Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs

In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.


INTRODUCTION
For the optimal dividend problem, one may adopt the objective of maximizing the expectation of the discounted dividends until possible ruin. This problem was first addressed by De Finetti [16] who considered a discrete time risk model with step sizes ±1 and showed that the optimal dividend strategy is a barrier strategy. Miyasawa [21] generalized the model to the case that periodic gains of a company can take on values −1, 0, 1, 2, 3, · · · , and showed that the optimal dividend strategy of the generalized model is a barrier one.
Subsequently, the problem of finding the optimal dividend strategy has attracted great attention in the literature of insurance mathematics. For nice surveys on this topic, we refer the reader to Avanzi [3] and Schmidli [22]. Besides insurance risk models, the optimal dividend problem in the so-called dual model has also been studied extensively in recent years. Among others, Avanzi et al. [6] discussed how the expectation of the discounted dividends until ruin can be calculated for the dual model when the gain amounts follow an exponential distribution or a mixture of exponential distributions, and showed how the exact value of the optimal dividend barrier can be determined; and Avanzi and Gerber [5] examined the same problem for the dual model that is perturbed by diffusion, and showed that the optimal dividend strategy in the dual model is also a barrier strategy. To make the problem more interesting, the issue of capital injections has also been considered in the study of optimal dividends in the dual model. Yao et al. [23] studied the optimal problem with dividend payments and issuance of equity in the dual model with proportional transaction costs, and derived the optimal strategy that maximizes the expected present value of dividend payments minus the discounted costs of issuing new equity before ruin. Yao et al. [24] considered the same problem with both fixed and proportional transaction costs. Dai et al. [14,15] investigated the same problem as in Yao et al. [23] for the dual model with diffusion with bounded gains and exponential gains, respectively. Avanzi et al. [7] derived an explicit expression for the value function in the dual model with diffusion when the gains distribution in a mixture of exponentials in the presence of both dividends and capital injections. Specifically, they showed that barrier dividend strategy is optimal, and conjectured that the optimal dividend strategy in the dual model with diffusion should be the barrier strategy regardless of the distribution of gains. Bayraktar et al. [11] examined the same cash injection problem, and used the fluctuation theory of spectrally positive Lévy processes to show the optimality of the barrier strategy for all positive Lévy processes. Bayraktar et al. [12] extended the study to the case with fixed transaction costs. Other related work can be found in Yin and Wen [26], Yin, Wen and Zhao [28], Avanzi et al. [8], Yao et al. [25] and Zhang [29].
In this paper, we provide a uniform mathematical framework to analyze the optimal control problem with dividends and capital injections in the presence of proportional transaction costs for the dual model with random return on investment. The associated value function is defined as the expected present value of dividends minus costs of capital injections until ruin. The rest of the paper is organized as follows. In Section 2, we give a rigorous mathematical formulation of the problem. Section 3 works on the model without capital injections, while Section 4 deals with the model with capital injections which never goes bankrupt. Finally, we solve the general stochastic control problem in Section 5.

Problem formulation
Assume that the surplus generating process P t at time t is given by where x > 0 is the initial assets, p and σ p are positive constants, {W p,t } t≥0 is a standard Brownian motion independent of the homogeneous compound Poisson process Nt i=1 X i , and {X i } is a sequence of independent and identically distributed random variables having common distribution function F with F (0) = 0. Let λ be the intensity of the Poisson process N t . We assume throughout the paper that E[X i ] < ∞ and λE[X i ] − p > 0. Here, we consider the return on investment generating process where {W R,t } t≥0 is another standard Brownian motion, and r and σ R are positive constants. It is assumed that W p,t and W R,t are correlated in the way that where ρ ∈ [−1, 1] is constant, and W 0 p,t is a standard Brownian motion independent of W p,t .
Define the risk process U t as the total assets of the company at time t, i.e., U t is the solution to the stochastic differential equation The solution to (2.3) is given by (see, e.g. Jaschke [19,Theorem 1]) Using Itô's formula for semimartingale, one can show that the infinitesimal generator L of U = {U t , t ≥ 0} is given by The model (2.3) is a natural extension of the dual model in Avanzi and Gerber [5] and Avanzi et al. [6]. As was mentioned in Avanzi et al. [6], the dual model is appropriate for companies that have deterministic expenses and occasional gains whose amount and frequency can be modelled by the jump process Nt i=1 X i . For example, for companies such as pharmaceutical or petroleum companies, the jump could be interpreted as the net present value of future gains from an invention or discovery. Another example is the venture capital investments or research and development investments. Venture capital funds screen out start-up companies and select some companies to invest in. When there is a technological breakthrough, the jump is generated. More examples can be found in Bayraktar and Egami [10] and Avanzi and Gerber [5].
In this paper, we denote by L t the cumulative amount of dividends paid up to time t with L 0− = 0, and by G t the total amount of capital injections up to time t with G 0− = 0.
A dividend control strategy ξ is described by the stochastic process ξ = (L t , G t ). A strategy is called admissible if both L and G are non-decreasing {F t }-adapted processes, and their sample paths are right-continuous with left limits. We denote by Ξ the set of all admissible dividend policies. The risk process with initial capital x ≥ 0 and controlled by a strategy ξ is given by U ξ = {U ξ t , t ≥ 0}, where U ξ t is the solution to the stochastic differential equation In words, the amount of dividends is smaller than the size of the available capitals. Let τ ξ = inf{t ≥ 0 : U ξ t = 0} be the ruin time. Then, the associated performance function is given by where δ > 0 is the discounted rate, 1 − α (0 < α ≤ 1) is the rate of proportional costs on dividend transactions, 1 ≤ β < ∞ is the rate of proportional transaction costs of capital injections. The notation E x represents the expectation conditioned on U ξ 0 = x and the integral is understood pathwise in a Lebesgue-Stieltjes sense. Our aim is to find the value and the optimal policy ξ * ∈ Ξ such that V (x; ξ * ) = V * (x) for all x ≥ 0.
The study of optimal dividends has been around many years. The commonly-used approach to solving these optimal control problems is to proceed by guessing a candidate optimal solution, constructing the corresponding value function, and subsequently verifying its optimality through a verification result. For the model of study, i.e., an upward jump-diffusion process with random return on investment, the optimal control problem remains to be solved. The problem of study can be seen as a natural extension of Bayraktar and Egami [10], and Avanzi, Shen and Wong [7]. In addition, one can see later that the method used in Bayraktar, Kyprianou and Yamazaki [11] cannot be applied to our model since their proof relies on certain characteristics of Lévy process. In order to solve the optimal control problem in this paper, we shall first consider two sub-optimal problems in the next two sections.

Optimal dividend problem without capital injections
In this section, we first consider the dividend problem without capital injections. We shall show that the barrier strategy solve the optimal dividend problem regardless of the jump distribution.
is the solution to the stochastic differential equation and the value function is given by is the time of ruin under the strategy ξ d . We next identify the form of the value function V d and the optimal strategy

HJB equation and verification lemma
For If v is twice continuously differentiable, then applying standard arguments from stochastic control theory (see Fleming and Soner [17]) or an approach similar to that in Azcue and Muler [9], we can show that the value function fulfils the dynamic programming principle for any stopping time T , and that the associated Hamilton-Jacobi-Bellman (HJB) equa- with v(0) = 0, where L is the the extended generator of U defined in (2.4). The HJB equation (3.2) can also be obtained by the heuristic argument of Avanzi et al. [7].
Proof. For any admissible strategy

Construction of a candidate solution
It is assumed that dividends are paid according to the barrier strategy ξ b . Such a strategy has a level of barrier b > 0. When the surplus exceeds the barrier, the excess is paid out immediately as dividends. Let L b t be the total amount of dividends up to time t. The controlled risk process when taking into account of the dividend strategy ξ b is is the solution to the following stochastic differential equation where δ > 0 is the force of interest and T The following result shows that V b (x) as a function of x satisfies an integro-differential equation with certain boundary conditions.

Lemma 3.2. For the risk process U of (2.3) and the infinitesimal generator
given by (3.6).
Proof. Applying Ito's formula for semimartingale to e −δt h b (U b t− ) gives where L c s is the continuous part of L s , and Note that N b t is a local martingale, and Thus, for any appropriate localization sequence of stopping times {t n , n ≥ 1}, we have Letting n → ∞ in (3.8) yields the result.
Proof. To prove the lemma, we use arguments similar to those in Kulenko and Schmidli [20]. Let x > 0, y > 0, and l ∈ (0, 1). Consider the strategies L x and L y for the initial capitals x and y.
Since the processes {P t , t ≥ 0} and {R t , t ≥ 0} have no negative jumps, we have τ L = τ L x ∨ τ L y . It follows and thus the concavity of V b follows. The increasingness of V b (x) is trivial

Verification of optimality
Define the barrier level by We conjecture that the barrier strategy ξ b * is optimal. Proof. Here, we follow the approach of Yao et al. [23] to prove the proposition. Suppose that b * = 0. Then, the associated value function is V d (x) = αx which satisfies the HJB equation (3.2). As a result, we obtain (Γ−δ)V d (x) ≤ 0 which in turn gives λ ∞ 0 yF (dy) ≤ p. On the other hand, suppose that λ ∞ 0 yF (dy) ≤ p. Then, w(x) = αx satisfies (3.2). By Lemma 3.1, we get w(x) ≥ V d (x). However, w(x) ≤ V d (x) since w(x) = αx is the performance function associated with the strategy that x is paid immediately as dividends.
In this case, ruin occurs immediately. Thus, w(x) = V d (x) and the optimal barrier level b * = 0.
Proof. Using the method of Avanzi and Gerber [5], it can be shown that Thus, the function V b * satisfies the HJB equation (3.2). Then, it follows from Lemma 3.1 that V b * (x) ≥ V d (x). However,

Two closed-form solutions
Owing to the complexity of the equation, the solution may not be available in explicit form in general. The following two examples show that one can derive closed-form solution in some special cases.
Example 3.1. Assume that r = 0 and σ R = 0. Then, V b * (x) satisfies the following integro-differential equation with the boundary conditions where Following the arguments of Laplace transform used in Yin, Wen and Zhao [28], one can show that the solution to (3.9)-(3.11) is given by Here, W (δ) is the so-called δ-scale function defined in the way that W (δ) (x) = 0 for all x < 0 and that its Laplace transform on [0, ∞) is given by For further details, the reader is referred to Yin and Wen [26].
Example 3.2. Let σ R = σ p = 0. Assume that X i is exponentially distributed with parameter µ. Then, by Theorem 3.1 and Lemma 3.2, it can be shown that V b * (x) satisfies the following integro-differential equation and with the boundary conditions (3.14) From equation (3.12), we find that Note that this is Kummer's confluent hypergeometric equation with the solution given by where C 1 and C 2 are constants, and M(a, b, x) is the standard confluent hypergeometric function with U(a, b, x) being its second form; see, for example, Abramowitz and Stugen [1, pp. 504-505]. Then, it follows that Using the boundary conditions (3.14) and the formulae we obtain the coefficients and b * is the maximizer of term 1/∆(b) with respect to b, i.e.,

Optimal dividend problem with capital injections
In this section, we consider the optimal dividend problem with capital injections. The set of admissible strategies is given by Ξ c = {ξ c = (L ξc , G ξc ) : (L ξc , G ξc ) ∈ Ξ and U ξc t ≥ 0}.
The controlled surplus process U ξc t satisfies dU ξc t = dP t + U ξc t− dR t − dL ξc t + dG ξc t , t ≥ 0, and the value function is defined as Since the controlled surplus process always stays positive, the company will never go bankrupt. We shall identify the form of the value function V c and the optimal strategy ξ * c such that V c (x) = V (x; ξ * c ).

HJB equation and verification lemma
Applying the techniques used in Section 3, we get the HJB equation and the verification Lemma.
for any admissible strategy ξ c ∈ Ξ c , and thus w(x) ≥ V c (x).

Construction of a candidate solution
We now construct a concave C 2 solution H to the HJB equation (4.2). Due to the effect of the discount factor, it is clear that the optimal strategy is the one that postpone capital injections as long as possible, i.e., we inject capital only when surplus become zero.
Consider the barrier strategy with the upper barrier B * and the lower barrier 0, and the strategy π * = (L π * , G π * ) where (U π * t , L π * ,x t , G π * ,x ) is a solution to the following system (P s − L π * s ), 0 , t > 0.
where the infinitesimal generator L is given by (2.4), then H(x) is given by Proof. For the strategy π * , define Λ = {s : L π * ,x s− = L π * ,x s }. Let L π * ,x,c t be the continuous part of L π * ,x t . Since the process is skip-free downward, G π * ,x t is continuous. In addition, we see from (4.6) that G π * ,x t ≥ 0 and that the support of the Stieltjes measure dG π * ,x t is contained in the closure of the set {t : U π * t = 0}. Applying Ito's formula for semimartingale to e −δt H(U π * t ) gives Note that (L − δ)H(U π * s ) = 0, and that Then, it follows that Since lim t→∞ E x [e −δt H(U π * t− )] ≤ lim t→∞ E x [e −δt H(B * )] = 0, letting t → ∞ in (4.9) and using the monotone convergence theorem yield Proof. Similar to the proof of Lemma 3.3, we use the arguments of Kulenko and Schmidli [20]. Let x > 0, y > 0, and l ∈ (0, 1). Consider the strategies (L π * ,x , G π * ,x ) and (L π * ,y , G π * ,y ) for the initial capitals x and y. Define L t = lL π * ,x t + (1 − l)L π * ,y t and G t = lG π * ,x t + (1 − l)G π * ,y t . Then, L t = L π * ,lx+(1−l)y t . So, we have This shows that the strategy (L t , G t ) is admissible and that It follows that which implies the concavity of V . The proof of increasingness of V (x; π * ) is routine.

Verification of optimality
Define the barrier level as We conjecture that the barrier strategy π * is optimal.

Two closed-form solutions
We now present two examples in which closed-form solution can be derived.
Example 4.1. Assume that r = 0 and σ R = 0. Then, H(x) satisfies the following integro-differential equation and with the boundary conditions where Proof. For any admissible strategy ξ ∈ Ξ, put Λ = {s : L ξ s− = L ξ s }. Applying Ito's formula for semimartingale to e −δt v(U ξ t ) gives where L ξ,c s is the continuous part of L ξ s . We see from (5.1) that (L − δ)v(U ξ d s− ) ≤ 0 and α ≤ v ′ (x) ≤ β. Thus, Finally, by letting t → ∞ in (5.6) and noting that (by Fatou's lemma) we prove the lemma.

Construction of a candidate solution
For any x ≥ 0, we set our candidate strategy to be and our candidate solution to be (5.8) where V d and V c are given by (3.1) and (4.1), respectively, and V b * and H are given by (3.6) and (4.7), respectively.

Verification of optimality
Theorem 5.1. The value function V ξ * defined in (5.8) satisfies and the joint strategy ξ * defined in (5.7) is optimal.