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On optimality of the barrier strategy for a general Lévy risk process aKAM CHUEN YUEN, and bCHUANCUN YIN 1 1 aDepartment of Statistics and Actuarial Science, The University of Hong Kong, 0 2 Pokfulam Road, Hong Kong n a e-mail: [email protected] J 3 bSchool of Mathematical Sciences, Qufu Normal University ] R Shandong 273165, P.R. China P . h e-mail: [email protected] t a m [ 1 Abstract We consider the optimal dividend problem for the insurance risk process in v 7 a general Lévy process setting. The objective is to find a strategy which maximizes the 4 expected total discounted dividends until the time of ruin. We give sufficient conditions 4 under which the optimal strategy is of barrier type. In particular, we show that if the 0 Lévy density is a completely monotone function, then the optimal dividend strategy is a . 1 barrier strategy. This approach was inspired by the work of Avram et al. (2007) [Annals 0 1 of Applied Probability 17, 156-180], Loeffen (2008) [Annals of Applied Probability 18, 1 1669-1680] and Kyprianou et al. (2010) [Journal of Theoretical Probability 23, 547-564] : v in which the same problem was considered under the spectrally negative Lévy processes i setting. X r a Keywords Lévy processes · Optimal dividend problem · Complete monotonicity · Bar- rier strategy · Scale function · Probability of ruin Mathematics Subject Classification (2000): 60J99 · 93E20 · 60G51 1 The model and problem setting Let X = {X : t ≥ 0} be a real-valued Lévy process define on a filtered probability t space (Ω,F,F,P) where F = (F) is generated by the process X and satisfies the usual t≥0 1 conditions of right-continuity and completeness. Denote by P the probability law of x X when it starts at x. For notational convenience, we write P = P . Let E be the 0 x expectation operator associated with P with E = E . For θ ∈ R, κ(θ) denotes the x 0 characteristic exponent of X given by 1 1 ∞ κ(θ) = logE(eiθXt) = iaθ − σ2θ2 + (eiθx −1−iθx1 )Π(dx), {|x|<1} t 2 Z−∞ where a,σ are real constants, and Π is a positive measure supported on (−∞,∞)\{0} which satisfies the integrability condition ∞ min{1,x2}Π(dx) < ∞. Z−∞ The characteristics (a,σ2,Π) are called the Lévy triplet of the process and completely determines its law; Π is called the Lévy measure; and σ is the Gaussian component of X. If Π(dx) = π(x)dx, then we call π the Lévy density. Such a Lévy process is of bounded ∞ variation if and only if σ = 0 and min{|x|,1}Π(dx) < ∞. It is well known that Lévy −∞ process X is a space-homogeneous strong Markov process. R For Reθ = 0, we can define the Laplace exponent of the process X by 1 1 ∞ Ψ(θ) = logE(eθXt) = aθ + σ2θ2 + (eθx −1−θx1 )Π(dx). (1.1) {|x|<1} t 2 Z−∞ That is, E(eθXt) = etΨ(θ), Reθ = 0, t ≥ 0. Using Ito’s formula, we find that the infinitesimal generator for X is given by 1 ∞ Γg(x) = σ2g′′(x)+ag′(x)+ [g(x+y)−g(x)−g′(x)y1 ]Π(dy), (1.2) {|y|<1} 2 Z−∞ for g ∈ C2 with compact support, where 1 is the indicator function of set A. A When Π{(0,∞)} = 0, i.e., the Lévy process X with no positive jumps, is called the spectrally negative Lévy process. In this case, we recall from Bertoin (1998) and Kyprianou (2006) that for each q ≥ 0, there exists a continuous and increasing function W(q) : R → [0,∞), called the q-scale function defined in such a way that W(q)(x) = 0 for all x < 0 and on [0,∞) its Laplace transform is given by ∞ 1 e−θxW(q)(x)dx = , θ > φ(q), Ψ(θ)−q Z0 where φ(q) = sup{θ ≥ 0 : Ψ(θ) = q} is the right inverse of Ψ. Smoothness of the scale function is related to the smoothness of the underlying paths of the associated process. The following facts are taken from Kyprianou et al. (2010) and Chan et al. (2010). It is known that if X has paths of bounded variation then, for all q ≥ 0, W(q)| ∈ C1(0,∞) (0,∞) if and only if Π has no atoms. In the case that X has paths of unbounded variation, it is known that, for all q ≥ 0, W(q)| ∈ C1(0,∞). Moreover, if σ > 0, then C1(0,∞) may (0,∞) be replaced by C2(0,∞); and if the Lévy measure has a density, then the scale function 2 is always differentiable. In particular, if π is completely monotone, then W(q)| ∈ (0,∞) C∞(0,∞). For general references onLévy processes, we refer the reader, among others, to Bertoin (1998), Sato (1999) and Kyprianou (2006). In this paper, we only consider the case that Π with density π is absolutely continuous with respect to Lebesgue measure. For the Lévy process X, we consider the following de Finetti’s dividend problem. Let ξ = {Lξ : t ≥ 0} be a dividend strategy consisting of a t left-continuous non-negative non-decreasing process adapted to the filtration {F } of t t≥0 X. Specifically, Lξ represents the cumulative dividends paid out up to time t under the t control ξ for an insurance company whose risk process is modelled by X. We define the controlled risk process Uξ = {Uξ : t ≥ 0} by Uξ = X −Lξ. Let τξ = inf{t > 0 : Uξ < 0} t t t t t be the ruin time when the dividend payments are taken into account. Define the value function of a dividend strategy ξ by τξ V (x) = E e−δtdLξ , ξ x t Z0 ! where δ > 0 is the discounted rate. The integral is understood pathwise in a Lebesgue- Stieltjes sense. A dividend strategy is called admissible if Lξ − Lξ ≤ Uξ for t < τξ. In words, the t+ t t lump sum dividend payment is smaller than the size of the available capitals. Let Ξ be the set of all admissible dividend policies. The de Finetti dividend problem consists of solving the following stochastic control problem: V (x) = supV (x), ∗ ξ ξ∈Ξ and, if it exists, we want to find a strategy ξ ∈ Ξ such that V (x) = V (x) for all x ≥ 0. ∗ ξ∗ ∗ This optimization problem goes back to de Finetti (1957), who considered a discrete timerandomwalkwithstepsizes±1andshowed thatacertainbarrierstrategymaximizes expected discounted dividend payments. Optimal dividend problem has recently gained great attention inthe actuarial literature. Forthe diffusion risk process, theoptimal problem has been studied by many authors including Asmussen et al. (2000), Paulsen (2003), and Decamps and Villeneuve (2007). It is well known that under some reasonable as- sumptions, the optimality in the diffusion process setting is achieved by a barrier strategy (see, for example, Shreve et al. (1984)). The general problem for the Cramér-Lundberg risk model was first solved by Gerber in (1969) via a limit of an associated discrete problem. Recently, Azcue and Muler (2005) used the technique of stochastic control theory and Hamilton-Jacobi-Bellman (HJB) equation to solve the problem. They also included a general reinsurance strategy as another possible control. For the Cramér-Lundberg risk model with interest, Yuen et al. (2007) investigated some ruin problems in the presence of a constant dividend barrier; and Albrecher and Thonhauser (2008) derived the optimal dividend strategy which is again of band type and for exponential claim sizes collapses to a barrier strategy. In fact, for various risk models, many results in the literature indicate that a band strategy turns out to be optimal among all admissible strategies. For a more general risk process, namely the spectrally negative Lévy process, Avram et al. (2007) 3 gave a sufficient condition involving the generator of the Lévy process for the optimality of the barrier strategy; Loeffen (2008) connected the shape of the scale function to the existence of an optimal barrier strategy and showed that the optimal strategy is a barrier strategy if the Lévy measure has a completely monotone density; and Kyprianou et al. (2010) further investigated the optimal dividend control problem and showed that the problem is solved by a barrier strategy whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex. Motivated by the work of Avram et al. (2007), Loeffen (2008) and Kyprianou et al. (2010) for a spectrally negative Lévy process, our objective is to consider the optimal dividend problem for a general Lévy process (not necessarily spectrally negative). In contrast to the previous works, our approach does not rely on the theory of scale function of a spectrally negative Lévy process. Instead, the derivation of our results requires an introduction of a new function and the use of the Wiener-Hopf factorization theory. The rest of this paper is organized asfollows. In Section 2, we present some preliminary results which are derived in Rogers (1983), Bertion and Doney (1994) and Kyprianou (2006). In Section 3, we discuss a barrier strategy for dividend payments for the risk model of study. In Section 4, we give the main results and their proofs. Finally, Section 5 presents some examples and ends with a remark. 2 Two lemmas on the probability of ruin In this section, we present two results on the probability of ruin which will be used later. The first result is about the complete monotonicity of the probability of ruin. By definition, an infinitely differentiable function f ∈ (0,∞) → [0,∞) is called completely monotone if (−1)nf(n)(x) ≥ 0 for all n = 0,1,2,···. Denote by τ(q) an exponential random variable with mean 1/q which is independent of the process X. For q = 0, τ(0) is understood to be infinite. Furthermore, let X = inf X and X = sup X , τ(q) t τ(q) t 0≤t≤τ(q) 0≤t≤τ(q) be the infimum and the supremum of the the Lévy process X killed at the random time τ(q), respectively. In the case with q = 0, we always assume that ∞ E(X ) = Ψ′(0) = a+ x1 π(x)dx > 0. (2.1) 1 {|x|≥1} Z−∞ We call the characteristic functions E(exp(αX )) and E(exp(αX )) the right and τ(q) τ(q) left Wiener-Hopf factors of X, respectively. According to Rogers (1983), the Wiener-Hopf factors are called mixtures of exponentials if there are probability measures H and H + − on (0,∞] such that ∞ λH (dλ) ∞ λH (dλ) + − E(exp(αX )) = , E(exp(αX )) = . τ(q) λ−α τ(q) λ+α Z0 Z0 4 Note that if Eexp(αX ) is a mixture of exponentials, then the probability distribution τ(q) function of X has the form τ(q) P (x) = eλxH (dλ), x < 0. − − Z(0,∞] Similarly, the probability distribution function of X has the form τ(q) P (x) = (1−e−λx)H (dλ), x > 0. + + Z(0,∞] For x ≥ 0, let τ = inf{t ≥ 0 : x+ X(t) ≤ 0}. Define its Laplace transform and the probability of ruin as ψ (x) = E(e−qτ), ψ(x) = P(τ < ∞), q respectively. Then, ψ (x) = P(X ≤ −x), q τ(q) and ψ(x) = limψ (x) = P(X ≤ −x). q ∞ q→0 Definition 2.1. (Rogers (1983)) The Lévy process X has completely monotone Lévy density if there exist measures µ ,µ on (0,∞) such that + − π(x) = 1 e−txµ (dt)+1 etxµ (dt), (2.2) {x>0} + {x<0} − Z(0,∞) Z(0,∞) where 1 (µ +µ )(dt) < ∞. t(1+t)2 + − Z Theorem 2 of Rogers (1983) states that the jump measure Π has a completely monotone density if and only if the Wiener-Hopf factors of X are mixtures of exponential distributions. So, we have the following lemma. Lemma 2.1. If the jump measure Π has a completely monotone density, then the functions ψ (x) and ψ(x) are completely monotone in (0,∞). q The second result is on Cramér’s estimate for ruin probability. Roughly speaking, under suitable conditions, the probability of ruin decays exponentially when the initial capitalbecomes larger. Thefollowing lemma isextracted fromKyprianou(2006, Theorem 7.6); see also Bertion and Doney (1994, Theorem). Lemma 2.2. Assume that X is a Lévy process which does not have monotone paths, for which (i) (2.1) holds; 5 (ii) there exists a R > 0 such that Ψ(−R) = 0 where Ψ is given by (1.1); (iii) the support of Π is not lattice if Π(R) < ∞. Then, −1 ∂κ(0,β) lim eRxP(τ < ∞) = κ(0,0) R , x→∞ ∂β (cid:12)β=R! (cid:12) where the limit is interpreted to be zero if the derivative on t(cid:12)he right-hand side is infinite. (cid:12) For more details, see Kyprianou (2006, Theorem 7.6). 3 Barrier strategy In this section, we consider a simple barrier strategy for dividend payments. Under a barrier strategy, if the controlled surplus reaches the level b, then the overflow will be paid as dividends; and if the surplus is less than b, then no dividends are paid out. Let ξ = {Lb : t ≥ 0} be a barrier strategy and Ub = {Ub : t ≥ 0} be the corresponding b t t controlled risk process. Note that Ub = X − Lb and ξ ∈ Ξ. Moreover, if Ub ∈ [0,b], t t t b 0 then the strategy ξ corresponds to a reflection of the process X − b at its supremum; b and if t ≤ τ = inf{t ≥ 0 : Ub ≤ 0}, the process Lb can be defined by Lb = 0 and b t t 0 Lb = sup [X − b] ∨ 0. Note that Lb is increasing, continuous and adapted such that t s≤t s t the support of the measure dLb is contained in the closure of the set {t : Ub = b}. If t t Ub = x > b, Lb has a jump at t = 0 of size x −b to bring Ub back to the level b and a 0 t similar structure afterward. LetV (x)denotethedividend-valuefunctionifabarrierstrategywithlevelbisapplied. b Then, τb V (x) = E e−δtdLb|Ub = x . (3.1) b t 0 Z0 ! The following result shows that V (x) as a function of x satisfies the following integro- b differential equations with certain boundary conditions. Note that Paulsen and Gjessing (1997) established a similar result for a very general jump-diffusion process. Theorem 3.1. Assume that the process X have Laplace exponent (1.1) and the infinitesimal generator Γ is given by (1.2). Let V (x) be bounded and twice continuously differen- b tiable on (0,b) with a bounded first derivative and with the understanding that we mean the right-hand derivatives at x = 0. (i) If V (x) solves b ΓV (x) = δV (x), 0 < x < b, b b together with the boundary conditions V (x) = 0, x < 0, b V (0) = 0, if σ2 > 0, b V′(b) = 1, b V (x) = V (b)+x−b, x > b, b b 6 then V (x) is given by (3.1). b Furthermore, let ψ (x) be bounded and twice continuously differentiable on (0,b) with b a bounded first derivative and with the understanding that we mean the right-hand derivatives at x = 0. (ii) If ψ (x) solves b Γψ (x) = 0, 0 < x < b, b together with the boundary conditions ψ (x) = 1, x < 0, b ψ (0) = 1, if σ2 > 0, b ψ′(b) = 0, b then ψ (x) = P (τ < ∞). b x b Proof. The proofof(i) canbedone by using the arguments used inPaulsen andGjessing (1997). If σ2 > 0, the process starting from 0 immediately has a negative value. Hence, V (0) = 0. Applying Ito’s formula to e−δ(t∧τb)V (Ub ) gives b b t∧τb t∧τb e−δ(t∧τb)V (Ub ) = V (x)+ e−δs(Γ−δ)V (Ub)ds b t∧τb b b s Z0 t∧τb − e−δsV′(Ub)dLb +M , b s s t Z0 where M is a martingale. Since V (Ub ) = 0 and the support of the measure dLb is t b τb t contained in the closure of the set {t : Ub = b}, taking expectation on both sides of the t equality above yields t∧τb E e−δ(t∧τb)V (Ub ) = V (x)−E e−δsdLb . (3.2) b t∧τb b s (cid:16) (cid:17) Z0 ! Therefore, (i) follows by letting t → ∞ in (3.2). The proof of (ii) is entirely analogous to (cid:3) the proof of (i). The integro-differential equation Γh(x) = δh(x), x > 0, has, apart from a constant factor, a unique nonnegative solution h(x). This together with Theorem 3.1 gives h(x), 0 ≤ x ≤ b, V (x) = h′(b) (3.3) b ( x−a+ h(b), x > b. h′(b) Inparticular, ifΠ{(0,∞)} = 0,thenhbecomestheδ-scalefunctionW(δ) and(3.3)reduces to Proposition 1 of Avram et al. (2007), which was proved by the excursion theory. See also Renaud and Zhou (2007) and Zhou (2005) for an alternative approach. 7 4 Main results and proofs Define a barrier level by b∗ = sup{b ≥ 0 : h′(b) ≤ h′(x) for all x ≥ 0}, where h′(0)is understoodto betheright-handderivative at0, andisnot necessarily finite. We now present the main results of the paper which concerns the optimal barrier strategy ξb∗ for a general Lévy processes. This is a continuation of the work of Avram et al. (2007), Loeffen (2008), and Kyprianou et al. (2010) in which only the spectrally negative Lévy process was considered. Theorem 4.1. Suppose that h belongs to C1(0,∞) if σ = 0 and ∞ min{|x|,1}Π(dx) < −∞ ∞ and otherwise belongs to C2(0,∞). If lim h′(x) = ∞ and h(x) is convex on [b∗,∞). x→∞ R Then, the barrier strategy at b∗ is an optimal strategy. For simplicity, we write the Lévy density π as π (x), x > 0, 1 π(x) = ( π (−x), x < 0, 2 where π ,π are Lévy measures concentrated on (0,∞). 1 2 Theorem 4.2. If π and π are completely monotone on (0,∞), then the barrier strategy 1 2 at b∗ is an optimal strategy. Before proving the main results, we present several lemmas which are similar to those for spectrally negative Lévy process. For δ > 0, we consider the following second order integro-differential equation: 1 ∞ σ2h′′(x)+ah′(x)+ [h(x+y)−h(x)−h′(x)y1 ]Π(dy) = δh(x), x > 0. (4.1) {|y|<1} 2 Z−∞ Set ρ(δ) = sup{θ : Ψ(θ) = δ}. We assume that ρ(δ) > 0. For such a ρ(δ), we denote by Pρ(δ) the exponential tilting of the measure P with Radom-Nikodym derivative dPρ(δ) = eρ(δ)X(t)−δt. dP (cid:12)Ft (cid:12) Under the measure Pρ(δ), the process X(cid:12) is still a Lévy process with Laplace exponent (cid:12) ψ given by ρ(δ) ψ (η) = ψ(η +ρ(δ))−δ. ρ(δ) Let the process X˜ has the Lévy triplet (a˜,σ˜2,Π˜) where σ˜2 = σ2, Π˜(dx) = π˜(x)dx = eρ(δ)xπ(x)dx, and ∞ a˜ = a+σ2ρ(δ)+ (eρ(δ)y −1)y1 π(y)dy. {|y|≤1} Z−∞ 8 Moreover, eρ(δ)xπ(x)dx < ∞. (4.2) Z|x|≥1 We refer the reader to Kyprianou (2006) for related discussions. Note that the law of X˜ is X under the new probability measure. Let ψ˜(x) be the ruin probability for the Lévy process X˜. Then, we have the following result. Lemma 4.1. The solutionsof equation(4.1)are proportionalto the function (1−ψ˜(x))eρ(δ)x. Proof. We first claim that E(X˜ ) = Ψ˜′(0) is always positive where Ψ˜ is the Lévy 1 exponent of X˜. In fact, −1 ∞ Ψ˜′(0) = a˜+ xπ˜dx+ xπ˜dx Z−∞ Z1 ∞ 1 = a+ρ(δ)σ2 + yeρ(δ)yπ(y)dy− yπ(y)dy Z−∞ Z−1 = Ψ′(ρ(δ)) > 0, since Ψ is strictly convex and increasing on [0,ρ(δ)]. If π is completely monotone, then it follows from Definition 2.1 that π has the form (2.2). By using the integrability condition ρ(δ) (4.2), we get µ (dt) = 0. Therefore, 0 + R π˜(x) ≡ eρ(δ)xπ(x) = 1 e−(t−ρ(δ))xµ (dt)+1 e(t+ρ(δ))xµ (dt), {x>0} + {x<0} − Z(ρ(δ),∞) Z(0,∞) ˜ which shows that π˜ is completely monotone. So, we see from Lemma 2.1 that ψ is also completely monotone, and hence ψ˜ ∈ C∞(0,∞). Furthermore, it follows from Theorem ˜ 3.1 (ii) that ψ solves Γ˜ψ˜(x) = 0, x > 0, together with the boundary conditions ψ˜(x) = 1, x < 0, and ψ˜(0) = 1 if σ2 > 0, where Γ˜ is the infinitesimal generator for X˜ given by 1 ∞ Γ˜g(x) = σ˜2g′′(x)+a˜g′(x)+ [g(x+y)−g(x)−g′(x)y1 ]π˜(y)dy. {|y|<1} 2 Z−∞ Then, it can be shown by straightforward calculations that the function h(x) = (1−ψ˜(x))eρ(δ)x (cid:3) satisfies integro-differential equation (4.1). Hence, the result follows. Moreover, using arguments similar to those in Avram et al. (2007) and Loeffen (2008), we have the following two results. 9 Lemma 4.2. Suppose that h belongs to C1(0,∞) if σ = 0 and ∞ min{|x|,1}Π(dx) < ∞, −∞ and otherwise belongs to C2(0,∞). If h is also convex on [b∗,∞), then R (Γ−α)Vb∗(x) ≤ 0, for x > b∗. Lemma 4.3. (Verification lemma) Suppose that ξ is an admissible dividend strategy such that V is twice continuously differentiable and for all x > 0 ξ max{(Γ−α)V (x),1−V′(x)} ≤ 0. ξ ξ Then, V (x) = V (x) for all x. ξ ∗ To end the section, we give the proofs of Theorems 4.1 and 4.2. Proof of Theorem 4.1. The condition lim h′(x) = ∞ implies that b∗ < ∞. Clearly, x→∞ it follows from the definition of Vb∗ and Lemma 4.2 that for x > 0 (Γ−α)Vb∗(x) ≤ 0, (1−Vb′∗(x))(Γ−α)Vb∗(x) = 0, V′ (x) ≥ 1, b∗ which in turn imply that for all x > 0 max{(Γ−α)Vb∗(x),1−Vb′∗(x)} = 0. (cid:3) Hence, the result is a direct consequence of Lemma 4.3. ˜ Proof of Theorem 4.2. If π is completely monotone, then ψ is also completely mono- ˜ tone. Hence, ψ(x) admits the following representation ∞ ψ˜(x) = e−sxµ(ds), (4.3) Z0 whereµisaBorelmeasureon[0,∞). MakinguseofCramér’sestimateforruinprobability (see Lemma 2.2), we have ψ˜(x) ∼ Ce−(R+ρ(δ))x, x → ∞, where R is a positive constant such that Ψ(−R) = 0, and C is a nonnegative constant. Consequently, we have lim eρ(δ)xψ˜(x) = 0. x→∞ ρ(δ) This together with (4.3) give µ(ds) = 0, and hence 0 R∞ ∞ eρ(δ)xψ˜(x) = e−sx+ρ(δ)xµ(ds) = e−txµ(ρ(δ)+dt), Zρ(δ) Z0 which is completely monotone. In particular, we have (eρ(δ)xψ˜(x))′′′ ≤ 0. Thus, h′′′(x) = ρ(δ)3eρ(δ)x −(eρ(δ)xψ˜(x))′′′ > 0, x > 0. That is, h′ is strictly convex on (0,∞). The result follows from Theorem 4.1 since lim h′(x) = ∞. (cid:3) x→∞ 10