The risk model with stochastic premiums, dependence and a threshold dividend strategy

The paper deals with a generalization of the risk model with stochastic premiums where dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by Farlie--Gumbel--Morgenstern copulas. In addition, dividends are paid to its shareholders according to a threshold dividend strategy. We derive integral and integro-differential equations for the Gerber--Shiu function and the expected discounted dividend payments until ruin. Next, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes. In particular, we find explicit formulas for the ruin probability in the model without either dividend payments or dependence as well as for the expected discounted dividend payments in the model without dependence. Finally, numerical illustrations are presented.


Introduction
In the actuarial literature, a lot of attention is paid to the investigation of the ruin measures such as the ruin probability, the surplus prior to ruin and the deficit at ruin (see, e.g., [6,34,39] and references therein). A unified approach to the study of these risk measures together by combining them into one function was proposed by Gerber and Shiu [22], who introduced the expected discounted penalty function for the classical risk model. The so-called Gerber-Shiu function has been investigated further by many authors (see, e.g., [13,23,29,42,44]) in more general risk models. In those risk models, claim sizes and inter-claim times are assumed to be mutually independent, which simplifies the investigation of the ruin measures. Nevertheless, this assumption has been proved to be very restrictive in some real applications. For instance, in modelling damages caused by natural catastrophic events, the intensity of the catastrophe and the time elapsed since the last catastrophe are expected to be dependent [10,38]. That is why more and more authors have concentrated on the investigation of risk models with dependence between claim sizes and inter-claim times recently.
Albrecher and Boxma [1] consider a generalization of the classical risk model, where the distribution of the time between two claims depends on the previous claim size (see also [2] for an extension). Albrecher and Teugels [3] apply the random walk approach and allow the inter-claim time and its subsequent claim size to be dependent according to an arbitrary structure. Boudreault, Cossette, Landriault and Marceau [11] consider a particular dependence structure between the inter-claim time and the subsequent claim size and derive the defective renewal equation satisfied by the expected discounted penalty function. In [33], the authors study the ruin probability in a model where the time between two claim occurrences determines the distribution of the next claim size. The ruin probability in a model with independent but not necessarily identically distributed claim sizes and inter-claim times is investigated in [5] (see also references therein).
Cossette, Marceau and Marri [15,16] deal with an extension of the classical compound Poisson risk model where a dependence structure between the claim size and the inter-claim time is introduced through a Farlie-Gumbel-Morgenstern copula and its generalization. They derive the integro-differential equation and the Laplace transform of the Gerber-Shiu discounted penalty function and concentrate on exponentially distributed claim sizes. Zhang and Yang [47] extend these results to the compound Poisson risk model perturbed by a Brownian motion. In [12,24,46], the authors deal with the Sparre Andersen risk model where the inter-claim times follow the Erlang distribution and extend results obtained in [15,16].
In all these papers, the dependence structure between the claim sizes and the interclaim times is described by the Farlie-Gumbel-Morgenstern copula. This copula is often used in applications to introduce dependence structures due to its tractability and simplicity. It allows positive and negative dependence as well as independence. Nevertheless, the Farlie-Gumbel-Morgenstern copula has been shown to be somewhat limited since it does not allow the modeling of high dependencies. Indeed, its dependence parameter is θ ∈ [−1, 1], so its Spearman's rho and Kendall's tau are ρ θ = θ/3 ∈ [−1/3, 1/3] and τ θ = 2θ/9 ∈ [−2/9, 2/9], respectively (see, e.g., [7,8,37] and references therein). This limited range of dependence restricts the usefulness of this copula for modeling. Note that the dependence parameter θ can be easily estimated from real data due to the simple relations between it and the measures of association ρ θ and τ θ . For more information on the Farlie-Gumbel-Morgenstern copula, we refer to, e.g., [21,37] (see also [16] and references therein for applications of this copula). Despite the popularity of the Farlie-Gumbel-Morgenstern copula, other copulas have been used in risk theory, for instance, an Archimedean copula [4], a Gaussian copula [19] and a Spearman copula [25].
Risk models where an insurance company pays dividends to its shareholders are of great interest in risk theory. Dividend strategies for insurance risk models were first proposed by De Finetti [20], who dealt with a binomial model. Barrier strategies for the classical risk model and its different generalizations have been studied in a number of papers (see, e.g., [14,27,28,30,31,43,45]). For optimal dividend problems in insurance risk models, see the monograph by Schmidli [40] and references therein.
Cossette, Marceau and Marri [17,18] consider the classical risk process with a constant dividend barrier and a dependence structure between claim sizes and interclaim times introduced through the Farlie-Gumbel-Morgenstern copula. They analyze the Gerber-Shiu function and the expected discounted dividend payments and then concentrate on exponentially distributed claim sizes investigating the impact of the dependence on ruin quantities. The same model is studied in [32], where, in particular, the authors show that the solution to the integro-differential equation for the Gerber-Shiu function is a linear combination of the Gerber-Shiu function with no barrier and the solution to the associated homogeneous integro-differential equation. For some earlier results in this direction, see also [26].
Shi, Liu and Zhang [41] consider the compound Poisson risk model with a threshold dividend strategy and a dependence structure modeled by the Farlie-Gumbel-Morgenstern copula. They derive integro-differential equations for the Gerber-Shiu function and the expected discounted dividend payments paid until ruin as well as renewal equations for these functions, which are used to obtain explicit formulas for them.
The present paper deals with a generalization of the risk model with stochastic premiums introduced and studied in [9] (see also [34]). In contrast to the classical compound Poisson risk model, where premiums arrive with constant intensity and are not random, in this risk model premiums also form a compound Poisson process, i.e. they arrive at random times and their sizes are also random (see also [35,36] for a generalization of the classical risk model where an insurance company gets additional funds whenever a claim arrives). In [9], claim sizes and inter-claim times are assumed to be mutually independent, and the same assumption is made concerning premium arrivals. In this paper, we suppose that the dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by the Farlie-Gumbel-Morgenstern copulas, which allows positive and negative dependence as well as independence. In addition, we suppose that the insurance company pays dividends to its shareholders according to a threshold dividend strategy.
To be more precise, this implies that when the surplus is below some fixed threshold, no dividends are paid, and when the surplus exceeds or equals the threshold, dividends are paid continuously at some constant rate. Our subjects of investigation are the Gerber-Shiu function, a special case of which is the ruin probability, and the expected discounted dividend payments until ruin.
The rest of the paper is organized as follows. In Section 2, we describe the risk model we deal with. In Section 3, we derive integral and integro-differential equations for the Gerber-Shiu function. In Section 4, we obtain corresponding equations for the expected discounted dividend payments until ruin. Section 5 deals with exponentially distributed claim and premium sizes in some special cases of the model. Namely, we consider the ruin probability in the model without either dividend payments or dependence, and the expected discounted dividend payments in the model without dependence. In these simpler models, we reduce the integral and integro-differential equations derived in Sections 2 and 3 to linear differential equations and find explicit solutions to these equations. Section 6 provides some numerical illustrations.
2 Description of the model Let (Ω, F, P) be a probability space satisfying the usual conditions, and let all the stochastic objects we use below be defined on it.
In the risk model with stochastic premiums introduced in [9] (see also [34]), claim sizes form a sequence (Y i ) i≥1 of non-negative independent and identically dis- The number of claims on the time interval [0, t] is a Poisson process (N t ) t≥0 with constant intensity λ > 0. Next, premium sizes form a sequence The number of premiums on the time interval [0, t] is a Poisson process (N t ) t≥0 with constant intensityλ > 0. Thus, the total claims and premiums on [ In what follows, we also assume that the r.v.'s (Y i ) i≥1 have a probability density function (p.d.f.) f Y (y) and a finite expectation µ > 0, and the r.v.'s (Ȳ i ) i≥1 have a probability density function (p.d.f.) fȲ (y) and a finite expectationμ > 0.
We denote a non-negative initial surplus of the insurance company by x. Let X t (x) be its surplus at time t provided that the initial surplus is x. Then the surplus process (X t (x)) t≥0 follows the equation In [9], the r.v.'s (Y i ) i≥1 and (Ȳ i ) i≥1 , and the processes (N t ) t≥0 and (N t ) t≥0 are assumed to be mutually independent. In this paper, we suppose that the claim sizes (Y i ) i≥1 and the inter-claim times are not independent but with a dependence structure modeled by a Farlie-Gumbel-Morgenstern copula, and we make the same assumption concerning premium arrivals.
To be more precise, let (T i ) i≥1 be a sequence of inter-arrival times of (N t ) t≥0 . In particular, T 1 is the time of the first claim. Thus, We assume that (Y i , T i ) i≥1 are i.i.d. random vectors and for every fixed i ≥ 1, the dependence structure between Y i and T i is modeled by a Farlie-Gumbel-Morgenstern copula with parameter θ ∈ [−1, 1], i.e.
(see, e.g., [21,37] for more information on copulas). In other words, a claim size depends on the time elapsed from the previous claim. Therefore, the bivariate c.d.f. of (Y i , T i ) i≥1 is defined by The corresponding bivariate p.d.f. of (Y i , T i ) i≥1 is given by where h Y (y) = f Y (y)(1 − 2F Y (y)), y ≥ 0. Note that the case θ = 0 corresponds to the situation where the claim sizes and the inter-claim times are independent. Next, let (T i ) i≥1 be a sequence of inter-arrival times of (N t ) t≥0 . In particular,T 1 is the time of the first premium. Therefore, We also suppose that (Ȳ i ,T i ) i≥1 are i.i.d. random vectors and for every fixed i ≥ 1, the dependence structure betweenȲ i andT i is modeled by a Farlie-Gumbel-Morgenstern copula with parameterθ ∈ [−1, 1]. So the bivariate p.d.f. of (Ȳ i ,T i ) i≥1 is given by where hȲ (y) = fȲ (y)(1 − 2FȲ (y)), y ≥ 0. The caseθ = 0 corresponds to the situation where the premium sizes and the inter-premium times are independent. The random vectors (Y i , T i ) i≥1 and (Ȳ i ,T i ) i≥1 are assumed to be mutually independent.
From (2) and (3) we obtain the conditional p.d.f.'s of the claim and premium sizes: and Moreover, we suppose that the insurance company pays dividends to its shareholders according to the following threshold dividend strategy. Let b > 0 be a threshold. When the surplus is below b, no dividends are paid. When the surplus exceeds or equals b, dividends are paid continuously at a rate d > 0. Let (X b t (x)) t≥0 denote the modified surplus process under this threshold dividend strategy. Then where 1(·) is the indicator function. Let (D t ) t≥0 denote the dividend distributing process. For the threshold dividend strategy described above, we have Next, let τ b (x) = inf{t ≥ 0 : X b t (x) < 0} be the ruin time for the risk process (X b t (x)) t≥0 defined by (6). In what follows, we omit the dependence on x and write τ b instead of τ b (x) when no confusion can arise.
For δ 0 ≥ 0, the Gerber-Shiu function is defined by where w(·, ·) is a bounded non-negative measurable function, X b τ b − (x) is the surplus immediately before ruin and |X b τ b (x)| is a deficit at ruin. Note that if w(·, ·) ≡ 1 and δ 0 = 0, then m(x, b) becomes the infinite-horizon ruin probability For δ > 0, the expected discounted dividend payments until ruin are defined by For simplicity of notation, we also write m(x) and v(x) instead of m(x, b) and v(x, b), respectively, when no confusion can arise. Moreover, we set and 3 Equations for the Gerber-Shiu function Theorem 1. Let the surplus process (X b t (x)) t≥0 follow (6) under the above assumptions with θ = 0 andθ = 0. Moreover, let the p.d.f.'s f Y (y) and fȲ (y) have the derivatives f ′ Y (y) and f ′ Y (y) on R + , which are continuous and bounded on R + , and let w(u 1 , u 2 ) have the second derivatives w ′′ u1u1 (u 1 , u 2 ), w ′′ u1u2 (u 1 , u 2 ) and w ′′ u2u2 (u 1 , u 2 ) on R 2 + , which are continuous and bounded on R 2 + as functions of two variables. Then the Gerber-Shiu function m(x) satisfies the equations and where Proof. It is easily seen that the time of the first jump of ( We first deal with the case x ∈ [0, b]. Considering the time and the size of the first jump of (X b t (x)) t≥0 and applying the law of total probability we obtain Substituting (4) and (5) into (11) and taking into account (7) give Separating the integrals on the right-hand side of (12) into integrals w.r.t. either t or y yields Taking the integrals w.r.t. t on the right-hand side of (13) we get which yields (9).
Let now x ∈ [b, ∞). Considering the time and the size of the first jump of (X b t (x)) t≥0 and applying the law of total probability we have Substituting (4) and (5) into (14) and taking into account (7) give where Changing the variable x − dt = s in the outer integral of the expression for I 1,2,3 (x) yields where Separating the integrals in the expression for I 4,5,6 (x) into integrals w.r.t. either t or y we get Taking the integrals w.r.t. t on the right-hand side of (17) we obtain where Thus, substituting (16) and (18) into (15) we have It is easily seen from (9) , and from (19) we conclude that m 2 (x) is continuous on [b, ∞). Indeed, the right-hand sides of (9) and (19) are continuous on [0, b] and [b, ∞), respectively, and so are the left-hand sides. Therefore, from (19) where the function β 1 (x) is defined in the assertion of the theorem. Multiplying (19) by (λ +λ + δ 0 )/d and adding (20) we get Since m 1 (x) and m 2 (x) are continuous and bounded on [0, b] and [b, ∞), respectively, and w(u 1 , u 2 ) is continuous and bounded on R 2 + as a function of two variables, from (21) we conclude that so is Taking into account that f ′ Y (y) and f ′ Y (y) are continuous and bounded on R + , and w ′ u1 (u 1 , u 2 ) and w ′ u2 (u 1 , u 2 ) are continuous and bounded on R 2 + , from (9) we conclude that so is m ′ (21) gives where which is continuous and bounded on [b, ∞), and the function β 2 (x) is defined in the assertion of the theorem. Here w ′ u1 (·, ·) and w ′ u2 (·, ·) stand for the partial derivatives of w(u 1 , u 2 ) w.r.t. the first and the second variables, respectively.
Multiplying (21) by (2λ +λ + δ 0 )/d and adding (22) we obtain It is easily seen from (23) that m ′′ 2 (x) is continuous and bounded on [b, ∞). Taking into account that f ′ Y (y) and f ′ Y (y) are continuous and bounded on R + , and w ′′ u1u1 (u 1 , u 2 ), w ′′ u1u2 (u 1 , u 2 ) and w ′′ u2u2 (u 1 , u 2 ) are continuous and bounded on R 2 + , from (9) we conclude that so is m ′′ Multiplying (23) by (λ + 2λ + δ 0 )/d and adding (24) yield (10), which completes the proof. (9) and (10), we need some boundary conditions. The first one is m 1 (b) = m 2 (b). Next, using standard considerations (see, e.g., [34,36,39]) we can show that lim x→∞ m 2 (x) = 0 provided that the net profit condition holds. Finally, we can substitute x = b into the intermediate equations (e.g., equation (21)) to get additional boundary conditions involving derivatives of m 2 (x). Furthermore, equations (9) and (10) are not solvable analytically in the general case, so we can use, for instance, numerical methods. Nevertheless, we can give explicit expressions for m(x) in some particular cases (see Section 5). The uniqueness of the required solutions to these equations should be justified in each case.

Remark 1. To solve equations
Remark 2. The corresponding model without dividend payments is obtained by b → ∞. In this case, the Gerber-Shiu function m(x) satisfies the integral equation Note that equation (25) for the ruin probability coincides with the equation derived in [9] (see also [34]) if θ = 0 andθ = 0. Remark 3. In Theorem 1, we assume that θ = 0 andθ = 0. Otherwise, we do not need to differentiate (19) three times and can obtain equations not involving the third derivative of m 2 (x) instead of (10).
Equation (9) holds in all possible cases. Furthermore, (9) involves no derivatives and holds under weaker assumptions than (10). To be more precise, we do not need the differentiability of f Y (y), fȲ (y) and w(u 1 , u 2 ) to get (9).
4 Equations for the expected discounted dividend payments until ruin Theorem 2. Let the surplus process (X b t (x)) t≥0 follow (6) under the above assumptions with θ = 0 andθ = 0. Moreover, let the p.d.f.'s f Y (y) and fȲ (y) have the derivatives f ′ Y (y) and f ′ Y (y) on R + , which are continuous and bounded on R + . Then the expected discounted dividend payments until ruin v(x) satisfy the equations and where Proof. The proof is similar to the proof of Theorem 1, so we omit detailed considerations.
Let x ∈ [0, b]. Considering the time and the size of the first jump of (X b t (x)) t≥0 and applying the law of total probability, we obtain Comparing (31) with (11) and applying arguments similar to those in the proof of Theorem 1 yield (29).
Let now x ∈ [b, ∞). By the law of total probability, we have Taking into account that substituting (4) and (5) into (32) and using (8) give v 2 (x) = I 7,8,9 (x) + I 10,11,12 (x) where Changing the variable x − dt = s in the outer integral of the expression for I 7,8,9 (x) yields where 0 v 2 (s + y) fȲ (y) −θhȲ (y) dy ds, Separating the integrals in the expression for I 10,11,12 (x) into integrals w.r.t. either t or y and then taking the integrals w.r.t. t we obtain where Thus, substituting (34) and (35) It is easily seen from (29) that v 1 (x) is continuous on [0, b], and from (36) we conclude that v 2 (x) is continuous on [b, ∞). Hence, from (36) where the function β 3 (x) is defined in the assertion of the theorem. Multiplying (36) by (λ +λ + δ)/d and adding (37) we get Since v 1 (x) and v 2 (x) are continuous and bounded on [0, b] and [b, ∞), respectively, from (38) we conclude that so is v ′ 2 (x) on [b, ∞). Taking into account that f ′ Y (y) and f ′ Y (y) are continuous and bounded on R + , from (29) we conclude that . From this and (38) it follows that v 2 (x) is twice differentiable on [b, ∞). Differentiating (38) gives where which is continuous and bounded on [b, ∞), and the function β 4 (x) is defined in the assertion of the theorem.

Multiplying (38) by (2λ +λ + δ)/d and adding (39) we obtain
It is easily seen from (40) that v ′′ 2 (x) is continuous and bounded on [b, ∞). Taking into account that f ′ Y (y) and f ′ Y (y) are continuous and bounded on R + , from (29) we conclude that so is v ′′ Moreover, applying similar arguments shows that β 4 (x) is differentiable on [b, ∞). From this and (40) it follows that v 2 (x) has the third derivative on [b, ∞). Differentiating (40) gives Multiplying (40) by (λ + 2λ + δ 0 )/d and adding (41) yield (30), which completes the proof. (29) and (30), we use the following boundary conditions. First of all, we have v 1 (b) = v 2 (b). Next, if the net profit condition holds, applying arguments similar to those in [40, p. 70] we can show that lim x→∞ v 2 (x) = d/δ. Moreover, we can substitute x = b into the intermediate equations (e.g., equation (38)) to get additional boundary conditions involving derivatives of v 2 (x). The uniqueness of the required solutions should also be justified. If θ = 0 andθ = 0, we can find explicit solutions to the equations (see Section 5). Remark 5. If at least one of the parameters θ andθ is equal to 0, we do not need to differentiate (36) three times and can obtain equations not involving the third derivative of v 2 (x) instead of (30).

Exponentially distributed claim and premium sizes
In this section, we deal with exponentially distributed claim and premium sizes, i.e. and

The ruin probability in the model without dividend payments
If no dividends are paid, equation (25) for the ruin probability ψ(x) takes the form Substituting (45) and (46) into (47) gives where x 0 ψ(u)e 2u/µ du, We now show that if either θ = 0 orθ = 0, integro-differential equation (48) can be reduced to a third-order linear differential equation with constant coefficients. Lemma 1. Let the surplus process (X t (x)) t≥0 follow (1) under the above assumptions, and let claim and premium sizes be exponentially distributed with means µ and µ, respectively.
If θ = 0 andθ = 0, then ψ(x) is a solution to the differential equation If θ = 0 andθ = 0, then ψ(x) is a solution to the differential equation Proof. First of all, note that
It is obvious that the minimal value of the expression on the left-hand side of the above inequality is attained when either θ = 1 or θ = −1 and equals either λμ(µ +μ) + 2λμ 2 or λμ(µ +μ) + 2λµμ, respectively. Both these expressions are positive. Thus, D 1 > 0 and (64) has two real roots z 2 and z 3 defined before the assertion of the theorem.
Note that letting x = 0 in (54) (and in (52) when C 2 = 0) gives no additional information about unknown constants. Nevertheless, the equalities must hold for the values of the constants found from (48) (and (52) when C 2 = 0). Consequently, differential equation (49) has the unique solution given by (59) or (60). Since we have derived (49) from (48) without any additional assumptions, we conclude that the function ψ(x) given by (59) or (60) is a unique solution to (48) satisfying the certain conditions. This guaranties that the solution we have found is the ruin probability and completes the proof.
The case θ = 0 andθ = 0 can be considered in a similar way by finding the required solution to equation (50).
To formulate the next theorem, we define the following constants: Theorem 4. Let the surplus process (X b t (x)) t≥0 follow (6) under the above assumptions with θ = 0 andθ = 0, and let claim and premium sizes be exponentially distributed with means µ andμ, respectively, and letλμ > λµ + d. Then we have and where the constants C 4 , C 5 , C 7 and C 8 are determined from the system of linear equations (80)-(83): and Proof. By Lemma 2, ψ 1 (x) and ψ 2 (x) are solutions to (70) and (71). We now find the general solutions to these equations. It is easily seen that the characteristic equation corresponding to (70) has two roots: z 4 = 0 and z 5 given before the assertion of the theorem. Hence, (78) is true with some constants C 4 and C 5 .
The characteristic equation corresponding to (71) has the form It is obvious that z 6 = 0 is a solution to (84). We now show that the equation has two negative roots. We first notice that its discriminant D 2 defined above is positive. Indeed, we have Therefore, (85) has two real roots. Next, by the conditions of the theorem, we haveλμ − λµ − d > 0 which shows that both roots are negative. Consequently, we get with some constants C 6 , C 7 and C 8 . Moreover, sinceλμ > λµ + d, using standard considerations (see, e.g., [34,36,39]) we can easily show that lim x→∞ ψ(x) = 0, which yields C 6 = 0. Thus, we obtain (79). The constants C 4 , C 5 , C 7 and C 8 are determined by letting x = 0 in (68) and (73), taking into account that ψ 1 (b) = ψ 2 (b) and letting x = b in (69).
Substituting (78) and (79)  We denote the determinant of the system of equations (80)-(83) by ∆ 2 . A standard computation shows that which is positive. Indeed, z 7 − z 8 > 0 by definition,λμ/µ − λ > 0 by the conditions of the theorem andμ 2 z 7 z 8 −μ(z 7 + z 8 ) + 1 > 0 since z 7 < 0 and z 8 < 0. Moreover, since z 5 < 0, we have Thus, since ∆ 2 = 0, the system of equations (80)-(83) has a unique solution. Furthermore, note that letting x = b in (76) gives no additional information about unknown constants, but the equality in (76) holds for the values of the constants found from the system of equations (80)-(83). Therefore, each of differential equations (70) and (71) has the unique solution given by (78) or (79), respectively. Since we have derived these equations from (68) and (69) without any additional assumptions, we conclude that the functions ψ 1 (x) and ψ 2 (x) given by (78) and (79) are unique solutions to (68) and (69) satisfying the certain conditions. This guaranties that the functions ψ 1 (x) and ψ 2 (x) we have found coincide with the ruin probability on the intervals [0, b] and [b, ∞), respectively, which completes the proof.

The expected discounted dividend payments until ruin in the model without de-
pendence We now also assume that θ = 0 andθ = 0. Then equations (29) and (42) for the expected discounted dividend payments v(x) in the case of exponentially distributed claim and premium sizes take the form and respectively. Lemma 3 below shows that integro-differential equations (86) and (87) can be reduced to linear differential equations with constant coefficients. Lemma 3. Let the surplus process (X b t (x)) t≥0 follow (6) under the above assumptions with θ = 0 andθ = 0, and let claim and premium sizes be exponentially distributed with means µ andμ, respectively. Then v 1 (x) and v 2 (x) are solutions to the differential equations The proof of the lemma is similar to the proof of Lemma 2.
Proof. The proof is similar to the proof of Theorem 4, so we omit detailed considerations. By Lemma 3, v 1 (x) and v 2 (x) are solutions to (88) and (89). It is easily seen that D 3 > 0. Hence the characteristic equation corresponding to (88) has two real roots z 9 and z 10 given before the assertion of the theorem. This yields (90) with some constants C 9 and C 10 .
The assumption D 4 > 0 guarantees that cubic equation (92) has three distinct real roots. Consequently, the general solution to (89) is given by v 2 (x) = C 11 e z11x + C 12 e z12x + C 13 e z13x , x ∈ [b, ∞) with some constants C 11 , C 12 and C 13 .
By Vieta's theorem, we conclude that (92) has either two or no negative roots. Sinceλμ > λµ + d, applying arguments similar to those in [40, p. 70] shows that lim x→∞ v 2 (x) = d/δ. Therefore, if (92) had no negative roots, the function v 2 (x) would be constant, which is impossible. From this we deduce that (92) has two negative roots. We denote them by z 11 and z 12 . Since z 13 > 0, we get C 13 = 0, which yields (91).
To determine the constants C 9 , C 10 , C 11 and C 12 , we apply considerations similar to those in the proof of Theorem 4 and obtain the system of linear equations (93)-(96), which has a unique solution provided that its determinant is not equal to 0. Finally, applying arguments similar to those in the proof of Theorem 4 guaranties that the functions v 1 (x) and v 2 (x) we have found coincide with the expected discounted dividend payments on the intervals [0, b] and [b, ∞).

Numerical illustrations
We now present numerical examples for the results obtained in Section 5. The claim and premium sizes are also assumed to be exponentially distributed. Set λ = 0.1, λ = 2.3, µ = 3 andμ = 0.2.
Let now the conditions of Theorems 4 and 5 hold. Set additionally b = 5, d = 0.1 and δ = 0.01. Applying Theorems 4 and 5 we can calculate the ruin probability ψ(x) and the expected discounted dividend payments until ruin v(x): The results of calculations for some values of x are given in Table 2. The results presented in Tables 1 and 2 show that the positive dependence between the claim sizes and the inter-claim times decreases the ruin probability and the negative dependence increases it. This conclusion seems to be natural. Indeed, in the case of negative dependence, the situation where large claims arrive in short time intervals is more probable, which obviously leads to ruin in the near future. Moreover, it is easily seen from Table 2 that dividend payments substantially increase the ruin probability, which is also an expected conclusion.