− r)Φ(Xu)du

]+ Φ(x), t > 0. (2.12)

Since Φ is not in C20(R), Dynkin’s formula is not directly applicable. Therefore, we need

to construct a sequence of C20 -functions that converges to Φ, and we may apply Dynkin’s

formula on them.To do this, we pick a sequence of uniformly bounded functions (gn) ⊂ C2

0((−∞, 0]) thatconverges to g except on a set N of Lebesgue measure 0 in R−− and gn(0) = g(0) forall n.(See Theorem A.1.) Therefore, we may consider a sequence of uniformly boundedfunctions (Φn;n ≥ 1) that satisfies the following conditions:

C1 Φn = gn on (−∞, 0];

C2 Φn = Φ on (1/n, n);

C3 Φn ∈ C20(R).

By (C1), (C2) and the fact that gn → g except on N , we have Φn → Φ pointwise on Rexcept on N .

Now, pick n large such that (ε, x + ε) ⊂ (1/n, n). Then Φn(x) = Φ(x) by (C2). ByDynkin’s formula, for all t > 0,

Ex[e−rτ∧tΦn(Xτ∧t)

]= Ex

[∫ τ∧t

0e−ru(L− r)Φn(Xu)du

]+ Φ(x). (2.13)

Since (Φn) is uniformly bounded, Φn → Φ pointwise on R+ and Φn → Φ a.e. on (−∞, 0),by Theorem A.2(1) and Dominated Convergence Theorem,

limn→∞

Ex[e−rt∧τΦn(Xt∧τ )

]= Ex

[e−rt∧τΦ(Xτ∧t)

]. (2.14)

On the other hand, for every u < τ ∧ t, Xu ∈ (ε, x+ ε), and hence Φn(Xu) = Φ(Xu) for alllarge n. These give

(L− r)Φ(Xu)− (L− r)Φn(Xu) =∫

[Φ(Xu − y)− Φn(Xu − y)] dF (y), (2.15)

and hence

|(L− r) [Φ(Xu)− Φn(Xu)]| ≤ supn‖Φn‖∞ + ‖g‖∞. (2.16)

By Dominated Convergence Theorem, (2.15) and the absolute continuity of F imply, for allu < t ∧ τ ,

(L− r)Φn(Xu) → (L− r)Φ(Xu), n→∞.

7

By (2.16) and Dominated Convergence Theorem, we have

limn→∞

Ex

[∫ τ∧t

0e−ru(L− r)Φn(Xu)du

]= Ex

[∫ t∧τ

0e−ru(L− r)Φ(Xu)du

]. (2.17)

By (2.14) and (2.17), and letting n→∞ for both sides of (2.13), we get (2.12).

Finally, we show that (L− r)Φ(x) = 0. By Lemma 2.1, we have Ex[e−rt∧τΦ(Xτ∧t)

]=

Φ(x). Hence, (2.12) implies that

Ex

[∫ t∧τ

0e−ru(L− r)Φ(Xu)du

]= 0, t > 0. (2.18)

On the other hand, by Lemma 2.2,

Ex[

supu<t∧τ

|e−ru(L− r)Φ(Xu)− (L− r)Φ(X0)|]→ 0, t ↓ 0 + . (2.19)

Therefore, using (2.18), we obtain

|(L− r)Φ(x)| =

∣∣∣∣∣1tEx

[∫ t∧τ

0e−ru(L− r)Φ(Xu)du

]− (L− r)Φ(x)

∣∣∣∣∣≤Ex

[supu<t∧τ

|e−ru(L− r)Φ(Xu)− (L− r)Φ(X0)|]

+ |(L− r)Φ(x)|Ex [(t− τ ∧ t)]t

.

Note that τ > 0 Px−a.s. and hence t − τ ∧ t = 0 for all sufficiently small t. Since further0 ≤ t−τ∧t

t ≤ 2, we have Ex[t−τ∧tt

]→ 0 as t → 0+. Together with (2.19) and the last

inequality, we establish that

|(L− r)Φ(x)| ≤ lim supt→0+

∣∣∣∣∣1tEx

[∫ t∧τ

0e−ru(L− r)Φ(Xu)du

]− (L− r)Φ(x)

∣∣∣∣∣ = 0.

3 Transformation of Integro-Differential Equation Into ODE

In this section, we consider phase-type jump distributions and the boundary value problem(L− r)Φ = 0, in R++,Φ = g, on R−,

(3.1)

where L is defined by (2.4) and r ≥ 0. Our main purpose is to show that if Φ satisfies(3.1), then it satisfies an ODE on R++. And from this, we obtain a general form of Φ andderive an explicit formula for Φ under some technical conditions. To begin with, we recallthe definition of phase-type distributions.

8

Definition 3.1 Let N ≥ 1. Assume B is an N × N nonsingular subintensity matrix,that is, bij ≥ 0 for i 6= j, bii ≤ 0 and b> = −Be> ∈ RN

−\0. Here, e = [1 1 · · · 1] and0 = [0 0 · · · 0]. Let α be an N -dimensional probability function. The probability distributionfunction F with the density function

f(x) =

αexBb>, x > 0,0, x ≤ 0.

is called a phase-type distribution. We denote this distribution by PH(α,B). We say thatthe representation (α,B) is minimal if there do not exist N0 < N , α′ of dimension N0,and a nonsingular subintensity matrix B′ dimension N0 such that f(x) = α′exB′

b′>1x>0.

A summary of analytic facts of phase-type distributions is given in Appendix B. It isworth noting that phase-type distributions are dense in the set of all probability distribu-tions on R++. Special cases of phase-type distributions include exponential distributions,Erlang distributions, mixtures of exponential distributions. In addition, since phase-typedistributions are closed under convolution, Gamma distributions with parameter n ∈ Nare also phase-type distributions. For details, see Asmussen (2000) or Rolski et al.(1999)Section 8.2.

We consider the process X defined in (2.1) and assume that its jump distribution F hasa probability density function f given by

f(x) =

pf(+)(x), x > 0,0, x = 0,qf(−)(−x), x < 0,

(3.2)

where p+q = 1, p, q ∈ R+ and f(±) are of PH(α±,B±). Here B+ and B− are not necessar-ily of the same dimension. We will write I for the identity matrices of the same dimensionsas those of B+ and B− if there is no confusion.

Remark. In Asumssen et al.(2004), they considered a Levy process X as in (2.1) exceptthat Z = (Zt; t ≥ 0) is given by

Zt =N+

t∑n=1

Y +n −

N−t∑

n=1

Y −n , t ≥ 0, (3.3)

where N± are both Poisson processes, (Y +n )(resp. (Y −

n )) are independently and identicallydistributed with distribution PH(α+,B+)(resp. PH(α−,B−)). Moreover, W , N+, N−,(Y +n ) and (Y −

n ) are independent. Their processes have the same characteristic exponent asours, and therefore the finite dimensional distributions of their processes coincide with thoseof ours. Since finite dimensional distributions determine law(see Jacod and Shiryayev(2003),Lemma VI.3.19), our definition of X is sufficient.

In the following, when we perform integration and differentiation with respect to amatrix of continuous parameter, these operations are meant to be performed termwise.Also, let ei be the row vector with a 1 in the ith component and zero otherwise , and set‖A‖ =

∑i,j |Aij | for any matrix A.

9

By assumption (3.2) and Theorem B.2, the characteristic exponent ψ in (2.2) is givenby

ψ(ζ) = Dζ2 + cζ + λψ1(ζ)− λ, ζ ∈ iR, (3.4)

where ψ1(ζ) =∫e−ζyf(y)dy is of the form:

ψ1(ζ) = pα+(ζI −B+)−1b>+ + qα−(−ζI −B−)−1b>−. (3.5)

Since the right hand side of (3.5) is a rational function in ζ on C(see Theorem B.2.), theright hand side of (3.4) is actually a rational function on C with a finite number of poles inC\iR. Accordingly, we consider ψ and ψ1 on C as analytic functions except at the poles inC\iR.

We denote by Z(−) the collection of zeros of ψ(ζ)−r(counting multiplicity) with strictlynegative real parts. We say that Z(−) is separable if its members are distinct. Let P0(ζ) bethe ”minimal” polynomial with leading coefficient 1 such that the zeros of P0(ζ) coincidewith the poles of ψ1(ζ), counting multiplicity. Write

P1(ζ) = P0(ζ)(ψ(ζ)− r). (3.6)

Then the zeros of ψ(ζ)−r coincide with those of the polynomial P1(ζ), counting multiplicity.On the other hand, the infinitesimal generator of the process X takes the form

Lh(x) = Dh′′(x) + ch′(x) + λpα+T+B+

h(x)b>+ + λqα−T−B−

h(x)b>− − λh(x), (3.7)

for all h ∈ C20(R). Here, for a nonsingular subintensity matrix B, the matrix-valued opera-

tors T±B are defined on the set of bounded measurable functions h by

T±Bh(x) =∫

R±h(x− y)e±Bydy.

(Note that Theorem B.1 (2) and (3) imply

limy→∞

|eieBye>j |eδy = 0, for some δ > 0.

Hence, T±B is well defined.)

Lemma 3.1 Let h be a bounded Borel measurable function. If h is continuous at x, thenT±Bh are both differentiable at x and

(ddx −B

)T+

B = −(B + d

dx

)T−B = I. Moreover, if

h ∈ C0,b(R++), each entry of T±Bh is in C0,b(R++).

Proof. Assume h is bounded and is continuous at x. By Theorem B.1 (1), ddxe

Bx = BeBx.On the other hand, since h is continuous at x, d

dx

∫ x−∞ h(y)e−Bydy = h(x)e−xB. Therefore,

d

dxT+

Bh(x) =d

dx

(eBx

∫ x

−∞h(y)e−Bydy

)= BT+

Bh(x) + h(x)I.

This shows that(ddx −B

)T+

Bh(x) = h(x)I. Similarly, −(B + d

dx

)T−Bh(x) = h(x)I. Fi-

nally, if h is bounded and h ∈ C0,b(R++), then it is clear that T±Bh is bounded on R.Moreover, by Dominated Convergence Theorem, T±Bh ∈ C0,b(R++). This proves the laststatement.

10

The following is a further refinement of Theorem 2.2.

Proposition 3.1 Assume that the jump density f is of (3.2) and E[X1] > 0 if r = 0. ThenΦ is in C∞0,b(R++) and for k ≥ 0, we have the recursive formula:

Φ(k+2)(x) =− c

DΦ(k+1)(x) +

(λ+ r)D

Φ(k)(x)− λ

DEk(x), (3.8)

where

Ek(x) =pα+

Bk+T

+B+

Φ(x) +k−1∑j=0

Bj+Φ(x)

b>+

+ qα−

(−B−)kT−B−Φ(x) +

k−1∑j=0

(−1)j+1Bj−Φ(x)

b>−. (3.9)

Proof. Fix x > 0. Then Φ is continuous at x by Theorem 2.1. Since (L − r)Φ(x) = 0, by(3.7), we get that (3.8) holds for k = 0. Moreover, from (3.8) for k = 0 and the fact thatΦ ∈ C1

0(R++), we obtain further that Φ ∈ C20,b(R++). By Lemma 3.1, T+

B+Φ is differentiable

at x andd

dxT+

B+Φ(x) = B+T

+B+

Φ(x) + Φ(x)I.

Similarly, we have ddxT

−B−

Φ(x) = −B−T−B−

Φ(x) − Φ(x)I. So, by the differentiability ofT±B±

Φ at x and (3.8) for k = 0, Φ′′ is differentiable at x and (3.8) holds for k = 1. SinceΦ ∈ C2

0,b(R++) and T±B±Φ ∈ C0,b(R++), Φ′′′ ∈ C0,b(R++) and hence Φ ∈ C3

0,b(R++). A similarargument holds for general k. The proof is completed.

The following theorem is crucial to our approach. It says that we can transform integro-differential equations into ordinary differential equations when the two sides of jump distri-bution are both phase-type distributions.

Theorem 3.1 Let D1 be the differential operator with the characteristic polynomial P1

given by (3.6). Then D1Φ ≡ 0 on R++.

Proof. Let D0 be the differential operator with the characteristic polynomial P0.(See(3.6).) For an operator A on L2 = L2(R), write A∗ as its adjoint. Also, set 〈f1, f2〉 =∫f1(x)f2(x)dx.

Let φ ∈ C∞c (R++) be a test function. Recall that by Theorem 2.2, (L−r)Φ ≡ 0 on R++.By integration by parts(see, e.g., Stein and Shakarchi(2005) Section 5.3.1) and Fubini’sTheorem, we have

0 = 〈D0(L− r)Φ, φ〉 = 〈Φ, (L∗ − r)D∗0φ〉. (3.10)

Here, L∗ is the infinitesimal generator of the dual process X = −X and is given by

L∗h(x) = Dh′′(x)− ch′(x) + λ

∫h(x− y)f(−y)dy − λh(x), h ∈ C2

0(R).

11

Write Th =∫h(x− y)f(y)dy and LD = L− r − λT . Then

(L∗ − r)D∗0φ = λT ∗D∗

0φ+ L∗DD∗0φ = λT ∗D∗

0φ+ (D0LD)∗φ. (3.11)

WriteD2 as the differential operator corresponding to the polynomial P2(ζ) = P0(ζ)ψ1(ζ).We prove that T ∗D∗

0φ = D∗2φ a.e. by showing that both T ∗D∗

0φ and D∗2φ are in L2 ∩ L1

and have the same Fourier transforms.First, we show that T ∗0D

∗0φ and D∗

2φ are in L1∩L2. Since φ ∈ C∞c (R++), D∗2φ ∈ L1∩L2.

Also, ∫|T ∗D∗

0φ(x)|dx ≤∫ ∫

|D∗0φ(x− y)|dxf(−y)dy ≤ ‖D∗

0φ‖L1‖f‖L1 <∞,

and hence T ∗0D∗0φ ∈ L1. We show that T ∗D∗

0φ ∈ L2. Since D∗0φ ∈ C∞c (R++) and T ∗h(x) =∫

h(x−y)f(−y)dy, it suffices to show that Tφ ∈ L2. Note by Theorem B.1, f has tails thatdecay exponentially. So,

∫f2 <∞. Hence, if we write H as the compact support of φ,∫

[Tφ(x)]2 dx =∫ (∫

φ(x− y)f(y)dy)2

dx =∫ (∫

Hφ(y)f(x− y)dy

)2

dx

≤∫ (∫

φ(y)2dy) (∫

Hf(x− y)2dy

)dx ≤ ‖φ‖2

L2‖f‖2

L2

∫Hdx <∞.

Next, we show that the Fourier transforms F(T ∗D∗0φ) and F(D∗

2φ) coincide, whereFh(θ) =

∫e−2πiθxh(x)dx. Recall the rational function ψ1(ζ) =

∫e−ζyf(y)dy. Also, note

that F(D∗0φ)(θ) = P0(−2πiθ)F(φ)(θ).(See Stein and Shakarchi(2005) Section 5.3.) Since

T ∗D∗0φ ∈ L1 ∩ L2, we have, for all θ ∈ R,

F(T ∗D∗0φ)(θ) =

∫e−2πiθx

(∫D∗

0φ(x− y)f(−y)dy)dx

=∫ (∫

D∗0φ(x− y)e−2πiθ(x−y)dx

)e−2πiθyf(−y)dy

= ψ1(−2πiθ)P0(−2πiθ)F(φ)(θ)= P2(−2πiθ)F(φ)(θ)= F(D∗

2φ)(θ).

Now, by the fact that T ∗D∗0φ,D

∗2φ ∈ L2, Fourier inversion formula gives that T ∗D∗

0φ = D∗2φ

almost everywhere.By (3.11) and the fact that T ∗D∗

0φ = D∗2φ a.e., (L∗−r)D∗

0φ = λD∗2φ+(D0LD)∗φ = D∗

1φa.e. Hence, by (3.10), we have

0 = 〈Φ, (L∗ − r)D∗0φ〉 = 〈Φ, D∗

1φ〉 = 〈D1Φ, φ〉.

Since φ ∈ C∞c (R++) is arbitrary, we have D1Φ = 0 a.s. on (0,∞). Since Φ ∈ C∞0,b(R++),D1Φ ≡ 0 on R++. This completes the proof.

12

Example 1. We assume r = 0 and consider the model (2.1) with dF (y) = ηe−ηy1y>0dyfor η > 0. Then

ψ(ζ) = Dζ2 + cζ + λη

η + ζ− λ.

In this case, P0(ζ) = η + ζ and

P1(ζ) = Dζ2(ζ + η) + cζ(ζ + η) + λη − λ(ζ + η) = Dζ3 + (Dη + c)ζ2 + (cη − λ)ζ.

(Note that P1(−η) = λη 6= 0.) Hence, Φ satisfies the following ODE:

DΦ′′′(x) + (Dη + c)Φ′′(x) + (cη − λ)Φ′(x) = 0, on x > 0.

This equation is consistent with Wang and Wu(2000) Eq.(IIIa) in which they consideredthe function g(y) = 1[−z,0](y).

Assume from now on that r > 0. We write x = [x1 x2 · · · xk]> for a sequence (xj)kj=1,k ≥ 1. Also, for Z(−) = (ρi)Si=1, write eρ(x) = [eρ1x eρ2x · · · eρSx]>.

Corollary 3.1 There exist polynomials Qi(x) for 1 ≤ i ≤ S such that Φ(x) = Q(x)>eρ(x),Q(x)> = [Q1(x), · · · ,QS(x)].

Proof. Denote by Z(+) the collection of all solutions to ψ(ζ)− r = 0 with nonnegative realparts, counting multiplicity. Since D1Φ ≡ 0 on R++ and ψ(ζ)− r and P1(ζ) have the samezero set, Φ is of the form

Φ(x) = Q(x)>eρ(x) +∑

ρ′∈Z(+)

Rρ′(x)eρ′x

for some polynomials Qi’s and Rρ′ ’s.(See Tenenbaum and Pollard(1985) Theorem 19.3 andLesson 20.) We show that Rρ = 0 for all ρ ∈ Z(+). Assume that Rρ(x) 6= 0 for someρ ∈ Z(+). Let a ≥ 0 be the maximum of all real parts of members in Z(+) and let ρ′1, · · · , ρ′kbe the set of all elements in Z(+) such that <(ρ′j) = a. Now, let m ≥ 0 be the smallestinteger such that the order of Rρ′j is ≤ m for all j, and select the ρ′j ’s such that the orderof Rρ′j is equal to m. Still call the selected members ρ′1, ρ′2, · · · , ρ′k. Let aj 6= 0 be thecoefficient of xm in Rρ′j for all j. Then

Φ(x)x−me−ax =∑j

ajei=(ρ′j)x + h(x), where h(x) → 0 as x→∞.

However,∑

j ajei=(ρ′j)x is the discrete Fourier transform of a nonzero function and hence

is not identically zero, and it has period 2π. So, Φ(x)x−me−ax 9 0, as x → ∞. SinceΦ ∈ C0(R+), this is impossible. Hence, Rρ′ ≡ 0 for all ρ′ ∈ Z(+).

Corollary 3.2 Suppose that Z(−) is separable. Then Φ(x) = Q>eρ(x), x ∈ R+, for someconstant vector Q.

13

Proof. By Corollary 3.1, we know that Φ(x) = Q(x)>eρ(x), where Qj(x) are polynomials.Since Z(−) is separable, standard theory of ordinary differential equation gives that Qj(x)must be a constant for all j. See Tenenbaum and Pollard(1985) Lesson 20B and 20D.

In the following, we provide a method to find Φ. By using the values Φ(k)(0+), weobtain Q by solving a system of linear equations.

Lemma 3.2 For k ≥ 0, the k−th derivative of Φ at 0+ is given by the recursive formula:

Φ(k+2)(0+) =1D

(−cΦ(k+1)(0+) + (λ+ r)Φ(k)(0+)− λEk(0)

). (3.12)

Proof. By Corollary 3.1, the right hand derivative Φ(k)(0+) exists for all k ≥ 1. Now,(3.12) follows from (3.8) by letting x→ 0+.

We write Uj = Φ(j−1)(0) for 1 ≤ j ≤ S. Then we can easily derive Q in terms of U andρi’s.

Theorem 3.2 Assume Z(−) is separable. Then Q = V −1U , where V is an S × S Van-dermonde matrix with V ij = ρi−1

j . In particular, if q = 0 in (3.2), then U depends only ong.

Proof. By Corollary 3.2, we know that V Q = U . Since ρi’s are distinct, the determinantof the Vandermonde matrix V is nonzero(See e.g. Friedberg et al.(1997) page 218) andhence V is invertible. This gives the first result. Assume further that q = 0. Then Φ′(0+)depends only on g(See Theorem C.1) and clearly so are Ek(0)’s. Hence, by (3.12), we haveexact formulae for Φ(k)(0+) in term of g.

Remark. Assume r > 0 and the representation (α+,B+) is minimal. Let

Ig =[g(0),

∫ ∞

0g(−z)e1e

B+ze>dz, · · · ,∫ ∞

0g(−z)eNeB+ze>dz

],

where N is the dimension of the matrix B+. Asmussen et al.(2004) showed that Φ(x) =

IgA−1eρ(x), where the i−th row of the matrix A is given by

[1

(ρiI −B+)−1b>+

]>. We

remark that they showed the invertibility of A when (α+,B+) is minimal and Z(−) isirreducible. Therefore, they did not give the general form of Φ, whereas we do this inCorollary 3.1.

On the other hand, we give a recursive formula for Φ(k)(0+) and this formula dependsonly on g if q = 0(Theorem C.1). Based on this result, we get an explicit formula for Φ(x)in terms of g and Z(−) and this formula does not require us to compute A−1.

The following proposition will play an important role in next section.

14

Proposition 3.2 If (α+,B+) is minimal, Z(−) is separable and p > 0, then the constantvector Q for Φ satisfies Q>e = g(0) and

α+

(∫ ∞

0g(−y)eB+ydy

)exB+b>+ = α+

[S∑i=1

Qi(ρiI −B+)−1

]exB+b>+, ∀x > 0.

(3.13)

Proof. The first statement follows from the fact that Q>e = Φ(0) = g(0).Consider the second statement. First, note that the minimality of the representation

(α+,B+) guarantees that the collection of all zeros η ∈ C of the rational function 1r−ψ(η)

is equal to that of all eigenvalues of B+. (For details, see Asmussen et al.(2004) Lemma 1and the paragraph above it.) Hence, ρi is not an eigenvalue of B+ for each i and ρiI −B+

is invertible for all i. By Theorem B.1(4) and B.2(2), we have∫ x

−∞Φ(x− y)f(y)dy =

∫ x

0Q>eρ(x− y)f(y)dy +

∫ 0

−∞Q>eρ(x− y)f(y)dy

=S∑i=1

Qieρixpα+(ρiI −B+)−1(I − ex(B+−ρiI))b>+

+S∑i=1

Qieρixqα−(−ρiI −B−)−1b>−,

Therefore, by (3.5),∫ x

−∞Φ(x− y)f(y)dy =

S∑i=1

Qieρixψ1(ρi)−

S∑i=1

Qieρixpα+(ρiI −B+)−1ex(B+−ρiI)b>+.

Recall ψ(ρi) = D(ρi)2 + cρi + λψ1(ρi) − (λ + r) = 0 for all i. Since Φ(x) = Q>eρ(x), wehave

0 =(L− r)Φ(x) = DΦ′′(x) + cΦ′(x) + λ

∫Φ(x− y)f(y)dy − (λ+ r)Φ(x)

=S∑i=1

Qiψ(ρi)eρix + λ

∫ ∞

xg(x− y)f(y)dy − λ

S∑i=1

Qieρixpα+(ρiI −B+)−1ex(B+−ρiI)b>+

=λpα+

(∫ ∞

0g(−y)eB+ydy

)exB+b>+ − λp

S∑i=1

Qieρixα+(ρiI −B+)−1ex(B+−ρiI)b>+.

So, the conclusion of the proposition follows.

4 The Constant Vector Q – Second Method

In this section, we consider a more general process whose upward jumps are determined bya general Levy measure and downward is by a compound Poisson process with a mixtureof exponential jump distributions. We show that if a vector Q satisfies a system of linear

15

equations, then a conjectured function must be the desired function Φ. And to find asolution to the system of equations, instead of directly inverting the system of equations,we exploit the technique in Dufresne and Gerber(1989). Unlike the previous method inSection 3, we do not need the knowledge of higher order derivatives at zero. Finally, weshow in Example 2 that our result is consistent with Theorem 3.1 and Corollary 3.1 in Kouand Wang(2003).

First, let X(+) = (X(+)t ; t ≥ 0) be a Levy process on R such that it starts at 0 and has

nontrivial diffusion part and no downward jumps. Let Z be a compound Poisson processthat is independent of X(+), and its jump distribution is given by f(+) =PH(α+,B+). Thenwe consider in this section the family of Levy processes (X, Pxx∈R) given by

Xt = X0 +X(+)t − Zt, t ≥ 0. (4.1)

Here, as before, under Px, X0 = x a.s. Clearly, the process in (4.1) is a generalization of(2.1).

The characteristic exponent ψ of X is given by

ψ(ζ) = Dζ2 + cζ +∫ ∞

0

[eζz − 1− ζz1|z|≤1

]ν(dz) + λ

∫ ∞

0e−ζyf(+)(y)dy − λ,

where ν is an arbitrary Levy measure on (0,∞) and∫∞0 min(1, z2)ν(dz) < ∞. Moreover,

the infinitesimal generator of X has a domain containing C20(R) and is given by

Lh(x) = Dh′′(x) + ch′(x) +∫ ∞

0[h(x+ z)−h(x)− h′(x)z1|z|≤1]ν(dz)

+ λ

∫ ∞

0h(x− z)f(+)(z)dz − λh(x).

Recall that Φ(x) = Ex [e−rτg(Xτ )], where τ = inft ≥ 0;Xt ≤ 0. The next theorem givesa converse of Theorem 2.2.

Theorem 4.1 If φ ≡ g on (−∞, 0], φ ∈ C20(R+) and (L− r)φ ≡ 0 on R++, then φ ≡ Φ on

R. In particular, the solution φ is unique.

Proof. Let x > 0. Similar to the proof of Theorem 2.2, we can pick a sequence of uniformlybounded functions (φn) ⊂ C2

0(R) such that φn ≡ φ on R+ and φn → φ a.e. on (−∞, 0). ByDynkin’s formula, we have

Ex[e−rt∧τφn(Xτ∧t)

]= Ex

[∫ t∧τ

0e−ru(L− r)φn(Xu)du

]+ φ(x), t > 0. (4.2)

Note that φn and φ only differ on (−∞, 0) and Xu > 0 for all u < t∧τ . Moreover, the jumpdistribution of Z is absolutely continuous. Hence, similar to the proof of (2.17), we have

limn→∞

Ex[∫ t∧τ

0e−ru(L− r)φn(Xu)du

]= Ex

[∫ t∧τ

0e−ru(L− r)φ(Xu)du

].

By Dominated Convergence Theorem and Theorem A.2(1), letting n → ∞ for both sidesof (4.2) gives

Ex[e−rt∧τφ(Xτ∧t)

]= Ex

[∫ t∧τ

0e−ru(L− r)φ(Xu)du

]+ φ(x) = φ(x). (4.3)

16

Note the last equality follows from the assumption that (L−r)φ ≡ 0 on R++. Let t→∞ forboth sides of the last equality. Since r > 0, the result follows by the fact that e−rt1[τ>t] → 0as t→∞. This completes the proof.

We consider below the special case that f(+)(y) is a mixture of exponential distributions.Namely, there exist constants (pj)mj=1 and (ηj)mj=1 such that pj > 0, ηj > 0,

∑mj=1 pj = 1

and

f(+)(y) =m∑j=1

pjηje−ηjy (4.4)

on y > 0 and f(+)(y) = 0 otherwise. Without loss of generality, assume that ηj ’s aredistinct.(Note that in Dufresne and Gerber(1993) and Moredecki(2002), they consideredthe jump distribution (4.4).)

It is worth noting that Z(−) = (ρi)Si=1 is separable and S = m + 1(see Asmussen etal.(2004) Lemma 1(1) or Mordecki(2002) Corollary 2). Now, based on Corollary 3.2, weconjecture that Φ(x) = Ex [e−rτg(Xτ )] = Q>eρ(x) for some Q. Moreover by Proposition3.2, we assume further that Q satisfies (3.13). More precisely, for f(+) given by (4.4),

m∑j=1

pjηje−ηjx

∫ 0

−∞g(y)eηjydy =

m+1∑i=1

Qi

m∑j=1

pjηje

−ηjx

ρi + ηj. (4.5)

(Note that f(+) =PH((p1, · · · , pm),diag(−η1, · · · ,−ηm)).) Recall that Q>e = g(0). Thenby comparing the coefficients of e−ηjx in (4.5), we get a system linear equations: ∑m+1

i=1 Qi = g(0),∑m+1i=1

Qiηj

ρi+ηj=

∫ 0−∞ g(y)ηjeηjydy, 1 ≤ j ≤ m.

(4.6)

The following theorem confirms our conjecture.

Theorem 4.2 (1) If Q satisfies (4.6) for some bounded Borel measurable function g, then

Φ(x) = Ex[e−rτg(Xτ )

]= Q>eρ(x), for x ≥ 0. (4.7)

(2) The system (4.6) admits at most one solution Q.

Proof. (1) Set φ(x) = Q>eρ(x) for x ≥ 0, and φ(x) = g(x) for x ≤ 0. Then, for everyx > 0,

(L− r)φ(x) =m+1∑i=1

Qi

[Dρ2

i + cρi +∫ ∞

0

(ezρi − 1− ρiz1|z|≤1

)ν(dz)− (λ+ r)

]eρix

+ λ

[m+1∑i=1

Qi

∫ x

0eρi(x−y)f(+)(y)dy +

∫ ∞

xg(x− y)f(+)(y)dy

]

=m+1∑i=1

Qi

[Dρ2

i + cρi +∫ ∞

0

(ezρi − 1− ρiz1|z|≤1

)ν(dz) + λ

m∑j=1

pjηjηj + ρi

− (λ+ r)

]eρix + λ

m∑j=1

e−ηjxpj

[−m+1∑i=1

ηjQi

ηj + ρi+ ηj

∫ 0

−∞g(y)eηjydy

]= 0,

17

since ψ(ρi) − r = 0 for all i and Q satisfies (4.6) and hence (4.5). Moreover, since r > 0,φ ∈ C2

0(R+) and φ ≡ g on R−, by Theorem 4.1, φ ≡ Φ on R. This completes the proof.(2) By (1), any solution Q to (4.6) induces a function φ such that φ(x) = Q>eρ(x) =Ex [e−rτg(Xτ )] and the conditions of Theorem 4.1 are satisfied. By Theorem 4.1, the solu-tion φ is unique. So there exists at most one solution to (4.6).

To solve (4.6), we exploit the technique in Dufresne and Gerber(1989).

Theorem 4.3 Assume X satisfies (4.1) and f(+)(y) is given by (4.4). Then for anybounded Borel function g : R− → R, we have Φ(x) = Ex [e−rτg(Xτ )] = Q>eρ(x) on R+,where QT = [Q1,Q2, ...,Qm+1] and, for 1 ≤ h ≤ m+ 1,

Qh =1ρh

m+1∑j=1

Rj

m+1∏k=1

(ηj + ρk)m+1∏

i=1,i6=j

−ρh − ηiηj − ηi

m+1∏p=1,p6=h

1(−ρh + ρp)

, (4.8)

where

ηm+1 = 0, Rj = g(0)−∫ ∞

0g(−y)ηje−ηjydy, for 1 ≤ j ≤ m, and Rm+1 = g(0). (4.9)

Proof. Clearly, the system (4.6) is equivalent to the following system of linear equationsQ>e = g(0),∑m+1

i=1Qiρi

ρi+ηj= Rj , 1 ≤ j ≤ m,

(4.10)

where Rj ’s are given by (4.9). Since the system (4.6) admits at most one solution, so does(4.10).

Consider the rational function

H(x) =m+1∑j=1

Rj

m+1∏k=1

ηj + ρkx+ ρk

m+1∏i=1,i6=j

x− ηiηj − ηi

. (4.11)

Note for each summand in (4.11), the numerator is a polynomial of degreem and the denom-inator is a polynomial of degree m+ 1. By the principle of partial fraction decomposition,there exist constants Dh, 1 ≤ h ≤ m+ 1, such that

m+1∑j=1

Rj

m+1∏k=1

ηj + ρkx+ ρk

m+1∏i=1,i6=j

x− ηiηj − ηi

= H(x) =m+1∑i=1

Diρiρi + x

. (4.12)

By multiplying both sides of (4.12) by (x + ρh) and then set x = −ρh, we get that Dh isgiven by the right hand side of (4.8). On the other hand, since

∏m+1i=1,i6=j

ηh−ηi

ηj−ηi= 0 for all

h 6= j,

m+1∑i=1

Diρiρi + ηh

= H(ηh) = Rh

m+1∏k=1

ηh + ρkηh + ρk

m+1∏i=1,i6=h

ηh − ηiηh − ηi

= Rh, 1 ≤ h ≤ m+ 1.

That is, D is a solution to (4.10). Since the solution to (4.10) is unique and (4.6) and (4.10)are equivalent, Q = D is the unique solution of (4.6). By Theorem 4.2, we have Φ ≡ Q>eρ

on R+. This completes the proof.

18

Remark. In fact, by approximation in (4.8) and (4.9), one sees that we still have (4.8)for Φ(x) if g is any Borel function on R− such that

∫ 0−∞ |g(y)|eηjydy <∞ for all 1 ≤ j ≤ m.

Example 2. Consider the special case that m = 1. Then Z(−) = ρ1, ρ2 and η2 = 0. Weconsider some special g’s:(1) Assume g ≡ 1. Then R1 = 0 and R2 = 1. Recall that η2 = 0. Then

Q1 =1ρ1

∑2j=1

Rj

∏2k=1(ηj + ρk)

∏2i=1,i6=j

−ρ1−ηi

ηj−ηi

∏2k=1,k 6=1(−ρ1 + ρk)

=1ρ1

R2(0 + ρ1)(0 + ρ2)−ρ1−η10−η1ρ2 − ρ1

=η1 + ρ1

η1

ρ2

ρ2 − ρ1,

and hence Q2 = 1−Q1 = −ρ2−η1η1

−ρ1ρ1−ρ2 . So,

Ex[e−rτ

]=η1 + ρ1

η1

−ρ2

ρ1 − ρ2eρ1x +

−ρ2 − η1

η1

−ρ1

ρ1 − ρ2eρ2x, x ≥ 0

See Eq.(3.1) in Kou and Wang(2003).

(2) Assume g ≡ 1(−∞,y) for some y ≤ 0. Then R1 = −eη1y and R2 = 0. Hence,

Q1 =1

ρ1(−ρ1 + ρ2)(−eη1y) (η1 + ρ1)(η1 + ρ2)

−ρ1

η1=eη1y(η1 + ρ1)(η1 + ρ2)

(ρ2 − ρ1)η1.

and Q2 = −Q1. Therefore, we obtain

Ex[e−rτ1(−∞,y)(Xτ )

]= eη1y

(η1 + ρ2)(η1 + ρ1)η1(ρ2 − ρ1)

[exρ1 − exρ2 ] .

See Eq.(3.2) in Kou and Wang(2003).(3) Assume g ≡ 10. Then R1 = R2 = 1. Hence,

Q1 =1ρ1

[(η1 + ρ1)(η1 + ρ2)

−ρ1

η1

1−ρ1 + ρ2

+ ρ1ρ2−ρ1 − η1

−η1

1−ρ1 + ρ2

]=

1ρ1(ρ2 − ρ1)

[(−ρ1)(η2

1 + (ρ1 + ρ2)η1 + ρ1ρ2)η1

+−ρ2

1ρ2 − ρ1ρ2η1

−η1

]=−η1 − ρ1

ρ2 − ρ1.

Also,

Q2 =g(0)−Q1 = 1−Q1 ==−η1 − ρ2

−ρ2 + ρ1.

This gives

Ex[e−rτ10(Xτ )

]=

η1 + ρ1

−ρ2 + ρ1eρ1x +

−ρ2 − η1

−ρ2 + ρ1eρ2x, x ≥ 0.

See Eq.(3.3) in Kou and Wang(2003).

19

5 Concluding Remarks

Expected discounted penalty is a generalized notion of ruin probability in insurance liter-ature. It has been widely studied and generalized since Gerber and Shiu(1998). On theother hand, this function has been a major concern in the pricing of perpetual financialsecurities, and examples include option theory and credit risk modelling.

While empirical studies have been indicating the failure of diffusion model, the semi-nal paper of Merton(1976) receives more attentions in recent years. However, unlike thediffusion case in which many functionals are available, this is no longer the case for jumpdiffusion process. This is where the difficulty in the pricing of securities under jump diffusionprocess arises.

In this paper, we consider the jump diffusion process as in Asmussen et al.(2004). Inthe case that jump distribution is a two-sided phase-type jump, by a Fourier transformargument, we transform the integro-differential equation of the expected discounted penaltyinto an ordinary differential equation. The present method could possibly be extended totransform more integro-differential equations into ordinary differential equations and hencegive an alternative approach to compute prices in jump diffusion model.

Next, by ODE theory, we know the solution for Φ is a linear combination of some knownexponential functions. Moreover, by using the asymptotic behavior of expected discountedpenalty and the integro-differential equation, we can determine these coefficients. All thesedistinguish not only our approach from that by Asmussen et al.(2004) but also itself from theclassical method to solve differential equations in which the knowledge of boundary valuesis required. This could be applied to solve other functionals once we have transformed itsintegro-differential into an ordinary differential equation.

Using our closed form solutions, especially Theorem 4.3, the pricing problems of per-petual securities in jump diffusion have improved answers. For example, the values of debtand firm in the extension model of Leland(1994) by Hilberink and Rogers(2002) have closedform solutions even in the two-sided jump case. And by imposing the smooth-pasting con-dition, the optimal default boundary can be derived in exact solutions. In addition, manystructural form models in credit risk can be reconsidered whether more phenomenon in em-pirical studies can be captured by jump diffusion model. In fact, this is one of our ongoingwork(see Chen et al.(2006c)).

Appendix A: Approximation of Bounded Function and Abso-lute Continuity of Distributions.

Theorem A.1 For every bounded Borel measurable function g on R−, there exists a se-quence of uniformly bounded functions (gn) in C2

0(R−) such that gn → g Lebesgue a.e. onR− and gn(0) = g(0) for all n.

Proof. Write dm for the Lebesgue measure on R. Let h be a normal density and setdG(y) = h(y)dm. Then by the strict positivity of h, dG and dm are equivalent.

Given ε > 0. By Dominated Convergence Theorem, we can find N large enough suchthat we have

∫ 0−∞ |g1[−N,0]−g|dG < ε. Also, since g1[−N,0] is bounded and is in L1(R, dm), a

modification of the proof of Theorem 2.4 in Stein and Shakarchi(2005) Chapter 2 shows thatwe can find a sequence of uniformly bounded C2

0(R−)−functions (g′n) such that g′n → g1[−N,0]dm−a.e. Since dG and dm are equivalent, g′n → g1[0,N ] dG−a.e. Hence, by Dominated

20

Convergence Theorem, we can pick n large such that∫|g′n − g1[−N,0]|dG < ε. We have

obtained that∫ 0−∞ |g′n− g|dG < 2ε. And this implies that there exists a sequence of C2

0(R−)functions (gn) such that gn → g dG−a.e. on R−. Since dG and dm are equivalent, wesee that gn → g dm-a.e. on R−. In addition, it is clear we can pick (gn) to be uniformlybounded.

Now, for each n, define a C20(R−)−function hn such that hn = gn on (−∞, 1/n), hn(0) =

g(0) and (hn) is uniformly bounded. Then it is obvious that (hn) is the desired sequence.This completes the proof.

Theorem A.2 The following distributions are absolutely continuous with respect to Lebesguemeasure:

1. The distribution Px [Xτ < 0, Xτ ∈ ·], where X is defined by (2.1) and τ is defined asin the proof of Theorem 2.2.

2. The distribution Px [Xτ < 0, Xτ ∈ ·] for x > 0, where X is defined by (4.1).

Proof. We only show (2) and the proof of (1) follows similarly. Let N be a Borel set in(−∞, 0) with Lebesgue measure zero. Write Jn as the n−th jump time of the compoundPoisson process Z. Observe that for x > 0,

Px [Xτ < 0, Xτ ∈ N ] =∞∑n=1

Px [Xτ < 0, Xτ ∈ N , τ = Jn] .

We first show that Px [Xτ ∈ N , Xτ < 0, τ = J1] = 0. Note since X(+), J1 and Y1 are inde-pendent, so are X(+)

J1and Y1. Hence, we have

Px [Xτ ∈ N , Xτ < 0, τ = J1] =Px[X

(+)J1

− Y1 ∈ N , X(+)J1

− Y1 < 0, τ = J1

]≤Px

[X

(+)J1

− Y1 ∈ N]

=∫

Px [z − Y1 ∈ N ] dG(z),

where we G is the distribution of X(+)J1

. Since Y1 has an absolute continuous distribution,we deduce from the last inequality that Px [Xτ ∈ N , Xτ < 0, τ = J1] = 0.

In general, for each n ≥ 2, we have by Strong Markov Property that

Px [Xτ < 0, Xτ ∈ N , τ = Jn] = Ex[1τ>Jn−1PXJn−1

[Xτ < 0, Xτ ∈ N , τ = J1]]

= 0,

since Jn−1 is an (Ft)−stopping time and XJn−1 > 0 on [τ > Jn−1]. This concludes thatPx [Xτ < 0, Xτ ∈ ·] is absolutely continuous with respect to Lebesgue measure for everyx > 0.

21

Appendix B: Toolbox For Phase-Type Distributions

We first give a summary of some facts of matrix algebra.

Theorem B.1 (1) The matrix exponential function ehA is differentiable on R and

dehA

dh= AehA = ehAA.

(2) Let θ1, · · · , θk be the eigenvalues of a matrix A. Then for each s > maxi<(θi),limt→∞ e−st exp(tA) = 0.(3) Let B be a subintensity matrix. Then B is nonsingular if and only if every eigenvalueof B has a strictly negative real part.(4) If A is nonsingular, then

∫ tv e

xAdx = A−1(etA − evA

); if further all eigenvalues of A

have strictly negative real parts, then∫∞0 exAdx = −A−1.

The following summarizes the main analytic properties of phase type distributions.

Theorem B.2 Suppose that F is PH(α,B). Then:(1) The n−th moment of F is (−1)nn!αB−ne>.(2) For any s ∈ C with <(s) ≥ 0,

∫∞0 e−sxdF (x) = α(sI − B)−1b> and is a rational

function.

Appendix C: First Order Derivative of Φ at Zero

We consider the process (2.1) and set

∆ = supζ ∈ R+;

∫e−ξydF (y) <∞,∀ξ ∈ [0, ζ]

. (C.1)

Throughout this section, we assume that ∆ > 0 and ψ(∆−) > 0. Let r > 0. Sinceψ(0) − r < 0 and ψ is strictly convex on [0,∆), there exists a unique number ρ∗ ∈ (0,∆)such that ψ(ρ∗)− r = 0. (ρ∗ is called the Lundberg’s constant in literatures.) We write

β =c

D+ ρ∗, α = β + ρ∗, (C.2)

and hρ∗(x) = e−ρ∗xh(x) for any function h. Recall that Φ(x) = Ex [e−rτg(Xτ )] for some

fixed bounded Borel measurable function g : (−∞, 0] → R and τ = inft ≥ 0;Xt ≤ 0. Thefollowing theorem gives the first order derivative of Φ at 0 in terms of g.Theorem C.1 Suppose r > 0. The derivative of Φ at 0 is given by

Φ′(0+) = ϑ0 −λ

D

∫ 0

−∞dF (y)

∫ −y

0dvΦρ∗(v)e−ρ

∗y, (C.3)

where

ϑ0 = −βg(0) +λ

D

∫ ∞

0dv

∫ ∞

vdF (y)e−ρ

∗ygρ∗(v − y). (C.4)

In particular, if∫(−∞,0] dF = 0, then Φ′(0+) = ϑ0.

22

Proof. By Theorem 2.2, we have (L− r)Φ(v) = 0 for all v > 0. Multiplying both sides ofthis equation by e−ρ

∗v gives

e−ρ∗v

[Dd2

dv2+ c

d

dv− (λ+ r)

]Φ(v) + λ

∫ ∞

−∞Φ(v − y)e−ρ

∗vdF (y) = 0. (C.5)

Note that Φ(v) = eρ∗vΦρ∗(v) for v ∈ R. Then

Φ′(v) = ρ∗eρ∗vΦρ∗(v) + eρ

∗vΦ′ρ∗(v) (C.6)

and

Φ′′(v) = (ρ∗)2eρ∗vΦρ∗(v) + 2ρ∗eρ

∗vΦ′ρ∗(v) + eρ

∗vΦ′′ρ∗(v).

Hence, (C.5) becomes

0 =DΦ′′ρ∗(v) +

(c+ ρ∗σ2

)Φ′ρ∗(v) + λ

∫ ∞

−∞Φρ∗(v − y)e−ρ

∗ydF (y) +[D(ρ∗)2 + cρ∗ − (λ+ r)

]Φρ∗(v).

Recall that ψ(ρ∗) = r. Then

0 =DΦ′′ρ∗(v) +

(c+ ρ∗σ2

)Φ′ρ∗(v) + λ

∫ ∞

−∞Φρ∗(v − y)e−ρ

∗ydF (y)− λΦρ∗(v)∫e−ρ

∗ydF (y).

(C.7)

By Theorem 2.1, Φ′(0+) exists in the sense of right hand derivative. Therefore, integrating(C.7) from v = 0 to v = z gives

D[Φ′ρ∗(z)− Φ′

ρ∗(0+)]+

(c+ ρ∗σ2

)[Φρ∗(z)− Φρ∗(0)] + λ

∫ z

0dv

∫ v

−∞dF (y)Φρ∗(v − y)e−ρ

∗y

− λ

∫ z

0dv

∫dF (y)Φρ∗(v)e−ρ

∗y + λ

∫ z

0dv

∫ ∞

vdF (y)gρ∗(v − y)e−ρ

∗y = 0. (C.8)

Note that we have∫ z

0dv

∫ v

−∞dF (y)Φρ∗(v − y)e−ρ

∗y

=∫ 0

−∞dF (y)

∫ z

0dvΦρ∗(v − y)e−ρ

∗y +∫ z

0dF (y)

∫ z

ydvΦρ∗(v − y)e−ρ

∗y (Fubini’s Theorem)

=∫ 0

−∞dF (y)

∫ z

0dvΦρ∗(v − y)e−ρ

∗y +∫ z

0dF (y)

∫ z−y

0dvΦρ∗(v)e−ρ

∗y (Change of Variable)

=∫ 0

−∞dF (y)

∫ z

0dvΦρ∗(v − y)e−ρ

∗y +∫ z

0dv

∫ z−v

0dF (y)Φρ∗(v)e−ρ

∗y. (Fubini’s Theorem)

So, (C.8) is equivalent to

0 =D[Φ′ρ∗(z)− Φ′

ρ∗(0)]+

(c+ ρ∗σ2

)[Φρ∗(z)− Φρ∗(0)] + λ

[ ∫ 0

−∞dF (y)

∫ z

0dvΦρ∗(v − y)e−ρ

∗y

−∫ z

0dv

(∫ ∞

z−v+

∫ 0

−∞

)dF (y)Φρ∗(v)e−ρ

∗y +∫ z

0dv

∫ ∞

vdF (y)e−ρ

∗ygρ∗(v − y)

]. (C.9)

23

By Theorem 2.1, Φ′(z) and Φ(z) tend to zero as z →∞. Hence, letting z →∞ in the lastequation gives

Φ′ρ∗(0) =

1D

[− (c+ ρ∗σ2)g(0)− λ

∫ 0

−∞dF (y)

∫ −y

0dvΦρ∗(v)e−ρ

∗y

+ λ

∫ ∞

0dv

∫ ∞

vdF (y)e−ρ

∗ygρ∗(v − y)

]. (C.10)

Since Φ′(0) = ρ∗g(0) + Φ′ρ∗(0), we get (C.3).

Proposition C.1 Assume (3.2) holds, and r > 0. Then the number ∆ defined by (C.1) isin (0,∞] and ψ(∆−) = +∞.

Proof. First, we show that ∆ > 0. Now, since all the eigenvalues of B− have strictlynegative real parts by Theorem B.1(3), so are the eigenvalues of B− + ζI for all ζ in anopen interval in (0,∞). For all these ζ, we obtain from Theorem B.1(4) that∫ ∞

0eζyf(−)(y)dy = α−(−ζI −B−)−1b>− <∞

and hence∫e−ζydF (y) <∞.

Next, we show that ψ(∆−) = +∞. If ∆ < ∞, ∆ must be a pole of ψ(ζ) on C so thatψ(∆−) = +∞ or −∞. Since ψ is strictly convex on [0,∆), we have ψ(∆−) = +∞. If∆ = ∞, we have ψ(∆−) = +∞ by the fact that D > 0. This completes the proof.

References

1. Asmussen, S.: Ruin probabilities. Singapore: World Scientific Publishing Co. Pte.Ltd. 2000

2. Asmussen, S., Avram F., Pistorius, M.R.: Russian and American put options underexponential phase-type Levy models. Stochastic Processes and their Applications109, 79-111 (2004)

3. Bertoin, J.: Levy processes. Cambridge Tracts in Mathematics, vol.121. Cambridge:Cambridge Universiry Press 1996

4. Cai, J., Yang H.: Ruin in the perturbed compound Poisson risk process under interestforce. Advances in Applied Probability 37, 819-835 (2005)

5. Chan, T.: Pricing perpetual American options driven by spectrally one-sided Levyprocesses. In: Kyprianou(eds.): Exotic option pricing and advanced Levy models.England: John-Wiley and Sons Inc. 2005, pp. 195-216

6. Chen, Y.T., Lee, C.F, Sheu, S.C.: An analytic study of bond price with jump risk,submitted, National Chiao Tung University, (2006a)

24

7. Chen, Y.T., Lee, C.F, Sheu, Y.C.: On Gerber Landry integral equation with two-sidedjumps, preprint, National Chiao Tung University, (2006b)

8. Chen, Y.T., Sheu, Y.C.: Optimal capital structure under phase-type jump diffusionasset return, preprint, National Chiao Tung University, (2006c)

9. Chen, N., Kou, S.: Credit spreads, optimal capital structure, and implied volatilitywith endogenous default and jump risk, preprint, University of Columbia, (2005)

10. Dao, B., Jeanblanc, M.: Double exponential jump diffusion process: A structuralmodel of endogenous default barrier with roll-over debt structure, preprint, Universited’Evry, (2006)

11. Dufresne, F., Gerber, H.U.: Three methods to calculate the probability of ruin.ASTIN Bulletin 19, 71-90(1989)

12. Dufresne, F., Gerber, H.U.: Risk theory for the compound Poisson process that isperturbed by diffusion. Insurance: Mathematics and Economics 10, 51-59 (1991)

13. Friedberg, S.H., Insel, A.J., Spence, L.E.: Linear algebra, 3rd edn. New Jersey:Prentice Hall 1997

14. Gerber, H.U.: An extension of the renewal equation and its application in the collec-tive theory of risk. Scandinavian Actuarial Journal, 205-210 (1970)

15. Gerber, H.U., Landry B.: On the discounted penalty at ruin in a jump-diffusionand the perpetual put option. Insurances: Mathematics and Economics 22, 263-276(1998)

16. Gerber, H.U., Shiu, E.S.W.: On the time value of ruin. North American ActuarialJournal 2, 48-78 (1998).

17. Hilberink, B., Rogers L.C.G: Optimal capital structure and endogenous default. Fi-nance and Stochastics 6, 237-263 (2002)

18. Jacod, J., Shiryayev, A.N.: Limit theorems for stochastic processes, 2nd edn. BerlinHeidelberg New York: Springer 2003

19. Kallenberg, O.: Foundations of modern probability, 2nd edn. Berlin Heidelberg NewYork: Springer 2002

20. Kou, S.G., Wang, H.: First passage times of a jump diffusion process. Advances inApplied Probability 35, 504-531 (2003)

21. Kou, S.G., Wang, H.: Option pricing under a double exponential jump diffusionmodel. Management Science 50, 1178-1192 (2004)

22. Leland, H.E.: Bond prices, yield spreads, and optimal capital strucutre with defaultrisk, working paper 240, IBER. Berkeley: University of California, (1994)

23. Merton R.: Option pricing when underlying stock returns are discontinuous. J. Fi-nancial Economics 3, 125-144 (1976)

25

24. Mordecki, E.: Optimal stopping and perpetual options for Levy processes. Financeand Stochastics 6, 473-493 (2002)

25. Revuz, D., Yor M.: Continuous martingales and Brownian motion, corrected 3rdprinting of the 3rd Edition. Berlin: Springer 2005

26. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic processes for insuranceand finance. England: John-Wiley and Sons Inc. 1999

27. Sato, K.I.: Levy processes and infinitely divisible distributions. Studies in advancesmathematics, vol.68. Cambridge: Cambridge University Press 1999

28. Stein, E.M., Shakarchi, R.: Real analysis. New Jersey: Princeton University Press2005

29. Tenenbaum, M., Pollard, H.: Ordinary differential equations. New York: Dover Pub-lications 1985

30. Tsai, C. Chi-Liang, Wilmott, G.E.: A generalized defective renewal equation for thesurplus process perturbed by diffusion. Insurance: Mathematics and Economics 30,51-66 (2002)

31. Wang, G., Wu, R.: Some distributions for classical risk process that is perturbed bydiffusion. Insurance: Mathematics and Economics 26, 15-24 (2000)

26