TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2015/354129 354129 Research Article Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process http://orcid.org/0000-0003-2539-5443 Yin Chuancun 1 Yuen Kam Chuen 2 Shen Ying 1 Hu Taizhong 1 School of Mathematical Sciences Qufu Normal University Shandong 273165 China qfnu.edu.cn 2 Department of Statistics and Actuarial Science The University of Hong Kong Pokfulam Road Hong Kong hku.hk 2015 1382015 2015 28 05 2014 09 11 2014 1382015 2015 Copyright © 2015 Chuancun Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.

1. Introduction

In the literatures of actuarial science and finance, the optimal dividend problem is one of the key topics. For companies paying dividends to shareholders, a commonly encountered problem is to find a dividend strategy that maximizes the expected total discounted dividends until ruin. The pioneer work can be traced to de Finetti  who considered a discrete-time risk model with step sizes ±1 and showed that a certain barrier strategy maximizes the expected discounted dividend payments. Since then, the problem of finding the optimal dividend strategy has become a popular topic in the actuarial literature. For diffusion models, see, for example, Jeanblanc-Picqué and Shiryaev , Asmussen and Taksar , Gerber and Shiu , Løkka and Zervos , Paulsen , He and Liang , and Bai and Paulsen . For the Cramér-Lundberg risk model, some related works on this subject include, among others, Gerber , Azcue and Muler [10, 11], Yuen et al. , Kulenko and Schmidli , Bai and Guo , and Hunting and Paulsen .

Analysis of optimal dividends for Lévy risk processes is of particular interest which have undergone an intensive development. For example, Avram et al.  considered a general spectrally negative Lévy process and gave a sufficient condition involving the generator of the Lévy process for the optimality of barrier strategy; Loeffen  showed that barrier strategy is optimal among all admissible strategies for general spectrally negative Lévy risk process with completely monotone jump density; Kyprianou et al.  relaxed this condition on the jump density; Yin and Wang  also studied the same problem and gave an alternate proof of the result; Loeffen [20, 21] considered the optimal dividend problem with transaction costs and a terminal value for the spectrally negative Lévy process. Recently, Bayraktar et al.  using the fluctuation theory of spectrally positive Lévy processes show the optimality of barrier strategies for all such Lévy processes. See Yin and Wen  for a different approach. All of the above mentioned works are based on spectrally one-sided models. There are, however, few papers that studied the analogous problems for Lévy process with two-sided jumps (cf. Bo et al. [24, 25]). Inspired by the works of Avram et al. , Loeffen , and Kyprianou et al. , Yuen and Yin  considered the optimal dividend problem for a special Lévy process with both upward and downward jumps and showed that the optimal strategy takes the form of a barrier strategy if the Lévy measure (both negative and positive jumps) has a completely monotone density. The purpose of the present paper is to extend the result of Yuen and Yin  to the case with less restrictive conditions on the Lévy measure. Although the broader case definitely makes the optimization problem more challenging and complex, recent results on the theory of potential analysis of subordinators can be applied to handle it. In particular, our main results show that the optimal dividend strategy is still of a barrier type if the Lévy process has certain positive jumps and Lévy density of negative jumps is completely monotone or log-convex.

The paper is organized as follows. In Section 2, we introduce the mathematical formulation of the problem. In Section 3, we give a brief review on ladder processes and potential measure for general Lévy processes. The convexities of the ruin probability and the scale function are discussed in Sections 4 and 5 and the main results and their proofs are given in Section 6.

2. The Model

To present the mathematical formulation of the problem of study, let us first introduce some notations and definitions. Let X={Xt}t0 be a real-valued Lévy process on a filtered probability space (Ω,F,F,P) where F=(Ft)t0 is generated by the process X and satisfies the usual conditions of right continuity and completeness. Denote by Px the law of X when X0=x. Let Ex be the expectation associated with Px. For notational convenience, we write P and E when X0=0. Write the Lévy triplet of X as (a,σ2,Π), where a,σ0 are real constants and Π is a positive measure on (-,){0} which satisfies the integrability condition (1)-1x2Πdx<.If Π(dx)=π(x)dx, then we call π the Lévy density. The characteristic exponent of X is given by (2)κθ=-1tlogEeiθXt=-iaθ+12σ2θ2+-1-eiθx+iθx1|x|<1Πdx,where 1A is the indicator of set A. Furthermore, define the Laplace exponent of X by(3)Ψθ=1tlogEeθXt=aθ+12σ2θ2+-eθx-1-θx1|x|<1Πdx.Such a Lévy process is of bounded variation if and only if σ=0 and -11|x|Π(dx)<. If Π{(0,)}=0, then the Lévy process X with no positive jumps is called the spectrally negative Lévy process; if Π{(-,0)}=0, then the Lévy process X with no negative jumps is called the spectrally positive Lévy process. It is usual to assume that P(limtXt=+)=1 which says nothing other than Ψ(0+)>0. For more information on Lévy processes we refer to the excellent book by Kyprianou .

Now, we consider an insurance company or investment company whose cash reserve process (also called risk process or surplus process) evolves according to the process X before dividends are deducted. Let ξ={Ltξ:t0} be a dividend policy consisting of a right-continuous nonnegative nondecreasing process adapted to the filtration {Ft}t0 of X with L0-ξ=0, where Ltξ represents the cumulative dividends paid up to time t. Given a control policy ξ, the controlled reserve process with initial capital x0 is given by Uξ={Utξ:t0} where(4)Utξ=Xt-Ltξ,with X0=x. Let τξ={t>0:Utξ<0} be the ruin time when dividend payments are taken into account. Define the value function associated to dividend policy ξ by (5)Vξx=Ex0τξe-δtdLtξ,where δ>0 is the discounted rate. The integral is understood pathwise in a Lebesgue-Stieltjes sense. Clearly, Vξ(x)=0 for x<0. A dividend policy is called admissible if Ltξ-Lt-ξUtξ for t<τξ and Lτξξ-Lτξ-ξ=0 for τξ<. Denote by Ξ the set of all admissible dividend policies. Our objective is to find (6)Vx=supξΞVξx,and an optimal policy ξΞ such that Vξ(x)=V(x) for all x0. The function V is called the optimal value function.

We denote by ξb={Ltb:t0} the barrier strategy at b and let Ub be the corresponding risk process; that is, Utb=Xt-Ltb. Note that ξbΞ. Also, if U0b[0,b], then the process Ltb can be explicitly represented by (7)Ltb=supstXs-b0.If U0b=x>b, then (8)Ltb=x-b1t=0+supstXs-b0.Denote by Vb(x) the dividend value function if barrier strategy ξb is applied; that is,(9)Vbx=Ex0τξbe-δtdLtb.Applying Ito’s formula for semimartingale, we can prove that Vb is the solution to (10)ΓVbx=δVbx,x>0,Vbx=0,x<0,Vb0=0,σ2>0,Vbb=1,Vbx=x-b+Vbb,where Γ is the infinitesimal generator of X with(11)Γgx=12σ2gx+agx+-gx+y-gx-gxy1y<1×Πdy.In the sequel, we assume that, for any δ>0, the equation Ψ(z)=δ has a unique solution on (0,), say ρ(δ). A typical example is that the Lévy measure of the positive jumps has the following gamma distribution Γ(r,1/γ); that is, (12)Px=0xr1/γΓ1/γy1/γ-1e-rydy,x>0,where r is a positive number and γ is an even number.

Following similar reasoning to Yuen and Yin , Vb can be expressed as(13)Vbx=hxhb,0xb,x-b+hbhb,x>b,where(14)hx=1-ψ~xeρδx.Here, let ψ~(u) be the ruin probability for a Lévy process X~ with Laplace exponent ψρ(δ) given by ψρ(δ)(η)=Ψ(η+ρ(δ))-δ. Note that the process X~ has the Lévy triplet (a~,σ~2,Π~), where σ~2=σ2, Π~(dx)=eρ(δ)xΠ(dx), and (15)a~=a+σ2ρδ+-keρδy-1y1|y|1Πdy.Moreover,(16)|x|1eρδxΠdx<.

3. Some Results on Ladder Processes and Potential Measure

In this section, we recap some basic facts about ladder processes and potential measure. Consider the dual process Y={Yt}t0, with Y0=0, where Yt=-Xt, t0. It is easy to see that the Lévy triplet of Y is (-a,σ2,ΠY), where ΠY(dx)=πX(-x)dx. Let (17)Yt_=inf0stYs,Y¯t=sup0stYsbe the processes of the first infimum and the last supremum of the Lévy process Y, respectively. Following Klüppelberg et al. , we now introduce the notion of ladder processes and potential measure. Let L={Lt:t0} denote the local time in the time period [0,t] that Y¯-Y spends at zero. Then L-1={Lt-1:t0} is the inverse local time such that Lt-1=inf{s0:Ls>t}, where we take the infimum of the empty set as . Define an increasing process H by {Ht=YLt-1:t0}, that is, the process of new maxima indexed by local time at the maximum. The processes L-1 and H are both defective subordinators, and we call them the ascending ladder time and ladder height process of Y, respectively. It is understood that Ht= when Lt-1=. Throughout the paper, we denote the nondefective versions of L, L-1, and H by L, L-1, and H, respectively. In fact, the pair (L-1, H) is a bivariate subordinator. Define (L^-1,H^) the descending ladder time and the ladder height processes in an analogous way. Note that H^ is a process which is negatively valued. Because Y drifts to -, the decreasing ladder height process is not defective. Associated with the ascending and descending ladder processes are the bivariate renewal functions U and U^. The former is defined by (18)Udx,ds=0PHtdx,Lt-1dsdt.Taking Laplace transforms shows that (19)0e-βx-αsUdx,ds=1kα,β,α,β0,where k(α,β) is its joint Laplace exponent such that (20)k0,β=q+cβ+0,1-e-βxΠHdx,q0 is the killing rate of H so that q>0 if and only if limtYt=-, c0 is the drift of H, and ΠH is its jump measure. Denote the marginal measure of U(·,·) by (21)Udx=Udx,0,=0PHtdxdt=0e-qtPHtdxdt,x0.The function U is called the potential/renewal measure. As for the descending ladder process, U^ and k^ are defined similarly. Write Π+ and Π- for the restrictions of Π(du) and Π(-du) to (0,). Furthermore, for u>0, define (22)Π¯Y+u=ΠYu,,Π¯Y-u=ΠY-,-u,Π¯Yu=Π¯Y+u+Π¯Y-u.

We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function ϕ:(0,)(0,) is called a Bernstein function if it admits a representation (23)ϕλ=a+bλ+01-e-λxμdx,where a0 is the killing term, b0 is the drift, and μ is the Lévy measure concentrated on (0,) satisfying 0(1x)μ(dx)<. A function ψ is called a special Bernstein function if the function ψ(λ)=λ/ϕ(λ) is again a Bernstein function. Let (24)ψλ=a~+b~λ+01-e-λxνdxbe the corresponding representation. It was shown in Song and Vondraček  that (25)b~=1a+μ0,1b=0,  a~=1b+0tμdt1a=0,μ0,<.A possibly killed subordinator is called a special subordinator if its Laplace exponent is a special Bernstein function. Song and Vondraček  showed that a sufficient condition for ϕ to be a special subordinator is that μ(x,) is log-convex on (0,). A function ϕ:(0,)R is called a complete Bernstein function if there exists a Bernstein function η such that (26)ϕλ=λ2Lηλ,λ>0,where L stands for the Laplace transform. It is known that every complete Bernstein function is a Bernstein function and that the following three conditions are equivalent:

ϕ  is a complete Bernstein function;

ψ(λ)=λ/ϕ(λ) is a complete Bernstein function;

ϕ is a Bernstein function whose Lévy measure μ is given by (27)μdt=dt0e-stνds,

where ν is a measure on (0,) satisfying (28)011sνds<,11s2νds<.To end the section, we present two results which are useful in potential theory and will be used in later sections of the paper. The first due to Kyprianou et al.  (see also Song and Vondraček ) is summarized in Lemma 1 while the second due to Kingman  and Hawkes  is given in Lemma 2.

Lemma 1.

Let H be a subordinator whose Lévy density, say μ(x), x>0, is log-convex. Then, the restriction of its potential measure to (0,) has a nonincreasing and convex density. Furthermore, if the drift of H is strictly positive, then the density is in C1(0,).

Lemma 2.

Suppose that H is a subordinator with Laplace exponent ϕ and potential measure U. Then, U has a density u which is completely monotone on (0,) if and only if the tail of the Lévy measure is completely monotone.

Remark 3.

Note that the tail of the Lévy measure μ is a completely monotone function if and only if μ has a completely monotone density. Thus, we have the following two equivalent statements: ϕ is a complete Bernstein function if and only if U has a density u which is completely monotone on (0,); or, equivalently, U has a density u which is completely monotone on (0,) if and only if μ has a completely monotone density.

4. Convexity of Probability of Ruin

Define the probability of ruin by (29)ψx=Pthere  exists  t0  such  that  x+Xt0=Pthere  exists  t0  such  that  Ytx.It follows from Bertoin and Doney  that ψ(x)=αU(x,), where α-1=U(0,)=0P(Ht<)dt, with U given in (21).

For simplicity, we write the Lévy measure Π as (30)Πdx=Π+dx,x>0,π--xdx,x<0.Recall that an infinitely differentiable function f(0,)[0,) is called completely monotone if (-1)nf(n)(x)0 for all n=0,1,2, and all x>0.

Lemma 4 (see Vigon [<xref ref-type="bibr" rid="B32">34</xref>]).

For the Lévy process X, one has (31)Π¯Hx=--0U^dyΠ¯Y+x-y=--0U^dyΠ¯X  -x-y,x>0,where Y=-X and U^ is the potential measure corresponding to H^.

Theorem 5.

(i) Suppose π- is completely monotone on (0,). Then, the probability of ruin ψ is completely monotone on (0,). In particular, ψC(0,).

(ii) Suppose π- is log-convex on (0,). Then,

ψ is convex on (0,);

ψ is concave on (0,);

if X has no Gaussian component, then ψ is twice continuously differentiable except at finitely or countably many points on (0,), else ψC2(0,).

Proof.

We first prove (i). Since π- is completely monotone on (0,), it follows from Lemma 4 that the tail ΠH(x,) of Lévy measure ΠH is a complete monotone function. Also, it follows from Lemma 2 that the potential measure U has a density u which is completely monotone on (0,). Thus, the probability of ruin ψ is completely monotone on (0,) as ψ(x)=αU(x,).

We now prove (ii). The log-convexity of π- implies the log-convexity of Π¯Y+, and hence Π¯H is log-convex on (0,) due to Lemma 4 as log-convexity is preserved under mixing. It follows from Lemma 1 that the potential measure U has a nonincreasing and convex density u. Thus, ψ=-αu is nondecreasing and concave on (0,), and hence (a) and (b) are proved. Since a convex function on (0,) is differentiable except at finitely or countably many points, we see that ψ is twice continuously differentiable except at finitely or countably many points on (0,) if X has no Gaussian component. On the other hand, if X has a Gaussian component or, equivalently, the drift of ascending ladder processes H strictly positive, then it follows from Lemma 1 that uC1(0,), and hence ψC2(0,). Therefore, (c) is proved.

5. Convexity of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M266"><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:math></inline-formula>

For h in (14), define a barrier level by (32)b=supb0:hbhxx0,where h0 is understood to be the right-hand derivative at 0.

For a spectrally negative Lévy process, that is, in the case of Π{(0,)}=0, it was shown in Loeffen  that the derivative of the δ-scale function Wδ(x) is convex for δ>0 if Π(x,) is completely monotone. This implies that there exists an a0 such that W(δ) is concave on (0,a) and convex on (a,). Also, Kyprianou et al.  showed that if Π(x,) has a density on (0,) which is nonincreasing and log-convex; then, for each δ0, the scale function W(δ)(x) and its first derivative are convex beyond some finite value of x.

Parallel to the results of Loeffen  and Kyprianou et al.  for spectrally negative Lévy processes, we have the following results.

Theorem 6.

(i) Suppose π- is completely monotone on (0,). Then, the derivative hu is strictly convex on (0,) and hC(0,).

(ii) Suppose π- is log-convex on (0,). Then, the h and its derivative h are strictly convex on (b,). Moreover, if X has no Gaussian component, h is twice continuously differentiable except at finitely or countably many points on (0,), else hC2(0,).

Proof.

Since π- is completely monotone on (0,), we have π~- which is also completely monotone on (0,), where Π~(dx)=π~-(-x)dx,  x<0. We can now apply Theorem 5 to deduce that the probability of ruin ψ~ is completely monotone on (0,). In particular, ψ~C(0,). It is easy to prove that hu is strictly convex on (0,) and hC(0,). Hence, (i) is proved.

Let H~ (H~^) be the ascending (descending) ladder height process of Y~=-X~. By Lemma 4, we have (33)Π¯H~x=--0U~^dyΠ¯X~-x-y,x>0,where U~^ is the renewal measure corresponding to H~^. Then, (34)ΠH~x=-eρδx-0U~^dye-ρδyπ-y-xeρδxν+x.The assumption of log-convexity of π- implies that ν+ is log-convex, and hence ΠH~(x) is also log-convex. It follows from Lemma 1 of Kyprianou and Rivero  that the restriction of its potential measure to (0,) of a subordinator with Lévy density ν+ has a nonincreasing and convex density, say fδ. Also, the restriction of its potential measure to (0,) of a subordinator with Lévy density ΠH~(x) has a nonincreasing and convex density, say hδ. Moreover, hδ(x)=eρ(δ)xfδ(x). Thus, ψ~(x)=-α~eρ(δ)xfδ(x), where α~-1=0P(H~t<)dt. Since h(x)=[1-ψ~(x)]eρ(δ)x, we have (35)hx=ρδhx+αfδx,x>0.This implies that hx tends to as x tends to as limxh(x)=. Thus b<. Applying the same arguments as those in Kyprianou et al. , we can prove that h and its derivative h are strictly convex on (b,). Finally, the smoothness of h is a direct consequence of Theorem 5. So, (ii) is proved.

6. Main Results and Proofs

We now present the main results of the paper about the optimality of the barrier strategy ξb for de Finetti’s dividend problem for general Lévy processes. This is a continuation of the work of Yuen and Yin  in which a special Lévy process with both upward and downward jumps and a completely monotone density was considered.

Theorem 7.

Suppose that ν is a nonnegative function on (0,) which is sufficiently smooth and satisfies the following:

(Γ-δ)ν(x)0, for almost every x>0;

ν is concave on (0,);

ν(x)1,   x>0.

Then, ν(x)V(x).

Theorem 8.

Suppose that Vb defined in (13) is sufficiently smooth and satisfies

Vbx>1, for all x0,b;

(Γ-δ)Vb(x)0, for all  x>b.

Then, Vb(x)=V(x). In particular, if (Γ-δ)Vb(x)0, for all x>b, then Vb(x)=V(x).

Theorem 9.

Suppose that π- is completely monotone. Then, Vb(x)=V(x); that is, the barrier strategy at b is the optimal strategy among all admissible strategies.

Theorem 10.

Suppose that π- is log-convex on (0,). Then, Vb(x)=V(x); that is, the barrier strategy at b is the optimal strategy among all admissible strategies.

Before proving the main results, we give two lemmas which are similar to those in Loeffen  for spectrally negative Lévy processes.

Lemma 11.

Suppose that h is sufficiently smooth and convex in the interval (b,). Then, the following statements hold:

b<;

Vb(x)1 for x[0,b] and Vb(x)=Vx(x)=1 for x>b;

(Γ-δ)Vb(x)=0 for x(0,b).

Proof .

As limxhx=, we have (i). For (ii), Vb(x)=h(x)/h(b) for x[0,b]; it follows from the definition of b that Vb(x)1 for x[0,b]; Vb(x)=Vx(x)=1 for x>b because of Vb(x)=x-b+Vb(b); and Vx(x)=1 since Vx(x)=h(x)/h(x). Finally, (iii) is due to (Γ-δ)h(x)=0 for x(0,b) and (13).

Lemma 12.

Suppose that h is sufficiently smooth and is convex in the interval (b,). Then, for x>b,

Vb(x)=0Vx(x-) if σ0;

Vb(y)Vx(y),y[0,x];

Vb(x)Vx(x);

(Γ-δ)Vb(x)0.

Proof .

If σ0, Vb(x)=0 is clear. Also, since hC2(0,) and is convex in the interval (b,), we have Vx(x-)=limyxVx(y)=limyxh(y)/h(x)0. Thus, (i) is proved.

For y[0,b], by the definition of b, we have (36)Vby-Vxy=hyhb-hyhx0.On the other hand, for y[b,x], by the convexity of h on (b,), we have (37)Vby-Vxy=1-hyhx0.These give (ii).

Note that Vb(b)=h(b)/h(b)h(b)/h(x)=Vx(b) and that (Vb-Vx) is nondecreasing on (b,) because of (ii). Thus, Vb(x)Vx(x); that is, (iii) holds.

For x>b, (Γ-δ)Vx(x-)=limyx(Γ-δ)Vx(y)=0. For xb, we have (38)Γ-δVbx=Γ-δVbx-Γ-δVxx-=12σ2Vbx-Vxx-+aVbx-Vxx+-Vbx+y-Vbx-Vbxy1y<1×πydy+-Vxx+y-Vxx-Vxxy1y<1×πydy-δVbx-VxxI1+I2+I3-I4.Lemmas 11(ii) and 12(i) imply that I10, and Lemma 12(iii) implies that I40. For I2+I3, we have (39)I2+I3=-Vb-Vxx+y-Vb-VxxVb-Vx-xy1y<1-Vb-Vxxy1y<1πydy=-0Vb-Vxx+y-Vb-VxxVb-Vx-xy1y<1-Vb-Vxxy1y<1πydy+0Vb-Vxx+y-Vb-VxxVb-Vx-xy1y<1-Vb-Vxxy1y<1πydyJ1+J2.Applying Lemmas 11(ii) and 12(ii) yields J10. For y>0, we obtain (40)Vb-Vxx+y=Vb-Vxx=x-b+hbhb-hxhx,which, together with Lemma 12(ii), imply that J2=0. These prove (iv).

We now present the proofs of Theorems 710.

Proof of Theorem <xref ref-type="statement" rid="thm6.1">7</xref>.

Define the jump measure of X by (41)μX=μXω,dt,dy=s1ΔXs0δs,ΔXsdt,dy,and its compensator by υ=υ(dt,dy)=dtΠ(dy). Then, the Lévy decomposition [36, Theorem 42] gives (42)Xt=σBt+0,t×Ry1y<1μX-υ+at+0,t×Ry-y1y<1μXMt+at+0stΔXs1y1,where B={Bt}t0 is a standard Brownian motion and Mt is a martingale with M0=0.

Note that ν is smooth enough for an application of the appropriate version of Itô’s formula and the change of variables formula. In fact, if X is of bounded variation, then νC1(0,) and we are allowed to use the change of variables formula [36, Theorem 31]; if X has a Gaussian exponent, then νC2(0,) and we are allowed to use Itô’s formula [36, Theorem 32]; and if X has unbounded variation and σ=0, then ν is twice continuously differentiable almost everywhere but is not in C2(0,) and we can use Meyer-Itô’s formula [36, Theorem 70] and product rule formula. In any cases, for any appropriate localization sequence of stopping times {tn,n1}, we get under Px(43)e-δtnτξνUtnτξξ-νU0ξ=0tnτξe-δsdMsξ+0tnτξe-δsΓ-δνUs-ξds+stnτξ1ΔLsξ>0e-δs×νUs-ξ+ΔXs-ΔLsξ-νUs-ξ+ΔXs+νUs-ξ+ΔXsΔLsξ-0tnτξe-δsνUs-ξdLsξ,where (44)Mtξ=st1|ΔXs|>0×νUs-ξ+ΔXs-νUs-ξ1|ΔXs|1-ΔXsνUs-ξ1|ΔXs|1-0t-νUs-ξ-y-νUs-ξ+yνUs-ξ1y1×πydyds+0tνUs-ξdMsis a local martingale. The concavity of ν implies that ν(x)-ν(y)+x-yν(y)0 for any xy. Taking expectations on both sides of (43) and using conditions (i)–(iii), we obtain(45)Exe-δtnτξνUtnτξξ-νx-Ex0tnτξe-δsdLsξ.Then, letting n in (45) and recalling that ξ is an arbitrary strategy in Ξ, we get (46)νxsupξΞVξx=Vx.This ends the proof of Theorem 7.

Proof of Theorem <xref ref-type="statement" rid="thm6.2">8</xref>.

It follows from (13) and conditions (i) and (ii) that (Γ-δ)Vb(x)0 for x(0,){b} and Vb(x)1 for x>0. Similar to (43), one can show that(47)e-δtVbUtξ-VbU0ξ=0te-δtdNsξ+0te-δsΓ-δVbUs-ξds+st1ΔLsξ>0e-δs×VbUs-ξ+ΔXs-ΔLsξ-VbUs-ξ+ΔXs-0,te-δsVbUs-ξdLsξ,c,where Lsξ,c is the continuous part of Lsξ, and (48)Ntξ=st1ΔXs>0×VbUs-ξ+ΔXs-VbUs-ξUs-ξ1|ΔXs|1-ΔXsVbUs-ξ1|ΔXs|1-0t-VbUs-ξ-y-VbUs-ξUs-ξ1y1+yVbUs-ξ1y1πydyds+0tVbUs-ξdMs.Note that P(ΔLsξ>0,ΔXs<0)=0 and that Us-ξ+ΔXsb on {ΔLsξ>0,ΔXs>0}. Consequently, Vb(Us-ξ+ΔXs)=1, and hence (49)st1ΔLsξ>0e-δsVbUs-ξ+ΔXs-ΔLsξ-VbUs-ξ+ΔXs=-st1ΔLsξ>0e-δsΔLsξ.Also, for any appropriate localization sequence of stopping times {tn,n1}, we have(50)Exe-δtnτξVbUtnτξξ-ExVbU0ξ-Ex0,tnτξe-δsdLsξ.Letting n in (50) yields (51)VbxsupξΞVξx=Vx.However, (52)VbxsupξΞVξx=Vx.This ends the proof of Theorem 8.

Proof of Theorem <xref ref-type="statement" rid="thm6.3">9</xref>.

If π- is completely monotone, it follows from Theorem 6(i) that hx is strictly convex on (0,). Then, Vb is concave on (0,) because of (13). From Lemmas 11(ii) and (iii) and 12(iv), we see that the conditions in Theorem 7 are satisfied. Thus, Vb(x)V(x). Consequently, Vb(x)=V(x) and the proof is complete.

Proof of Theorem <xref ref-type="statement" rid="thm6.4">10</xref>.

If π- is log-convex on (0,), it follows from Theorem 6(ii) that h(x) is strictly convex on (b,). Then, applying Lemma 12(iv) gives (Γ-δ)Vb(x)0 for all x>b. The result follows from Theorem 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Chuancun Yin was supported by the National Natural Science Foundation of China (no. 11171179) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20133705110002).

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