AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 675202 10.1155/2013/675202 675202 Research Article The Exit Time and the Dividend Value Function for One-Dimensional Diffusion Processes http://orcid.org/0000-0001-8524-2348 Li Peng 1 http://orcid.org/0000-0003-2539-5443 Yin Chuancun 1 Zhou Ming 2 Zhou Yong 1 School of Mathematical Sciences Qufu Normal University Shandong 273165 China qfnu.edu.cn 2 China Institute for Actuarial Science Central University of Finance and Economics Beijing 100081 China cufe.edu.cn 2013 20 11 2013 2013 30 08 2013 27 10 2013 2013 Copyright © 2013 Peng Li 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 investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.

1. Introduction

Diffusion processes have extensive applications in economics, finance, queueing, mathematical biology, and electric engineering. See, for example,  and the references therein. The main tool for studying various properties of diffusion is the result on exit times from an interval. Motivated by Yin et al. in , who considered the exit problems for jump processes with applications to dividend problems. In this paper, we consider the Laplace transforms of some random variables involving the exit time for the general one-dimension diffusion processes with applications to dividend problems.

Let U={Ut,t0} be a one-dimensional time-homogeneous diffusion process, which is defined by the following stochastic differential equation: (1)dUt=μ(Ut)dt+σ(Ut)dBt,U0=u(a,b), where Bt is a Brownian motion and a<b are constants. It is well known that under certain conditions on the coefficients μ(u) and σ(u), the SDE (1) has a unique strong solution for each starting point. The solution Ut is a time-homogeneous strong Markov process with infinitesimal generator as follows: (2)𝒜g(u)=12σ2(u)g′′(u)+μ(u)g(u),u(a,b), for any twice continuously differentiable function g.

Define (3)τa-=inf{t0:Uta},τb+=inf{t0:Utb},(4)τab=τa-τb+.

For δ>0, we consider the following Laplace transforms: (5)φ1(u)=Eu[e-δτa-,τa-<τb+],(6)φ2(u)=Eu[e-δτb+,τb+<τa-],(7)φ(u)=Eu[e-δτab]=φ1(u)+φ2(u).

We study the differential equations satisfied by the Laplace transforms and some applications of the popular dividend strategy in risk theory.

The rest of the paper is organized as follows. Section 2 studies the Laplace transforms of exit times and considers some popular diffusions. Some applications in the calculation of dividend value functions for the barrier strategy and the threshold strategy are considered in Section 3.

2. Laplace Transform

In this section, we consider the Laplace transform of the exit time for the general diffusion process {Ut;t0} defined by (1).

Theorem 1.

The function φ1(u) defined by (5) satisfies the following differential equation: (8)12σ2(u)φ1′′(u)+μ(u)φ1(u)=δφ1(u),u(a,b), with the boundary conditions φ1(a)=1, φ1(b)=0.

Proof.

We assume that f(u) is twice continuously differentiable and satisfies the following differential equation: (9)𝒜f(u)=δf(u),u(a,b),(10)f(a)=1,f(b)=0. Applying Dynkin’s formula to h(t,Ut)=e-δtf(Ut), we obtain (11)Eu[h(t,Ut)]=h(0,u)+Eu(0t(𝒜-δ)h(s,Us)ds)=f(u)+Eu(0te-δs(𝒜-δ)f(Us)ds). Since τab< is a stopping time, it follows from the optional sampling theorem that (12)Eu[e-δ(τabt)f(U(τabt))]=f(u)+Eu(0(τabt)e-δs(𝒜-δ)f(Us)ds0(τabt)), and letting t, we get (13)Eu[e-δτabf(Uτab)]=f(u)+Eu(0τabe-δs(𝒜-δ)f(Us)ds). By the definitions of τab, we get (14)Eu[e-δτabf(Uτab)]=f(a)Eu[e-δτa-;τa-<τb+]+f(b)Eu[e-δτb+;τb+<τa-]. Substituting (9) and (10) into (13) and (14), we get (15)f(u)=Eu[e-δτa-;τa-<τb+]=φ1(u). This completes the proof.

Theorem 2.

The function φ2(u)=Eu[e-δτb+;τb+<τa-] satisfies the following differential equation: (16)12σ2(u)φ2′′(u)+μ(u)φ2(u)=δφ2(u),u(a,b), with the boundary conditions φ2(a)=0, φ2(b)=1.

Proof.

The proof of this theorem is similar to that of Theorem 1. We first assume that f(u) is twice continuously differentiable and satisfies the following differential equation: (17)𝒜f(u)=δf(u),u(a,b),f(a)=0,f(b)=1. Applying Dynkin’s formula to e-δtf(Ut), and after the same discussion as of Theorem 1, we also can obtain (13) and (14). Substituting (17) into (13) and (14), we get (18)f(u)=Eu[e-δτb+;τb+<τa-]=φ2(u). This completes the proof.

According to the definition of (7), we can lead to the following theorem from Theorems 1 and 2.

Theorem 3.

The function φ(u)=Eu[e-δτab] satisfies the following differential equation: (19)12σ2(u)φ′′(u)+μ(u)φ(u)=δφ(u),u(a,b), with the boundary conditions φ(a)=1, φ(b)=1.

Now, we consider some examples.

Example 4.

The Bessel process: dUt=((d-1)/2Ut)dt+dBt, where d>1 is a real number. We assume that b>a>0 in this process.

First, we consider the following differential equation: (20)12f′′(u)+d-12uf(u)=δf(u). It is well known that the increasing and decreasing solutions are, respectively, as follows: (21)f+(u)=u-vIv(2δu),f-(u)=u-vKv(2δu), where v=(d-2)/2, and Iv(·) and Kv(·) are the usual modified Bessel functions.

Then, from Theorem 1, we can give φ1(u) as follows: (22)φ1(u)=C1f+(u)+C2f-(u),φ1(a)=1,φ1(b)=0, where the constants C1 and C2 are to be determined. From the boundary conditions (22), we can obtain the expression of the constants C1 and C2 as follows: (23)C1=avKv(2δb)Iv(2δa)Kv(2δb)-Iv(2δb)Kv(2δa),C2=avIv(2δb)Iv(2δb)Kv(2δa)-Iv(2δa)Kv(2δb). So, we get (24)φ1(u)=(au)vKv(2δb)Iv(2δu)-Iv(2δb)Kv(2δu)Iv(2δa)Kv(2δb)-Iv(2δb)Kv(2δa).

According to Theorem 2 and (21), we can give φ2(u) as follows: (25)φ2(u)=C3f+(u)+C4f-(u),φ2(a)=0,φ2(b)=1, where the constants C3 and C4 are to be determined. From the boundary conditions (25), we can determine the constants and obtain the expression of φ2(u) as follows: (26)φ2(u)=(bu)vKv(2δa)Iv(2δu)-Iv(2δa)Kv(2δu)Iv(2δb)Kv(2δa)-Iv(2δa)Kv(2δb).

According to Theorem 3, the expression of φ(u) can be obtained from solving the following differential equation: (27)12φ′′(u)+d-12uφ(u)=δφ(u),φ(a)=1,φ(b)=1. Furthermore, from the definition of φ(u)=φ1(u)+φ2(u), we also can get the expression of φ(u). The two methods can lead to the same results as follows: (28)φ(u)=(1u)v×([avKv(2δb)-bvKv(2δa)]+[bvIv(2δa)-avIv(2δb)]Kv(2δu)[avKv(2δb)-bvKv(2δa)]Iv(2δu)+[bvIv(2δa)-avIv(2δb)]Kv(2δu))×((2δa)Iv(2δa)Kv(2δb)-Iv(2δb)Kv(2δa))-1.

Example 5 (the square root process (see [<xref ref-type="bibr" rid="B6">6</xref>])).

(29) d U t = v ( k - U t ) d t + σ U t d B t , v , σ > 0 , k [ a , b ] .

We assume that b>a>0 and consider the following differential equation: (30)12σ2u2f′′(u)+(vk-vu)f(u)=δf(u),δ>0. We assume that (2ν/σ2)k is not an integer, the two linear independent solutions are (31)f+(u)=M(δv,2vσ2k,2vσ2u),f-(u)=U(δv,2vσ2k,2vσ2u), where M and U are the confluent hypergeometric functions of the first and second kinds, respectively. Then, as the way at used in Example 4, and from Theorems 1 and 2, we get that the expressions of φ1(u) and φ2(u) are as follows: (32)φ1(u)=g(b,u)-g(u,b)g(b,a)-g(a,b),φ2(u)=g(a,u)-g(u,a)g(a,b)-g(b,a), where (33)g(x,y)=U(δv,2vσ2k,2vσ2x)M(δv,2vσ2k,2vσ2y). So, we can get (34)φ(u)=g(a,u)-g(b,u)+g(u,b)-g(u,a)g(a,b)-g(b,a).

Example 6 (the Ornstein-Uhlenbeck process (see [<xref ref-type="bibr" rid="B5">7</xref>])).

(35) d U t = v ( k - U t ) d t + σ d B t , v , σ > 0 , k [ a , b ] . The Ornstein-Uhlenbeck process above is the only process that is simultaneously Gaussian, Markov, and stationary, and has been discussed extensively, see, for example [24, 8].

We consider the following differential equation: (36)12σ2f′′(u)+v(k-u)f(u)=δf(u).

In the case of k=0, σ=1, the two independent solutions to (37)12f′′(u)-vuf(u)=δf(u) are (38)f+(u)=H-δ/v(-vu)=2-δ/2ve(1/2)vu2D-δ/v(-2vu),f-(u)=H-δ/v(vu)=2-δ/2ve(1/2)vu2D-δ/v(2vu), where Hv(·) and Dv(·) are, respectively, the Hermite and parabolic functions. We obtain the expressions of φ1(u) and φ2(u) as follows: (39)φ1(u)=e(1/2)v(u2-a2)h(u,b)-h(b,u)h(a,b)-h(b,a),φ2(u)=e(1/2)v(u2-b2)h(a,u)-h(u,a)h(a,b)-h(b,a), where (40)h(x,y)=D-δ/v(-2vx)D-δ/v(2vy). By the definition of φ(u), we get (41)φ(u)=(e(1/2)v(u2-a2)(h(u,b)-h(b,u))+e(1/2)v(u2-b2)(h(a,u)-h(u,a)))×(h(a,b)-h(b,a)e(1/2)v(u2-a2))-1.

For the general k and σ, the two independent solutions of (36) are, respectively, as follows: (42)f+(u)=H-δ/v(-vσ(u-k))=2-δ/2ve(1/2)(v/σ2)(u-k)2D-δ/v(-2vσ(u-k)),f-(u)=H-δ/v(vσ(u-k))=2-δ/2ve(1/2)(v/σ2)(u-k)2D-δ/v(2vσ(u-k)). Then, we obtain the expressions of φ1(u) and φ2(u) as follows: (43)φ1(u)=e(1/2)(v/σ2)[(u-k)2-(a-k)2]h0(u,b)-h0(b,u)h0(a,b)-h0(b,a),φ2(u)=e(1/2)(v/σ2)[(u-k)2-(b-k)2]h0(a,u)-h0(u,a)h0(a,b)-h0(b,a), where (44)h0(x,y)=D-δ/v(-2vσ(x-k))D-δ/v(2vσ(y-k)). Finally, we get (45)φ(u)=e(1/2)(v/σ2)[(u-k)2-(a-k)2]h0(u,b)-h0(b,u)h0(a,b)-h0(b,a)+e(1/2)(v/σ2)[(u-k)2-(b-k)2]h0(a,u)-h0(u,a)h0(a,b)-h0(b,a).

Example 7 (the Gompertz Brownian motion process (see [<xref ref-type="bibr" rid="B12">9</xref>])).

(46) d U t = v U t ( ln k - ln U t ) d t + σ U t d B t , v , σ > 0 , k [ a , b ] . We assume that b>a>0.

Now, consider the differential equation (47)12σ2u2f′′(u)+vu(lnk-lnUt)f(u)=δf(u),δ>0. It is well known that the increasing and decreasing solutions are, respectively, as follows: (48)f+(u)=M(δ2v,12,vσ2(lnuk+σ22v)2),f-(u)=U(δ2v,12,vσ2(lnuk+σ22v)2), where M and U, as in Example 5, are the first and second Kummer’s function, respectively. From the boundary conditions φ1(a)=1 and φ1(b)=0, we get (49)φ1(u)=g0(b,u)-g0(u,b)g0(b,a)-g0(a,b), where (50)g0(x,y)=U(δ2v,12,vσ2(lnxk+σ22v)2)×M(δ2v,12,vσ2(lnyk+σ22v)2). From the boundary conditions φ2(a)=0 and φ2(b)=1, we get (51)φ2(u)=g0(u,a)-g0(a,u)g0(b,a)-g0(a,b). Then, we obtain (52)φ(u)=g0(b,u)-g0(a,u)+g0(u,a)-g0(u,b)g0(b,a)-g0(a,b).

3. Applications to Dividend Value Function 3.1. Barrier Strategy

In this subsection, we consider the barrier strategy for dividend payments which are discussed in various model, see, for example, . More specifically, we assume that the company pays dividends according to the following strategy governed by parameter b>0. Whenever the surplus is above the level b, the excess will be paid as dividends, and when the surplus is below b nothing is paid out. We denote the aggregate dividends paid in the time interval [0,t] by Dr(t), the modified risk process by Ur(t)=Ut-Dr(t), the ruin time by Tr=inf{t0:Ur(t)=0}, and the present value of all dividends until ruin Tr by Dr=0Tre-δtdDr(t), here, δ>0 is the discount factor, and the expectation of Dr by (53)Vr(u,b)=Eu[Dr].

Now, we want to derive the dividend value function by the Laplace transform of exit time. We denote (54)τ0-=inf{t0:Ut0},(55)τ0b=τ0-τb+,(56)ψ2(u)=E[e-δτb+;τb+<τ0-], where τb+ is defined by (3). Let a=0 in the function φ2(u) be defined by (6), we get the definition of ψ2(u). So, we get the following lemma from Theorem 2.

Lemma 8.

The function ψ2(u) defined by (56) satisfies the following differential equation: (57)12σ2(u)ψ2′′(u)+μ(u)ψ2(u)=δψ2(u),u(0,b), with the boundary conditions ψ2(0)=0, ψ2(b)=1.

Then, we have the following theorem.

Theorem 9.

For 0ub, one has (58)Vr(u,b)=ψ2(u)ψ2(b), where ψ2(u) is defined by (56).

Proof.

The one-dimensional diffusion model defined by (1) is a time-homogeneous strong Markov process. Then, when 0<u<b, we have (59)Vr(u,b)=Eu(0Tre-δtdDr(t))=Eu(0τ0be-δtdDr(t)+τ0bTre-δtdDr(t))=Eu(τ0bTre-δtdDr(t))=Eu[e-δτ0b(0Tre-δtdDr(t)θτ0b)]=Eu[e-δτ0bEUτ0b(0Tre-δtdDr(t))]=Eu[e-δτ0bVr(Uτ0b,b)], where θτ0b is the shift operator. By the definition of τ0b, we get (60)Eu[eτ0bVr(Uτ0b,b)]=Vr(0,b)Eu[e-δτ0-;τ0-<τb+]+Vr(b,b)Eu[e-δτb+;τb+<τ0-]=Vr(b,b)ψ2(u). From (59) and (60), we obtain (61)Vr(u,b)=Vr(b,b)ψ2(u), where ψ2(u) can be determined from Lemma 8. From , for the barrier strategy, we have the following boundary condition: (62)V(u,b)u|u=b=1. Then, we have (63)Vr(b,b)=1ψ2(b). So, we get the result. This completes the proof.

Now, we consider the examples discussed in Section 2.

Example 10.

The Bessel process discussed in Example 4. Reference  gives the following helpful formulas: (64)I0(u)1,Iv(u)(u/2)2Γ(1+v)(v-1,-2,),K0(u)-lnu,Kv(u)Γ(v)2(u2)-v(Re(v)>0). Letting a=0 in Example 4, we get ψ2(u) from φ2(u) as follows: (65)ψ2(u)=(bu)vIv(2δu)Iv(2δb). Using the following formula: (66)Iv(u)=vuIv(u)+Iv+1(u), we have (67)ψ2(b)=(bu)v2δIv+1(2δb)Iv(2δb). Then, from Theorem 9, and substituting ψ2(u) and ψ2(b) into (58), we obtain (68)Vr(u,b)=Iv(2δu)2δIv+1(2δb).

Example 11.

We consider the square root process discussed in Example 5. Let a=0 in Example 5, according to M(x,y,0)=1, we obtain (69)ψ2(u)=1C[U(δv,2vσ2k,0)M(δv,2vσ2k,2vσ2u)-U(δv,2vσ2k,2vσ2u)], where (70)C=U(δv,2vσ2k,0)M(δv,2vσ2k,2vσ2b)-U(δv,2vσ2k,2vσ2b). Using the following helpful formulas (see ): (71)M(c1,c2,u)u=c1c2M(c1+1,c2+1,u),U(c1,c2,u)u=-c1U(c1+1,c2+1,u), we get (72)ψ2(b)=1C[δvkU(δv,2vσ2k,0)M(δv+1,2vσ2k+1,2vσ2b)+2δσ2U(δv+1,2vσ2k+1,2vσ2b)]. Finally, we have (73)Vr(u,b)=[U(δv,2vσ2k,0)M(δv,2vσ2k,2vσ2u)-U(δv,2vσ2k,2vσ2u)]×[δvkU(δv,2vσ2k,0)M(δv+1,2vσ2k+1,2vσ2b)+2δσ2U(δv+1,2vσ2k+1,2vσ2b)]-1.

Example 12.

We consider the Ornstein-Uhlenbeck process considered in Example 6, and let a=0 in Example 6. From , we have the following formulas: (74)Dv(0)=2v/2πΓ((1/2)-(v/2)),Dv(u)=-u2Dv(u)+vDv-1(u), where Γ(x) is the gamma function.

In the case of k=0, σ=1, we have (75)ψ2(u)=e(v/2)(u2-b2)D-δ/v(2vu)-D-δ/v(-2vu)D-δ/v(2vb)-D-δ/v(-2vb),ψ2(b)=δ2vD(-δ/v)-1(2vb)+D(-δ/v)-1(-2vb)D-δ/v(-2vb)-D-δ/v(2vb). Then, we get (76)Vr(u,b)=1δv2e(v/2)(u2-b2)×D-δ/v(-2vu)-D-δ/v(2vu)D(-δ/v)-1(2vb)+D(-δ/v)-1(-2vb).

For the general k, σ, we have (77)ψ2(u)=1Ge(v/2σ2)((u-k)2-(b-k)2)×[D-δ/v((2vσ)k)D-δ/v((2vσ)(u-k))-D-δ/v(-(2vσ)k)×D-δ/v((-2vσ)(u-k))], where (78)G=D-δ/v((2vσ)k)D-δ/v((2vσ)(b-k))-D-δ/v(-(2vσ)k)D-δ/v(-(2vσ)(u-k)). We can get (79)ψ2(b)=-1Gδσ2v×[D-δ/v((2vσ)k)D(-δ/v)-1((2vσ)(b-k))+D-δ/v(-(2vσ)k)×D(-δ/v)-1(-(2vσ)(b-k))]. Finally, we obtain (80)Vr(u,b)=1Ge(v/2σ2)((u-k)2-(b-k)2)σδv2×[(-(2vσ)(u-k))D-δ/v(-(2vσ)k)×D-δ/v(-(2vσ)(u-k))-D-δ/v((2vσ)k)×D-δ/v((2vσ)(u-k))]×[((2vσ)(b-k))D-δ/v((2vσ)k)×D(-δ/v)-1((2vσ)(b-k))+D-δ/v(-(2vσ)k)×D(-δ/v)-1(-(2vσ)(b-k))]-1.

Example 13.

We consider the Gompertz Brownian motion process discussed in Example 7. In , the authors point out as u+, (81)U(c1,c2,u)=u-c1[1+o(|u|-1)],M(c1,c2,u)=Γ(c2)Γ(c1)euuc1-c2[1+o(|u|-1)]. Letting a=0 in Example 7, and using (81), we lead to the expression of ψ2(u) as follows: (82)ψ2(u)=U(δ2v,12,vσ2(lnuk+σ22v)2)×[U(δ2v,12,vσ2(lnbk+σ22v)2)]-1. Using (71), we get (83)ψ2(b)=-δσ2bkln(bk+σ22v)×U(δ2v+1,12+1,vσ2(lnbk+σ22v)2)×[U(δ2v,12,vσ2(lnbk+σ22v)2)]-1. Then, from Theorem 9, we have (84)Vr(u,b)=-σ2bkδU(δ2v,12,vσ2(lnuk+σ22v)2)×[U((lnbk+σ22v)2δ2v+1,12ln(bk+σ22v)×U(δ2v+1,12+1,vσ2(lnbk+σ22v)2)].

3.2. Threshold Strategy

We consider the company pays dividends according to the threshold dividend strategy; that is, dividends are paid at a constant rate α whenever the modified surplus is above the threshold b, and no dividends are paid whenever the modified surplus is below b. For recent publications on threshold strategy, see, for example, [3, 16, 17]. We define the modified risk process by (85)Ud(t)=Ut-Dd(t), where Dd(t)=α0tI(Ud(s)>b)ds. Let Dd denote the present value of all dividends until ruin as follows: (86)Dd=α0Tde-δsI(Ud(s)>b)ds, where Td=inf{t0:Ud(t)=0}. We denote by Vd(u,b) the expected discounted value of dividend payments; that is, (87)Vd(u,b)=Eu[Dd]. We denote (88)τb-=inf{t:Ud(t)b},φ3(u)=E[e-δτb-].

We can mimic the discussion of Theorem 1 to give the differential equation and the boundary conditions satisfied by φ3(u).

Lemma 14.

The function φ3(u) defined by (88) satisfies the following differential equation: (89)12σ2(u)φ3′′(u)+(μ(u)-α)φ3(u)=δφ3(u),ub, with the boundary conditions φ3(b)=1, φ3()=0.

We have the following theorem.

Theorem 15.

For u[0,b], one has (90)Vd(u;b)=αδφ3(b)ψ2(u)φ3(b)-ψ2(b), and, for u>b, one has (91)Vd(u;b)=αδ+αδψ2(b)φ3(u)φ3(b)-ψ2(b).

Proof.

When u[0,b], in view of the strong Markov property, we obtain (92)Vd(u;b)=Vd(b;b)ψ2(u). When u>b, since τb- is a stopping time, it follows from the strong Markov property of Ud(u) that (93)Vd(u,b)=Eu(0τb-αe-δtdt)+Eu(τb-Tdαe-δtI(Ud(t)>b)dt)=αδ(1-Eue-δτb-)+Eu[e-δτb-EUτb-(0Tde-δtdDd(t))]=αδ+(Vd(b,b)-αδ)φ3(u). Using the continuity of the function Vd(u;b) at u=b, we get (94)Vd(b;b)ψ2(b)=(Vd(b,b)-αδ)φ3(b). So, we get (95)Vd(b;b)=αδφ3(b)φ3(b)-ψ2(b). Substituting the above expression into (92) and (93), we can get the results (90) and (91). This completes the proof.

We just consider the square root process.

Example 16.

We consider the square root process discussed in Example 11. We first solve the differential equation satisfied by φ3(u) as follows (96)12σ2u2φ3′′(u)+(vk-vu-α)φ3(u)=δφ3(u),δ>0. We assume that 2((vk-α)/σ2) is not an integer, the two linear independent solutions are (97)φ+(u)=M(δv,2(vk-α)σ2,2vσ2u),φ-(u)=U(δv,2(vk-α)σ2,2vσ2u). Using (81), and from the boundary conditions φ3(b)=1, φ3()=0, we get (98)φ3(u)=[U(δv,2(vk-α)σ2,2vσ2u)]×[U(δv,2(vk-α)σ2,2vσ2b)]-1. From (71), we have (99)φ3(b)=-2δσ2[U(δv+1,2(vk-α)σ2+1,2vσ2b)]×[U(δv,2(vk-α)σ2,2vσ2b)]-1. Furthermore, ψ2(u) and ψ2(b) have been given in Example 11. From Theorem 15, we obtain (100)Vd(u,b)=αδU(δv+1,2(vk-α)σ2+1,2vσ2b)×[U(δv,2vσ2k,0)M(δv,2vσ2k,2vσ2u)-U(δv,2vσ2k,2vσ2u)]×{CU(δv+1,2(vk-α)σ2+1,2vσ2b)+U(δv,2(vk-α)σ2,2vσ2b)×[σ22vkU(δv,2vσ2k,0)×M(δv+1,2vσ2k+1,2vσ2b)+U(δv+1,2vσ2k+1,2vσ2b)]}-1,u[0,b], and for u>b, we get (101)Vd(u,b)=αδ-αδU(δv,2(vk-α)σ2,2vσ2u)×[σ22vkU(δv,2vσ2k,0)×M(δv+1,2vσ2k+1,2vσ2b)+U(δv+1,2vσ2k+1,2vσ2b)]×{CU(δv+1,2(vk-α)σ2+1,2vσ2b)+U(δv,2(vk-α)σ2,2vσ2b)×[σ22vkU(δv,2vσ2k,0)×M(δv+1,2vσ2k+1,2vσ2b)+U(δv+1,2vσ2k+1,2vσ2b)]}-1, where C is defined in Example 11.

Conflict of Interests

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

Acknowledgments

The authors are grateful to the anonymous referee’s careful reading and detailed helpful comments and constructive suggestions, which have led to a significant improvement of the paper. The research was supported by the National Natural Science Foundation of China (no. 11171179), the Research Fund for the Doctoral Program of Higher Education of China (no. 20133705110002), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

Asmussen S. Taksar M. Controlled diffusion models for optimal dividend pay-out Insurance 1997 20 1 1 15 2-s2.0-0031161138 10.1016/S0167-6687(96)00017-0 ZBL1065.91529 Cai J. Gerber H. U. Yang H. Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest North American Actuarial Journal 2006 10 2 94 119 2-s2.0-52749090722 10.1080/10920277.2006.10596250 Fang Y. Wu R. Optimal dividends in the Brownian motion risk model with interest Journal of Computational and Applied Mathematics 2009 229 1 145 151 2-s2.0-65049084014 10.1016/j.cam.2008.10.021 ZBL1162.91012 Yin C. C. Wang H. Q. The first passage time and the dividend value function for one-dimension diffusion processes between two reflecting barriers International Journal of Stochastic Analysis 2012 2012 15 971212 10.1155/2012/971212 Yin C. C. Shen Y. Wen Y. Z. Exit problems for jump processes with applications to dividend problems Journal of Computational and Applied Mathematics 2013 245 30 52 10.1016/j.cam.2012.12.004 Ditlevsen S. A result on the first-passage time of an Ornstein-Uhlenbeck process Statistics and Probability Letters 2007 77 18 1744 1749 2-s2.0-35548977214 10.1016/j.spl.2007.04.015 ZBL1133.60314 Cox J. C. Ingersoll J. E. Jr. Ross S. A. A theory of the term structure of interest rates Econometrica 1985 53 2 385 407 10.2307/1911242 Madec Y. Japhet C. First passage time problem for a drifted Ornstein-Uhlenbeck process Mathematical Biosciences 2004 189 2 131 140 2-s2.0-1942503238 10.1016/j.mbs.2004.02.001 ZBL1047.92027 Wang L. Pötzelberger K. Crossing probabilities for diffusion processes with piecewise continuous boundaries Methodology and Computing in Applied Probability 2007 9 1 21 40 2-s2.0-33847785606 10.1007/s11009-006-9002-6 ZBL1122.60070 Gerber H. U. Shiu E. S. W. On the time value of ruin North American Actuarial Journal 1998 2 1 48 78 2-s2.0-16844381682 10.1080/10920277.2005.10596197 ZBL1085.62508 Gerber H. U. Shiu E. S. W. Optimal dividends: analysis with Brownian motion North American Actuarial Journal 2004 8 1 1 20 Wang C. W. Yin C. C. Li E. Q. On the classical risk model with credit and debit interests under absolute ruin Statistics and Probability Letters 2010 80 5-6 427 436 2-s2.0-74849095706 10.1016/j.spl.2009.11.020 ZBL1183.91078 Yuen K. C. Yin C. On optimality of the barrier strategy for a general Lévy risk process Mathematical and Computer Modelling 2011 53 9-10 1700 1707 2-s2.0-79951942228 10.1016/j.mcm.2010.12.042 ZBL1219.91076 Zhang S. J. Jin J. M. Computation of Special Functions 2011 Nanjing, China Nanjing University Press Abramowitz Stegun M. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables 1972 Washington, DC, USA United States Department of Commerce, US Government Printing Office Chi Y. Lin X. S. On the threshold dividend strategy for a generalized jump-diffusion risk model Insurance 2011 48 3 326 337 2-s2.0-78651545502 10.1016/j.insmatheco.2010.11.006 ZBL1218.91072 Ng A. C. Y. On a dual model with a dividend threshold Insurance 2009 44 2 315 324 2-s2.0-63449097159 10.1016/j.insmatheco.2008.11.011 ZBL1163.91441