Dynamic risk-sharing game and reinsurance contract design

Insurance: Mathematics and Economics 86 (2019) 216–231

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Dynamic risk-sharing game and reinsurance contract design✩ ∗

Shumin Chen a , Yanchu Liu b , , Chengguo Weng c ,

∗

a

School of Management, Guangdong University of Technology, Guangzhou 510006, PR China Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, PR China c Department of Statistics and Actuarial Science, University of Waterloo, Waterloo N2L 3G1, Canada b

highlights • • • •

The risk-sharing game problem for an insurer and a reinsurer is studied for the first time. A system of coupled HJB equations with delicate analyses is presented. Closed-form solutions are derived for two concrete cases. Sensitivity analysis and numerical examples are provided.

article

info

Article history: Received June 2018 Received in revised form November 2018 Accepted 13 March 2019 Available online 18 March 2019 JEL classification: C73 G22 Keywords: Reinsurance Ruin probability Nash equilibrium Stochastic control HJB equation

a b s t r a c t This paper studies the optimal risk-sharing between an insurer and a reinsurer. The insurer purchases reinsurance for risk-control and decides her retention level with an objective to minimize her ruin probability. The reinsurer has control over the reinsurance price and aims to maximize her expected discounted profits up to the time when the insurer goes bankrupt. In a stochastic differential game-theoretic framework, we determine the insurer’s optimal reinsurance strategy and specify the reinsurance contract by solving a system of coupled Hamilton–Jacobi–Bellman equations. We obtain explicit solutions for the game problem when both the insurance and the reinsurance premiums are calculated according to the standard-deviation principle or the expected value principle, respectively. Our results show that, depending on the model parameters, the reinsurance contract is either provided with a peak price when the insurer has sufficient cash reserve and with a minimum price when otherwise, or is always provided with a peak price. We also perform some numerical analyses and provide economic interpretations for the results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the past decades, the optimal reinsurance problem of insurers has been extensively studied in the fields of insurance and optimal control theory, see Golubin (2008), Chiu and Wong (2014), Jang and Kim (2015), Bi et al. (2016), Deng et al. (2018), among others. The growing literature coincides with increasing practical challenges for insurers to use reinsurance to manage their risk exposure to insurance claims in stochastic environments. The ruin probability is deemed as the insurer’s central concern among these works. Schmidli (2001) pioneered the study of dynamic optimal proportional reinsurance for the minimization of ruin probability under the classical risk model. This result ✩ This research is partially supported by grants of the National Natural Science Foundation of China (grant numbers 71671047, 71501196, 71721001, 11761141007) and the National Social Science Fund of China (grant number 18ZDA092). ∗ Corresponding authors. E-mail addresses: [email protected] (S. Chen), [email protected] (Y. Liu), [email protected] (C. Weng). https://doi.org/10.1016/j.insmatheco.2019.03.004 0167-6687/© 2019 Elsevier B.V. All rights reserved.

is extended to excess-of-loss reinsurance contracts by Hipp and Vogt (2003), to a zero-sum framework by Taksar and Zeng (2011), and to the case with a bivariate reserve risk process by Bai et al. (2013). More results of optimal reinsurance under a ruin probability minimization framework include Schmidli (2008), Luo et al. (2008), Zhang et al. (2016), among others. While the aforementioned literature considers the optimal reinsurance problem only from an insurer’s point of view, more recent articles investigate the optimal reinsurance problem by taking into account both the insurer and the reinsurer’s interests. For example, Zeng and Luo (2013) consider a stochastic cooperation game between an insurer and a reinsurer, and study the Pareto-optimal reinsurance strategy that maximizes a weighted average of the utilities of the insurer and the reinsurer. When the reinsurance premium is calculated according to the variance principle and the mean-value principle, Zeng and Luo (2013) obtain explicit solutions for certain specific utility functions. Li et al. (2014) consider a model where the insurer invests in a risky asset and the reinsurer invests in a risk free asset. With the objective to maximize the product of the utilities of the insurer’s and the

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

reinsurer’s terminal wealth, they obtain explicit solutions for the case of exponential utility. Following this line, Li et al. (2016), Zhao et al. (2017) consider the optimal reinsurance strategy to optimize the expected exponential utility and the mean–variance objective of the weighted average of the insurer and reinsurer’s wealth at terminal time. It is worth noting that these papers all assume a constant reinsurance premium rate (i.e., a pre-fixed reinsurance price) in their models. Nevertheless, the reinsurance market is not competitive and a reinsurance program is typically customized for the buyer (i.e., the insurer) with a price adjusted according to the risk level of the insurer. In this paper we consider the optimal reinsurance contract design problem by assuming that the reinsurer can dynamically adjust the reinsurance safety loading, which uniquely specifies the reinsurance premium and thus reflects the price level of the reinsurance contract. In line with Zeng and Luo (2013), Hu et al. (2018a), we model the dynamics of the insurer’s cash reserve as a diffusion approximation process, consisting of a general reinsurance protection and a risk-free investment. The insurer’s objective is to minimize her ruin probability, and the reinsurer’s objective is to maximize her profits up to the time when the insurer goes bankrupt. We formulate the insurer and the reinsurer’s optimization problems as a stochastic differential game. By using the dynamic programming approach, we come up with a system of coupled Hamilton–Jacobi–Bellman (HJB) equations and a verification theorem for the stochastic differential game. Based on some intriguing observations on the system of HJB equations, we show that under certain conditions the reinsurer acts according to a bang–bang control. More specifically, the reinsurer would either charge the reinsurance contract at a peak price or at a minimal price, depending on the insurer’s cash reserve level. Thus, the system of HJB equations can be reduced into two sets of ordinary differential equations (ODEs), which can be solved much simply. To illustrate the applicability of our theoretical results, we respectively consider the cases where both the insurer and the reinsurer’s premiums are calculated according to standard-deviation principle and expected value principle. Explicit solutions are derived for both cases. Our results show that the equilibrium solution depends on the model parameters and either of following cases holds: (1) the reinsurance price is state-dependent, i.e. it is at the peak price when the insurer’s cash reserve is larger than certain level and is provided at the minimal price when the insurer’s cash reserve is under that level; (2) the reinsurance is always provided at the peak price. Based on these results, a reinsurance contract is specified. To show the impact of model parameters on the reinsurance contract, we perform extensive numerical examples under the standard-deviation premium principle. Our numerical results show that the reinsurance price is negatively related to the insurer’s business performance and is positively related to the reinsurer’s discount rate. That is, when the insurer has a strong business performance, the reinsurance contract is provided more cheaply to reflect the large demand for reinsurance protection and the reinsurer’s long-term interest. Otherwise, when the insurer has a weak business performance, the reinsurance contract is highly priced to reflect the insurer’s high risk level. Finally, for the reinsurer, a larger discount rate means a higher capital cost, and thus a higher reinsurance price that should be charged. The rest of this paper is organized as follows. Section 2 describes the dynamics of the insurer’s cash reserve and formulates the game problem. Section 3 derives a system of coupled HJB equations for the game problem and performs some delicate analyses on the equilibrium strategies. Sections 4 and 5 deal with the game problem by concentrating on the standard-derivation premium principle and the expected value premium principle, respectively. Section 6 presents numerical examples. Section 7 concludes the paper. Technical details and proofs are all relegated to the Appendices.

217

2. The model In this section, we rigorously formulate a Nash game problem for an insurer aiming at risk-control and a reinsurer aiming at profit-seeking. To this end, define a filtrated probability space (Ω , F , {Ft }t ≥0 , P) and consider an insurer with cash reserve described by the classical risk model: Xt = x + ct −

Nt ∑

Zi ,

i=1

where x ≥ 0 is the initial cash reserve, c > 0 is the insurer’s premium rate, Nt is the number of claims up to and including time t, and Zi is the amount of the ith claim. {Nt }t ≥0 is a Poisson process with intensity λ > 0. {Zi }i∈N are positive, independent and identically distributed (i.i.d.) random variables with distribution function FZ (·). We assume that {Zi }i∈N have finite first moment µ0 and second moment σ02 , and that c fulfills the net-profit condition c > λ E[Z ], where Z := Z1 . The net-profit condition indicates that the insurer is in profit. It is widely adopted in the literature of optimal reinsurance, see for example, Bai et al. (2013), Li et al. (2014, 2016). We assume that the insurer purchases reinsurance for risk control. She pays a fraction of her premium to the reinsurer in exchange for the reinsurer’s coverage on a part of the claims {Zi }i∈N . The insurer determines the reinsurance strategy by a selfreinsurance function να (·) : [0, +∞) → [0, +∞). Here, α ≥ 0 is a control parameter that uniquely characterizes ν such that 0 ≤ να (Z ) ≤ Z and ν0 (Z ) ≡ 0. Thus, for the ith claim Zi the insurer pays the amount να (Zi ) and cedes the rest Zi − να (Zi ) to the reinsurer. The reinsurer provides the reinsurance contract up to the time when the insurer goes bankrupt, and is in a privileged position to determine the price of reinsurance. Given the selfreinsurance function να , in each unit of time the insurer has to transfer a part of her premium proceeds, λHθ (Z − να (Z )), to the reinsurer. Here, Hθ (·) : [0, +∞) → [0, +∞) is called the reinsurance premium function, θ ∈ [θ1 , θ2 ] is the safety loading of the reinsurer that specifies the reinsurance premium rate, θ1 and θ2 satisfying 0 < θ1 < θ2 are the exogenously given lower and upper bounds. In addition, we assume that Hθ (Z ) is strictly increasing in θ and Z with Hθ (0) = 0, and that the reinsurance contract is non-cheap, i.e. λHθ (Z ) > c for θ ∈ [θ1 , θ2 ]. Since the self-reinsurance function and the premium function are uniquely characterized by parameters α and θ , we call them the insurer’s reinsurance strategy and the reinsurer’s pricing strategy, respectively. Besides, we assume that the insurer’s cash reserve is invested with a risk-free rate of return r ≥ 0. Thus, the insurer’s cash reserve denoted as Xt at time t, satisfies t

∫

rXs + c − λHθ (Z − να (Z )) ds −

[

Xt = x +

]

0

Nt ∑

να (Zi ), t ≥ 0.

i=1

According to Grandell (1991), Højgaard and Taksar (2001), Siu et al. (2016), this process can be approximated by the following diffusion process:

∫ Xt = x +

t

rXs + c − λ E[να (Z )] − λHθ (Z − να (Z )) ds

[

∫0 t √ + λ E[να2 (Z )]dBs , 0

]

(1)

218

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

where {Bt }t ≥0 is a standard Brownian motion. We assume that the insurer and the reinsurer adjust their strategies dynamically, and rewrite their strategies as α = {αt }t ≥0 and θ = {θt }t ≥0 , where αt and θt denote the reinsurance strategy and reinsurance pricing strategy at time t, respectively. We call the pair of strategies (α, θ ) admissible if (i) {αt }t ≥0 and {θt }t ≥0 are Ft -progressively measurable, and (ii) the corresponding process (1) has a strong solution. Let us denote the sets of admissible reinsurance and pricing strategies by V and Θ , respectively. Given the reinsurer’s pricing strategy θ ∈ Θ , the insurer’s objective is to minimize her ruin probability: JI (x; α, θ ) = Px (τ0 < ∞),

(2)

where Px denotes a conditional probability measure given X0 = x ≥ 0 and τ0 := inf{t ≥ 0 : Xt ≤ 0} is the ruin time of the insurer. In the meanwhile, given the insurer’s reinsurance strategy α ∈ V , the reinsurer’s objective is to maximize her expected aggregate discounted net profits up to the time of the insurer’s bankruptcy: JR (x; α, θ ) := Ex

τ0

[∫

]

e−ρ t λf (αt , θt )dt ,

(3)

0

where Ex denotes the expectation operator conditional on the event X0 = x, λf (αt , θt ) represents the reinsurer’s net profit rate at time t with

and ρ > r is the discount rate of the reinsurer.1 Problems (2)–(3) lead to a stochastic differential game between the insurer and the reinsurer, as formulated below. Find a Nash equilibrium (α ∗ , θ ∗ ) ∈ V × Θ such

{

JI (x; α ∗ , θ ∗ ) ≤ JI (x; α, θ ∗ ),

JR (x; α , θ ) ≥ JR (x; α , θ ), ∗

∗

∗

for all α ∈ V , for all θ ∈ Θ .

xs :=

λHθ1 (Z ) − c

κi := λ

r

f (0, θi )

ρ

,

,

xS :=

λHθ2 (Z ) − c r

,

i = 1, 2.

(5)

Due to our assumptions on H and f , it is clear that 0 < xs < xS and 0 < κ1 < κ2 . Inspired by Luo et al. (2008), we have the following result. Lemma 2.1. We have V (x) = 0, W (x) = κ2 , and (α ∗ , θ ∗ ) = (0, θ2 ) for x ≥ xS . Proof. See Appendix A.1. □ Lemma 2.1 shows that, if the insurer has a sufficient cash reserve, say at least xS , she is able to afford the reinsurance at any price to avoid bankruptcy by ceding all claim risks to the reinsurer. In this case, since the reinsurance is in high demand, the reinsurer charges the contract at the peak price to boost her profits. Thus, xS is a critical level of the insurer’s reserve to the optimal reinsurance contract. We call xS the insurer’s safe level. In the sequel, we focus on the more interesting case when 0 < x < xS . 3. HJB equations and verification theorems

f (α, θ ) := Hθ (Z − να (Z )) − E(Z − να (Z )),

Problem 2.1. that

To simplify our notations, we denote

(4)

V (x) := JI (x; α ∗ , θ ∗ ) and W (x) := JR (x; α ∗ , θ ∗ ) are called the insurer and the insurer’s equilibrium value functions, respectively. Problem 2.1 captures the insurer and the reinsurer’s interests simultaneously and provides a reinsurance contract, which is the compromise between the two parties. In equilibrium, V is the risk measure for the insurer and W can be seen as the value of the reinsurance contract for the reinsurer. Remark 2.1. (i) Since our model setup has a Markovian structure, we expect the equilibrium strategies (α ∗ , θ ∗ ) to be functions of the insurer’s cash reserve. In the sequel, we confine ourselves to Markovian controls. (ii) While it may also be interesting to consider other objectives for the insurer and the reinsurer, for example, maximizing the utility of their wealth at terminal time (see Zeng and Luo, 2013; Hu et al., 2018a,b; Deng et al., 2018), we tend to believe that the insurer is primarily concerned about risk mitigation in entering a reinsurance contract, whereas the reinsurer is mainly concerned about profitability. Thus, in this paper we consider the ruin probability minimization problem for the insurer and the profit maximization problem for the reinsurer. 1 In fact, ρ is the risk-adjusted discount rate and can be determined by the capital asset pricing model (CAPM), i.e. ρ = r+‘‘risk premium". Thus, it is reasonable to assume ρ > r. This assumption is also adopted by Dixit and Pindyck (1994), Grenadier and Wang (2007), Bai et al. (2012), Chen et al. (2018), etc.

In this section, we tackle Problem 2.1 by using the stochastic dynamic programming approach. To this end, for g ∈ C2 (0, xS ), define the operators Lα,θ and Aα,θ as

⎧ [ ] α,θ ′ ⎪ ⎨L g(x) := g (x) rx + c − λHθ (Z − να (Z )) − λ E[να (Z )] + 1 λ E[να2 (Z )]g ′′ (x), ⎪ ⎩ α,θ 2 A g(x) := Lα,θ g(x) − ρ g(x) + λf (α, θ ),

(6)

where g ′ and g ′′ are the first- and second-order derivatives of g with respect to x, respectively. By standard argument in stochastic dynamic programming theory (see Krylov, 2008; Schmidli, 2008), if V , W ∈ C2 (0, xS ) and the Nash equilibrium (α ∗ , θ ∗ ) exist, they satisfy the following system of coupled HJB equations: inf Lα,θ v (x) = 0, ∗

α∈V

sup Aα θ ∈Θ

∗ ,θ

w (x) = 0,

v (0) = 1, v (xS ) = 0, w(0) = 0, w(xS ) = κ2 .

(7) (8)

The following verification theorem states that smooth solutions to HJB equations (7) and (8) coincide with the value functions (V , W ), and characterizes the equilibrium strategies. Theorem 3.1. If (α ∗ , θ ∗ ) ∈ V × Θ and v, w ∈ C2 (0, xS ) satisfy HJB equations (7) and (8), then V (x) = v (x), W (x) = w (x), and (α ∗ , θ ∗ ) is the equilibrium strategy. Moreover, (α ∗ , θ ∗ ) are characterized by

{ ∗ α ∗ (x) := arg infα Lα,θ v (x), θ ∗ (x) := arg supθ ∈[θ1 ,θ2 ] Aα

∗ ,θ

w(x).

(9)

Proof. The proof of this theorem is standard, and is thus omitted for simplicity. □ According to Theorem 3.1, we just need to find solutions (v, w ) along with strategies (α ∗ , θ ∗ ) such that (7)–(8) hold. However, since the self-reinsurance function ν and the premium function H are rather general, we cannot obtain the solutions without further specification on the model. Below, based on careful observations on the system of equations, we provide some heuristic results that will help in constructing solutions to the equations in Sections 4 and 5 where a certain concrete functional form is specified for the premium function H.

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

Remark 3.1. Eq. (9) implies that at equilibrium the reinsurer’s pricing strategy is a feed-back control of the insurer’s cash reserve:

θ ∗ (x) = arg sup

{

θ ∈[θ1 ,θ2 ]

( )} λ(1 − w′ (x))Hθ Z − να∗ (Z ) .

Therefore, when( the insurer ) purchases reinsurance (i.e. να∗ (Z ) < Z ), we have Hθ Z − να ∗ (Z ) > 0 and

θ ∗ (x) =

{

θ2 , θ1 ,

if w ′ (x) ≤ 1, if w ′ (x) > 1.

Corollary 3.1 suggests that xs is the level of cash reserve above which the insurer will transfer all the claims to the reinsurer if the reinsurance contract is priced at θ = θ1 . Nevertheless, when the insurer’s cash reserve is no less than xs , the large demand for reinsurance indicates that a contract with price θ1 does not serve the reinsurer’s best interest. To increase her profits the reinsurer will increase the price to the higher level θ2 (see Remark 3.2). Thus, if w ′ (x) > 1 for x ∈ [0, xw ), we must have 0 ≤ xw ≤ xs . In this case, the equilibrium strategies are given by

(10)

{ (α (x), θ (x)) = ∗

Thus, we expect that at equilibrium the reinsurer’s pricing strategy is a bang–bang control.2 It is worth noting that θ ∗ can take any value in [θ1 , θ2 ] when w ′ (x) = 1. However, for notational convenience, in this case we still let θ ∗ (x) = θ2 . Similarly, we let θ ∗ (x) = θ2 when the insurer purchases no reinsurance (i.e. να ∗ (Z ) = Z ). Remark 3.2. Since θ ∗ (xS ) = θ2 (see Lemma 2.1), from Eq. (10) we have w ′ (xS −) ≤ 1. Let

∗

xw := sup{x ∈ [0, xS ] : w (x) = 1}

x ∈ [0, xw ),

(α2∗ (x), θ2 ),

x ∈ [xw , xS ],

P1 : Aα1 ,θ1 v (x) = 0, ∗

α2∗ ,θ2

A

v (x) = 0,

θ ∗ (x) =

θ1 , θ2 ,

x ∈ [0, xw ), x ∈ [xw , xS ].

(11)

In this case, xw is called the switching-over point at which the price of the reinsurance contract is adjusted from a minimal (maximal) level to a maximal (minimal) level. Accordingly, the insurer’s optimal strategy is given by

α ∗ (x) =

{

α1∗ (x), α2∗ (x),

x ∈ [0, xw ), x ∈ [xw , xS ). α,θi

where αi (x) := arg infα L v (x) is the insurer’s optimal strategy when the reinsurance safety loading θ ≡ θi , i ∈ {1, 2}. ∗

Eq. (11) indicates that the reinsurer charges more (less) for the reinsurance contract when the insurer has a sufficient (an insufficient) cash reserve. Intuitively, when the insurer has a sufficient cash reserve, she can afford more reinsurance protection. The large demand for reinsurance leads to a high reinsurance price θ = θ2 . Otherwise, when the insurer has an insufficient cash reserve and is faced with high bankruptcy risk, she cannot afford too much reinsurance. The small demand leads to a decrease in the reinsurance price. Moreover, for the reinsurer’s long-term interest, the reinsurer also tends to decrease the reinsurance price to protect the insurer from going into bankruptcy. In this sense, the reinsurance contract provides a mechanism to protect both the insurer and the reinsurer’s long-term interests. We present the following two corollaries to have an estimation over the value of xw . Corollary 3.1. We have α1∗ (x) = 0 for x ≥ xs and α2∗ (x) = 0 for x ≥ xS , where xs and xS are defined in (5). Proof. The proof is similar to part (i) of Appendix A.1, and is thus omitted. □ Corollary 3.2. Assume w ′ (x) > 1, x ∈ [0, xw ). Then xw ≤ xs , where xs is defined in (5). Proof. See Appendix A.2. □ 2 Because of its simplicity or convenience, bang–bang control is widely adopted in economical decision-making, e.g. see Evstigneev et al. (2004).

x ∈ (0, xw ), v (0) = 1,

(13)

x ∈ (xw , xS ), v (xS ) = 0,

(14)

′

(15)

w(x) = 0,

x ∈ (0, xw ), w (0) = 0, w (xw −) = 1, (16)

Lα2 ,θ2 w (x) = 0,

x ∈ (xw , xS ), w ′ (xw +) = 1, w (xS ) = κ2 , (17)

′

{

(12)

v (xw −) = v (xw +), v (xw −) = v (xw +), ′

L

with the convention that xw = 0 if w (x) < 1, ∀x ∈ [0, xS ]. If w′ (x) > 1 for x ∈ [0, xw ), from Eq. (10) we have

(α1∗ (x), θ1 ),

with the convention (α ∗ (x), θ ∗ (x)) = (α2∗ (x), θ2 ) if xw = 0. With strategy (12), if xw > 0, the system of coupled HJB equations (7) and (8) can be rewritten as

α1∗ ,θ1

′

219

∗

′

w(xw −) = w(xw +).

(18)

Here, (15) and (18) are the smooth-pasting and value-matching conditions; w ′ (xw −) = w ′ (xw +) = 1 arise from (10) and guarantee the optimality of xw ; w (0) = 0, w (xS ) = κ2 , v (0) = 1 and v (xS ) = 0 are due to the boundary conditions. Otherwise, if xw = 0, the HJB equations (7) and (8) become P2 : Lα2 ,θ2 v (x) = 0, ∗

A

α2∗ ,θ2

w(x) = 0,

x ∈ (0, xS ), v (0) = 1, v (xS ) = 0, x ∈ (0, xS ), w (0) = 0, w (xS ) = κ2 ,

(19) (20)

where (19) characterizes a ruin probability problem and is commonly observed in the literature of ruin probability optimization, see e.g. Schmidli (2008) and Bai et al. (2013). Remark 3.3. If P1–P2 admit solutions (v, w ), since w is continuously differential and w ′ (xw −) = w ′ (xw +) = 1, from ∗ ∗ Aα1 ,θ1 w (xw −) = Aα2 ,θ2 w (xw +) = 0 we have

w′′ (xw −) E[να2∗ (xw ) (Z )] = w ′′ (xw +) E[να2∗ (xw ) (Z )]. 1

2

Thus, if E[να2∗ (x ) (Z )] ̸ = E[να2∗ (x ) (Z )], w is not twice continuously 1 w 2 w differentiable at xw . Based on the above observations, for specific να and Hθ , we may first construct solutions (v, w ) to P1–P2. If (v, w ) satisfy HJB equations (7) and (8) and (θ ∗ , α ∗ ) are admissible, according to Theorem 3.1, (V , W ) = (v, w ) and (θ ∗ , α ∗ ) is the equilibrium strategy. Note that if xw > 0, v and w are C1 but not necessarily C2 at xw (see Remark 3.3). However, since the proof of Theorem 3.1 is based on the application of Itô’s formula, which is valid for functions whose second derivatives have a finite number of discontinuities (see Øksendal and Sulem, 2005), we may apply Theorem 3.1 to v and w in the same way as if they are twice continuously differentiable and prove the optimality of (v, w ). In this case, from (12) we can also see that (α ∗ , θ ∗ ) are continuous except at xw . In actuarial science, many commonly used premium principles are based on moments, particularly the mean and variance, of claims, e.g. expected value principle (Højgaard and Taksar, 2001), variance principle (Liang and Yuen, 2016), exponential principle (Cheng et al., 2016), standard deviation principle (Schmidli, 2008), and so on. In the sequel, for illustration we consider both

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the insurer and the reinsurer determine their premiums based on the standard-deviation principle or the expected value principle.3 For both cases, we show that either P1 or P2 admits solutions (v, w ), and rigorously show that (v, w ) are indeed the equilibrium value functions. Remark 3.4. In real applications, the price of a reinsurance contract may not be continuously adjusted. However, in practice reinsurance agreement can be signed on a term basis and the price can be renegotiated for each term with a built-in contract period (e.g. 1 year). In this sense, our model can be viewed as a continuous approximation to reality and our results provide useful insights for the decision makings for both parties. 4. The standard-deviation principle

4.1. Case (1) x0 < xs

In this section, we assume that both the insurer and the reinsurer charge premiums according to the standard-deviation √ principle, i.e., Hθ (Z ) = E[Z ] + θ E[Z 2 ].4 Thus, the insurer and the reinsurer’s premium rates are given by

{

c = λ(µ0 + ησ0 ),

λHθ (Z ) = λ(µ0 + θ σ0 ), where η ∈ (0, θ1 ) and θ ∈ [θ1 , θ2 ] are the safety loadings of the insurer and the reinsurer. Accordingly, from (5) we have xs = λσ0 θ1r−η , xS = λσ0 θ2r−η , and κi = λθi σρ0 , i ∈ {1, 2}. Inspired by Zeng and Luo (2013), we have the following result. Lemma 4.1. Assume V (x) ≥ 0, x ∈ (0, xS ). Then the optimal selfreinsurance function has the form να (Z ) = α Z , where α ∈ [0, 1] is the self-retention level. ′′

This lemma is a modified version of Proposition 1 in Zeng and Luo (2013). The proof is similar and thus omitted. Lemma 4.1 shows that, under the standard-deviation principle, at equilibrium the reinsurance strategy is a proportional reinsurance strategy. Thus, in this section we only need to focus on proportional reinsurance strategies for the insurer. Given a proportional reinsurance strategy {αt }t ≥0 , the dynamics of the insurer’s cash reserve (1) can be rewritten as dXt = rXt + θt αt − (θt − η) λσ0 dt +

[

(

)

α,θ

and the operators L

α,θ

and A

]

√

λαt σ0 dBt ,

X0 = x,

′

λ

2

2 ′′ 0 g (x)

αi∗ (x) =

2r

λθi σ0

,

Assume v ′ (x) < 0 and v ′′ (x) > 0. Then (

(θi − η)λσ0 r

− x) ∧ 1,

i = 1, 2.

Lemma 4.3. With xw = y ∈ (0, xs ], ODE (16) admits a solution w1 (·; y) ∈ C2 (0, y) and ODE (17) admits a solution w2 (·; y) ∈ C2 (y, xS ). Moreover, (i) w1 (y; y) and w2 (y; y) are continuous in y ∈ [0, xs ]; (ii) w2 (xs ; xs ) < κ1 and w2 (0; 0) = q0 , where q0 is defined in (A.12). Proof. See Appendix A.4. □ According to Lemma 4.3, if there exists a switching-over point xw ∈ (0, xs ] such that w1 (xw ; xw ) = w2 (xw ; xw ), then the solution to Eqs. (16)–(18) is given by

w(x) =

{

w1 (x; xw ), w2 (x; xw ),

x ∈ [0, xw ), x ∈ [xw , xS ].

By substituting strategies (21) into (13)–(15), v is also determined, and thus problem P1 is solved. Otherwise, we consider problem P2. The following theorem rigorously shows our results. Assume xs > x0 . Let 𭟋 be as defined in (A.18).

(i) If q0 > 0, P1 has a solution. Moreover,

We now proceed to construct solutions (v, w ) to P1–P2. First, we characterize the strategy αi∗ (x), i ∈ {1, 2}. Lemma 4.2.

θ

First, we consider the case where x0 < xs (i.e. θ1 > 22 ). Since xw ≤ xs (see Corollary 3.2), depending on the model parameters there are three scenarios for the value of xw : (a) x0 ≤ xw ≤ xs ; (b) 0 < xw < x0 ; and (c) xw = 0 (i.e. θ ∗ (x) ≡ θ2 ). Fig. 1 illustrates the insurer and the reinsurer’s strategies for each scenario.

Theorem 4.4.

defined in (6) become

α,θ

L g(x) = [rx + (θ α − (θ − η))λσ0 ]g (x) + 2 α σ Aα,θ g(x) = Lα,θ g(x) − ρ g(x) + λσ0 θ (1 − α ).

{

insurance business is less profitable with a smaller η. The reason is that, with insufficient cash reserve or income, the insurer cannot afford too much reinsurance and thus needs to pay more claims by herself. Moreover, it is clear that α2∗ (x) ≥ α1∗ (x). That is, with the reinsurance contract becoming more costly, the insurer tends to retain more claim risk and the demand for reinsurance decreases. To make our results concise, in this section we further assume that θ1 < 2η < θ2 .5 Thus, from (21) it is clear that 0 ≤ α1∗ (x) < 1 for x ∈ [0, xs ], 0 ≤ α2∗ (x) < 1 for x ∈ (x0 , xS ], and α2∗ (x) = 1 λσ θ for x ∈ [0, x0 ], where x0 := r 0 ( 22 − η) is another critical point. Depending on the values of xs and x0 , we consider two different cases in detail: Case (1) x0 < xs and Case (2) x0 ≥ xs . In both cases we can obtain explicit solutions to Problem 2.1.

(a) if 𭟋(xs − x0 ) < 0, the equilibrium strategies are given by

⎧( ) ⎪ ⎨ 2r ( (θ1 −η)λσ0 − x), θ1 , λθ σ r (α ∗ (x), θ ∗ (x)) = ( 1 0 ) ⎪ ⎩ 2r ( (θ2 −η)λσ0 − x), θ2 , λθ2 σ0 r

Note that αi∗ is the decreasing function of η and x. Thus, with θ = θi , the insurer retains more claim risk and purchases less reinsurance when she has a smaller amount of cash reserve or the 3 In fact, since our framework presented in this paper is rather general, our results in Sections 4 and 5 can be extended to other premium functions and self-reinsurance functions. 4 Commonly, it is assumed that c is calculated according to the expected value principle, see Chen et al. (2018). However, to compare the safety loadings of insurer and reinsurer, we assume that both insurer and reinsurer determine their premiums using the same premium principle.

x ∈ [xw , xS ]; (22)

(21)

Proof. See Appendix A.3. □

x ∈ [0, xw ),

(b) otherwise, if 𭟋(xs − x0 ) ≥ 0, the equilibrium strategies are given by

⎧( ) (θ1 −η)λσ0 2r ⎪ ( − x) , θ , ⎪ 1 ⎨ λθ1 σ0 r ∗ ∗ (1 , θ ) , (α (x), θ (x)) = ( 2 ) ⎪ ⎪ ⎩ 2r ( (θ2 −η)λσ0 − x), θ2 , λθ2 σ0 r

x ∈ [0, xw ), x ∈ [xw , x0 ], x ∈ (x0 , xS ]. (23)

5 Our results in this section can be easily extended to the cases 2η < θ 1 and θ2 < 2η. However, since in both cases the basic structures and the basic ∗ ∗ properties of (α , θ ) and (v, w ) are similar to the case θ1 < 2η < θ2 , for saving space we choose not to include these two cases in this paper.

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

221

Fig. 1. The equilibrium strategies (α ∗ , θ ∗ ) for the scenarios (a) x0 ≤ xw ≤ xs , (b) 0 < xw < x0 , and (c) xw = 0 when x0 < xs .

(ii) If q0 ≤ 0, P2 has a solution. The equilibrium strategies are given by (α ∗ (x), θ ∗ (x)) =

{ (1 ( , θ2 ),

(θ −η)λσ ( 2 r 0 λθ2 σ0 2r

) − x), θ2 ,

x ∈ [0, x0 ),

Table 1 Benchmark model parameters for the standard-deviation principle case. Parameters

η

θ1

θ2

µ0

σ0

r

ρ

λ

Values

0.3

0.5

0.8

1.0

0.4

0.03

0.1

1.0

x ∈ [x0 , xS ]. (24)

4.2. Case (2) xs ≤ x0 θ

Proof. See Appendix A.5. □ Theorem 4.4 verifies our postulation in Remark 3.1 that, when the insurer has a sufficient cash reserve (> xw ), she purchases more reinsurance protection, leading to a maximal reinsurance price θ2 . Otherwise, when the insurer has an insufficient cash reserve (< xw ), the insurer prefers to purchase less reinsurance, leading to a minimal reinsurance price θ1 . An interesting feature for the case q0 > 0 and 𭟋(xs − x0 ) ≥ 0 is that the insurer does not purchase reinsurance (i.e., α ∗ (x) = 1) when she has a medium cash reserve (xw ≤ x ≤ x0 ), and otherwise, she buys some reinsurance. Intuitively, when the insurer has a medium cash reserve, knowing that the reinsurance price will be cut down to θ1 when she is in financial distress, the insurer will purchase reinsurance only if reinsurance becomes cheap (x < xw ) or if she has a sufficient cash reserve (x > x0 ). Remark 4.1. (i) If q0 > 0, since α1∗ (xw −) ̸ = α2∗ (xw +), from Remark 3.3 we can see that w ′′ (xw −) ̸ = w ′′ (xw +). Similarly, from (A.2) we see that v ′′ (xw −) ̸ = v ′′ (xw +). (ii) When xw < x0 , from Aα1 ,θ1 w2 (x0 −) = Aα2 ,θ2 w2 (x0 +) = 0 and α2∗ (x0 ) = 1, we have w ′′ (x0 −) = w ′′ (x0 +). Similarly, we have v ′′ (x0 −) = v ′′ (x0 +). That is, v and w are twice continuously differentiable at x0 . ∗

∗

When xs ≤ x0 (i.e. θ1 ≤ 22 ), akin to the analyses in Section 4.1, we are able to construct solutions (v, w ) to P1–P2 along with the equilibrium strategies (α ∗ , θ ∗ ), and then verify their optimality. An illustration of the equilibrium strategies is provided in Fig. 2. The results are presented in the following theorem. Theorem 4.5. Assume xs ≤ x0 . If q0 > 0, the equilibrium strategy is given by (23); otherwise, if q0 ≤ 0, the equilibrium strategy is given by (24). Proof. Since the proof of this theorem is very similar to that of Theorem 4.4 (ii), we omit it for saving space. □ As compared to the case xs > x0 , in this case the reinsurance contract has a larger upper bound θ2 for the safety loading, indicating that the reinsurance contract becomes more costly. Thus, with other model parameters being fixed, there is always a noreinsurance zone [xw , x0 ] in which the insurer will not purchase reinsurance. Consider the benchmark model parameters as given in Table 1. Let η = 0.38, 0.30 and 0.255, we have xw = 0.3108, 0.24 and 0, respectively. Fig. 3 plots the equilibrium value functions (V , W ). As expected, V is strictly decreasing and convex, and W is strictly increasing. When the insurance safety loading increases (i.e., the insurance business is more profitable), the insurer has a smaller ruin probability and the reinsurance contract becomes

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Fig. 2. Optimal equilibrium strategies (α ∗ , θ ∗ ) for the scenarios (a) 0 < xw < xs and (b) xw = 0, when x0 > xs .

Fig. 3. The equilibrium value functions V and W for η = 0.38, 0.30, and 0.255.

more valuable. These results are consistent with our common sense. 5. The expected value principle

see Højgaard and Taksar (2001), Bai et al. (2013), Siu et al. (2016). Thus, in this section we confine ourself to excess-of-loss reinsurance and call α as the insurer’s reinsurance strategy. Given a dynamic strategy {αt }t ≥0 , Eq. (1) becomes

[

In this section, we consider the case where both the insurer and the reinsurer’s premiums are determined according to the expected value principle, i.e., Hθ (Z ) = (1 + θ ) E[Z ]. In this case, the insurer and the reinsurer’s premium rates are given by

]

dXt = rXt + λ[θµ(αt ) + (η − θ )µ0 ] dt +

√ λσ (αt )dBt ,

X0 = x,

where

c = (1 + η)λµ0 ,

⎧ ⎨µ(α ) := E[(Z ∧ α )] = ∫ α F¯ (y)dy, Z 0 √ √ ∫α ⎩σ (α ) := E[(Z ∧ α )2 ] = 2yF¯Z (y)dy,

λHθ (Z ) = (1 + θ )λµ0 ,

and F¯Z (x) := 1 − FZ (x). For any test function g ∈ C2 (0, xS ), the

{

where the positive parameters η ∈ (0, θ1 ) and θ ∈ [θ1 , θ2 ] are the insurance and reinsurance safety loadings, respectively. θ −η θ −η Accordingly, we have xs = λµ0 1 r , xS = λµ0 2 r , and κi = θi λµ0 ρ , i ∈ {1, 2}. Similar to Lemma 4.1, we present the following result without proof. Lemma 5.1. Assume V ′′ (x) ≥ 0, x ∈ (0, xS ). Then the optimal reinsurance strategy is in the form να (Z ) = min{Z , α0 Z + α}, where α0 ∈ [0, 1] and α > 0 are constants. In particular, when α0 = 0, the insurer’s strategy is an excessof-loss reinsurance strategy that is uniquely specified by the retention level α . In this case, for each claim Zi , the insurer pays the amount min{Zi , α} and the reinsurer covers the rest Zi − min{Zi , α}. With a larger α , the insurer pays a larger proportion for each claim and takes more risk by herself. The excess-of-loss reinsurance is well adopted in practice and theoretical research,

0

operators Lα,θ and Aα,θ become

⎧ α,θ ⎪ ⎪ ⎨L g(x)

[ ] := rx + λ[θµ(α ) − (θ − η)µ0 ] g ′ (x)

⎪ ⎪ ⎩

:= Lα,θ g(x) − ρ g(x) + λθ (µ0 − µ(α )),

Aα,θ g(x)

+ λ2 σ 2 (α )g ′′ (x),

To construct solutions (v, w ) to P1–P2, we first characterize the insurer’s reinsurance strategy αi∗ (x), i ∈ {1, 2}. Lemma 5.2. Assume v ′′ (x) > 0 and v ′ (x) < 0. Then, αi∗ (x) ∈ [0, α¯ i∗ ] is uniquely determined by the following non-linear equation:

] λ α[ rx + λ[θi µ(α ) + (η − θi )µ0 ] + σ 2 (α ) = 0, (25) θi 2 ∗ where α¯ i is the solution of Eq. (25) when r = 0. α1∗ (x) = 0 if and only if x ≥ xs and α2∗ (x) = 0 if and only if x ≥ xS . Moreover, when αi∗ (x) > 0, αi∗ (x) is strictly decreasing in x, η and r, and α1∗ (x) < α2∗ (x). Pi (α ) := −

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

Proof. See Appendix A.6. □

Table 2 Benchmark model parameters for the expected value principle case.

Lemma 5.2 shows that, for the same reason as in the standarddeviation principle case, the insurer purchases less reinsurance and pays more claims by herself when (1) the cash reserve is insufficient; (2) the investment return rate is poor; (3) the insurance business is less profitable; and (4) the reinsurance contract becomes costly. Lemma 5.3. With xw = y ∈ (0, xs ], ODE (16) admits a solution w1 (·; y) ∈ C2 (0, y) and ODE (17) admits a solution w2 (·; y) ∈ C2 (y, xS ). Moreover, w1 (y; y) and w2 (y; y) are continuous in y ∈ [0, xs ] with w2′′ (y+; y) < 0 and limy→xs w2 (y; y) < κ1 . Proof. See Appendix A.7. □ Thus, if there exists xw ∈ (0, xs ] such that w1 (xw ; xw ) = w2 (xw ; xw ), a solution to the system of Eqs. (16)–(18) is determined. Substituting αi∗ (x) into (13)–(15), we can also determine the insurer’s ruin probability. Otherwise, if xw = 0, we consider problem P2. The following theorem rigorously shows this. Theorem 5.4. (i) If w1 (0; 0) > 0, the equilibrium strategy is given by (12), where xw solves w1 (y; y) = w2 (y; y). Moreover, the equilibrium value functions are given by

w(x) =

{

w1 (x; xw ), w2 (x; xw ),

x ∈ [0, xw ), x ∈ [xw , xS ],

(26)

and

⎧ ∫ ∫ x − yxw α∗θ1(z) dz ⎨ 1 dy, v (xw ) − v ′ (xw ) x w e ∫ v (x) = θ ∫ − xyw ∗2 dz ⎩ x α2 (z) v (xw ) + v ′ (xw ) xw e dy,

x ∈ [0, xw ), x ∈ [xw , xS ], (27)

where

⎧ ′ ⎪ ⎪ ⎨v (xw ) = −

∫ xw − 0

⎪ ⎪ ⎩

∫ xw y

e

v (xw ) = 1 + v ′ (xw )

θ1 dz α1∗ (z)

∫ xw 0

1

∫y θ2 ∫ x − xw α ∗ (z) dz 2 dy+ xwS e dy ∫ xw θ 1 − y dz ∗ α1 (z)

e

,

dy.

(ii) If w1 (0; 0) ≤ 0, the equilibrium strategy is given by (12) with xw = 0. Moreover,

[ ⎧ ⎪ w (x) = Ex κ2 e−ρτS 1{τS <τ0 } ⎪ ⎪ ⎪ ∫ τ ∧τ ( ) ] ⎪ ⎨ + 0 S 0 e−ρ t λθ2 µ0 − µ(α2∗ (Xt )) dt , ∫ ⎪ ∫ xS − yxS α∗θ2(z) dz ⎪ 1 ⎪ 2 e dy, v (x) = ⎪ ∫ xS θ 2 ⎪ x ⎩ ∫ xS − y α ∗ (z) dz 0

e

2

223

(28)

dy

where τS := inf{t ≥ 0 : Xt ≥ xS }. Proof. See Appendix A.8. □ With Theorem 5.4, we may determine the solutions to Problem 2.1 numerically by using the Monte Carlo simulation or the finite difference method. As an example, we suppose that the claims {Zi }i∈N follow a US-Pareto distribution with density dFZ (y) = 3(1 + y)−4 , y ≥ 0 (see Zeng and Luo, 2013), so that dy µ0 = 0.9706, σ0 = 0.9852. The other default parameters are given in Table 2. When η = 0.18, we have w1 (0; 0) = 0.0405 and xw = 0.069. According to Theorem 5.4, at equilibrium the reinsurance is priced with θ ∗ = 0.3 when insurer’s cash reserve is more than xw = 0.069, and is adjusted down to θ ∗ = 0.2 when the

Parameters

η

θ1

θ2

r

ρ

λ

Values

0.18

0.2

0.3

0.05

0.12

0.2

insurer’s cash reserve is less than xw = 0.069. When η = 0.10, we have w1 (0; 0) = −0.0958 and xw = 0; thus, the reinsurance contract is provided at the peak price with θ ∗ = 0.3. Fig. 4 plots the equilibrium strategy and the equilibrium value functions. In line with our findings in Section 4, we see that, with a larger η the insurance business becomes more profitable, the insurer is able to afford for more insurance protection to decrease her ruin probability. Correspondingly, the contract becomes more valuable. 6. Numerical analyses In this section, we conduct some numerical studies to investigate the effects of model parameters on the equilibrium strategy. To simplify calculation, we assume that both the insurer and the reinsurer adopt the standard-deviation principle, and set the default model parameters as given in Table 1. Since the insurer’s optimal reinsurance strategy provided in Theorem 4.4 has a simple structure and is easy to understand, in this section we mainly focus on the impact of model parameters on the reinsurance contract, which is uniquely characterized by the switching-over point xw . Fig. 5 shows the impacts of η, r , λ and ρ on xw . When the insurer’s cash reserve is more (or less) than xw , as stated in Section 4, the reinsurance contract is in high (or low) price region. Fig. 5(a) plots xw as the function of the insurer’s safety loading η ∈ θ θ ( 21 , 22 ) = (0.26, 0.4). It clearly shows that the safety loading 0.30, where q0 = 0, is critical for the reinsurance contract. For η > 0.3, we have q0 > 0. In this case, the reinsurance contract is provided at a state-dependent price. Moreover, xw is an increasing function of η, indicating that the reinsurance contract becomes cheaper as the insurance business becomes more profitable. In fact, as η increases, the insurer becomes more profitable and is at a lower risk level, thus less risk premium is imposed on the reinsurance and the reinsurance contract becomes cheaper. We can also see that, when η > 0.382, since the price of reinsurance is low, the insurer always chooses to purchase some reinsurance for risk control. When the insurer is less profitable with η ∈ (0.3, 0.382), with reinsurance becoming more costly, the insurer will purchase some reinsurance if the price is low (x < xw ) or she has a sufficient cash reserve (x > x0 ). Finally, when η < 0.3, q0 < 0, the insurance business is in a bad condition such that the reinsurance contract is provided at peak price θ2 . Paying the reinsurance poses a heavy financial burden on the insurer, thus she will purchase reinsurance only if she has a sufficient cash reserve (x > x0 ). Due to the same reason, we can see from Fig. 5(b) that r has similar impacts on the switching over point xw . That is, when the insurer gains a higher yield from her investment, the reinsurance contract becomes cheaper and the insurer has a larger demand for it. Fig. 5(c) shows that, when λ < 1.01, q0 < 0 and hence xw = 0; when λ > 1.01, q0 > 0 and xw is positive and slightly increases as λ increases. Since λ can be used to measure the insurer’s business scale, this result indicates that the insurer’s business scale has light impact on the reinsurance price. Moreover, we see that, as λ increases, the no reinsurance purchase region enlarges. That is, with a larger business scale the insurer has less desire to seek reinsurance protection. A possible reason is that, with a larger

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Fig. 4. Equilibrium strategies and value functions when η = 0.18 and η = 0.10.

business scale, the insurer becomes more stable and has less inclination to purchase reinsurance. Fig. 5(d) shows that, when ρ < 0.1, q0 > 0 and xw is positive and is a decreasing function of ρ . Moreover, the no reinsurance purchase region enlarges as λ increases. Since ρ represents the capital cost of the reinsurer, this result indicates that, with a higher capital cost the reinsurer focuses more on the short-term interest and tends to raise the reinsurance price, which leads to less demand for reinsurance from the insurer. Fig. 6 illustrates the impact of θ1 and θ2 on the reinsurance price. We can see that xw is increasing in θ1 and decreasing in θ2 . This can be explained as follows. When θ1 increases and gets closer to θ2 , a reinsurance contract with minimal price becomes more acceptable to the reinsurer and thus the low price region enlarges. Moreover, as θ2 increases, the reinsurer naturally prefers to enlarge the high price region for more profits. We also note that the slope of the line connecting xw is much steeper in panel (a) than it is in panel (b). This result indicates that a change in the peak price has less impact on the insurer when she has a small amount of cash reserve (note that the price in low price region remains unchanged). Otherwise, if xw decreases sharply in panel (b), insurer with an insufficient cash reserve will probably give up the purchase of reinsurance, which is not in the reinsurer’s interest. 7. Conclusion This study simultaneously considers the optimal reinsurance strategy for an insurer and the optimal reinsurance contract design for a reinsurer. The insurer determines her reinsurance

strategy to minimize her ruin probability, and the reinsurer determines the safety loading of the reinsurance contract to maximize her profits in the long run. We model the insurer and the reinsurer’s problems as a stochastic differential game for which we come up with a system of coupled HJB equation and a verification theorem for its equilibrium solution. Based on the HJB equations, we provide a procedure for solving the game problem. In particular, when both the insurer and the reinsurer adopt the mean-deviation principle or the expected value principle, we obtain explicit (for the former) and semiexplicit (for the latter) solutions for the problem. Our results show that, depending on the model parameters and the insurer’s cash reserve, the reinsurance safety loading is set to be either the highest or the lowest level in the equilibrium solution. There are several interesting directions to extend our model in the paper. First, one may consider other objective functions for the game problem in the paper, e.g., the expected utility maximization based on both the insurer and the reinsurer’s wealth at certain terminal time. Second, it would be interesting to study the game problem when both the insurer and the reinsurer are investing over the equity and bond markets, and to analyze how the price of the reinsurance contract is adjusted to the performance of their investment. Third, it is also interesting to consider the game problem under the classical risk model for the insurer’s cash reserve while a diffusion approximation model is adopted in the present paper.

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

225

Fig. 5. The impact of η, r , λ and ρ on the thresholds xw and x0 .

Appendix. Proof A.1. Proof of Lemma 2.1 Proof. We prove the lemma in two steps. (i) Let (αt , θt )t ≥0 ≡ (0, θ2 ). Then να (Z ) ≡ 0 and the insurer’s cash reserve (1) becomes

JR (x; α , θ ) = Ex

X 0 = x.

(A.1)

That is, Xt = (x − xS )ert + xS . When x ≥ xS , we have Xt ≥ 0, and hence τ0 = +∞ and JI (x; 0, θ2 ) = 0 ≤ JI (x; α, θ2 ) for all α ∈ V . (ii) On the other hand, since f (α, θ ) is increasing in θ , when x ≥ xS we have JR (x; 0, θ2 ) = Ex

≥ Ex

[∫

∞

0 [∫ τ0

e

−ρ t

s

∗

dXt = [rXt − (λHθ2 (Z ) − c)]dt

= r(Xt − xS )dt ,

θ ∗ (x) = θ1 , x ∈ [xs , xw ), and the insurer’s optimal reinsurance strategy is given by α1∗ (x) with α1∗ (x) = 0, x ∈ [xs , xw ) (see Corollary 3.1). For x ∈ (xs , xw ), similar to (i) of Appendix A.1 we have Xt = (x − xs )ert + xs for t < τw , where τw := inf{t ≥ 0 : Xt = xw } is the first time when Xt arrives xw , i.e. τw = 1r ln xxw−−xxs . Thus,

]

λf (0, θ2 )dt = λ

]

f (0, θ2 )

ρ

e−ρ t λf (0, θt )dt = JR (x; 0, θ ),

∀θ ∈ Θ ,

0

where τ0 is the ruin time of the insurer. Therefore, when x ≥ xS , from (4) we can see that (0, θ2 ) is the equilibrium strategy and V (x) = JI (x; 0, θ2 ) = 0, W (x) = JR (x; 0, θ2 ) = κ2 . □ A.2. Proof of Corollary 3.2 Proof. We prove by contradiction and assume that xw > xs . Since w ′ (x) > 1, x ∈ [0, xw ), according to Remark 3.2 we have

∗

= Ex

[∫

τ0

e

0 [∫ τw

+ Ex

0 [∫ τ0

τw

−ρ t

λf (α , θ )dt ∗

∗

e−ρ t λf (0, θ1 )dt

]

] ]

e−ρ t λf (α2∗ , θ2 )dt .

On the other hand, let x˜ := xs2+x ∈ (xs , xS ). Given θ ∈ (θ1 , θ2 ), since λHθ1 (Z ) − c λHθ2 (Z ) − c λHθ (Z ) − c xs := < < xS := , r r r there exists θ¯ ∈ (θ1 , θ2 ) such that

⎧ ⎨(α1∗ (x), θ1 ), (α˜ (x), θ˜ (x)) = (0, θ¯ ), ⎩(α ∗ (x), θ ), 2 2

λHθ¯ (Z )−c r

= x˜ . Consider strategy

x ∈ [0, xs ), x ∈ [xs , xw ), x ∈ [xw , xS ].

With strategy (α, ˜ θ˜ ), similar to (A.1) we have X˜ t = (x − x˜ )ert + x˜ for t < τ˜w , where X˜ is the controlled process and τ˜w is the time when −˜x X˜ arrives at xw for the first time, i.e. τ˜w = 1r ln xxw−˜ . It is clear that x

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Fig. 6. The impact of θ1 and θ2 on the thresholds xw and x0 .

τ˜w > τw . Since X and X˜ have the same trajectory after arriving xw , under the two strategies the insurer has[the same ruin probability ] ∫ τ˜ (i.e. JI (x; α, ˜ θ˜ ) = JI (x; α ∗ , θ ∗ )) and Ex τ˜w0 e−ρ t λf (α2∗ , θ2 )dt = ] [∫ τ Ex τ 0 e−ρ t λf (α2∗ , θ2 )dt , where τ˜0 is the ruin time with strategy w (α, ˜ θ˜ ). However, since f (α, θ ) is strictly increasing in θ and τ˜w > τw , ] [∫ τ˜w e−ρ t λf (0, θ¯ )dt JR (x; α, ˜ θ˜ ) = Ex 0 ] [∫ τ˜0 e−ρ t λf (α2∗ , θ2 )dt + Ex τ˜w ] [∫ τw e−ρ t λf (0, θ1 )dt > Ex 0 ] [∫ τ0 e−ρ t λf (α2∗ , θ2 )dt = JR (x; α ∗ , θ ∗ ), + Ex τw

which is a contradiction to the optimality of (α ∗ , θ ∗ ). This completes our proof. □

λθ12

where γ1 :=

w1 (x; y) := B11 (xs − x)t1 + B12 (xs − x)t2 +ℓ(xs − x) +κ1 ,

(A.3) where ℓ := − r +ρ ∈ (−1, 0), t1 > 1 and t2 < 0 are the roots

t 2 − ( γ1 + 1)rt − ρ = 0. B11 and B12 are determined by 1 w1 (0; y) = 0 and w1′ (y; y) = 1, i.e. of

{

r

γ1

t

t

B11 xs1 + B12 xs2 + ℓxs + κ1 = 0, −t1 B11 (xs − y)t1 −1 − t2 B12 (xs − y)t2 −1 − ℓ = 1.

It is clear that w1 (·; y) ∈ C2 (0, y), B1 and B2 are continuous in y, thus w1 (y; y) is also continuous in y. Moreover, direct calculation shows limy→0 w1 (y; y) = 0 and limy→xs w1 (y; y) = κ1 . (ii) Second, we determine the solution to ODE (17). (a) If y ∈ [x0 , xs ], using the boundary condition w ′ (y; y) = 1 and w (xS ; y) = κ2 , inspired by (16) we can see that the ODE (17) has solution

w2 (x; y) := B2 (xS − x)s + ℓ(xS − x) + κ2 , λθ22

where γ2 := r

s :=

−θi v (x) ∧ 1. σ0 v ′′ (x)

−

λ 2r

θi2

[v (x)] (θi − η)λσ0 ′ + (x − )v (x) = 0. v ′′ (x) r

x ∈ [y, xS ],

(A.4)

,

2r

√

( γr + 1)2 + 4ρ γr

+1+

2

2

2r

γ2

>

γ1

+1+

√

( γ + 1)2 + 4ρ γr r

1

1

2r

= t1 > 1

γ1

If αi∗ (x) < 1, the HJB equation (7) can be rewritten as ′

γ2

r

′

αi∗ (x) :=

x ∈ [0, y],

2r

A.3. Proof of Lemma 4.2 Proof. If v ′′ (x) > 0, Lα,θi v (x) is convex in α . Using the first order condition we have

. This ODE admits solution

2r

is the positive root of

2

(A.2)

B2 = −

1+ℓ s

r

γ2

s2 − ( γ1 + 1)rs − ρ = 0, and 2

(xS − y)1−s = −m(xS − y)1−s < 0,

(A.5)

Combining the above two equations we have (21). This completes our proof. □

where m =

A.4. Proof of Lemma 4.3

is continuous in y. Moreover, since w2 (x; y) = B2 s(s − 1)(xS − ∗ x)s−2 < 0, from Aα2 ,θ2 w2 (xs ; xs ) = 0 and w2′ (xs +; xs ) = 1, we have

Proof. (i) First, we determine the solution to (16). Substituting (21) into (16) leads to r(xs − x)w ′ (x) +

r

(xs − x)2 w ′′ (x) − ρw (x) γ1 + λθ1 σ0 − 2r(xs − x) = 0, x ∈ [0, y],

1+ℓ . s

It is clear that w2 (·; y) ∈ C2 (y, xS ) and

w2 (y; y) = (ℓ − m)(xS − y) + κ2

(A.6) ′′

r

w2 (xs ; xs ) =

γ2

(xS − xs )2 w2′′ (xs +; xs ) + λσ0 θ2 − r(xS − xs )

ρ λσ0 θ2 − r(xS − xs ) < = κ1 . ρ

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

(b) If y ∈ [0, x0 ), it is clear that α2∗ (x) = 1, ∀x ∈ [y, x0 ], and (17) can be rewritten as (rx + λσ0 η)w ′ (x) +

λ 2

σ02 w′′ (x) − ρw (x) = 0,

x ∈ (y, x0 ).

(A.7)

This ODE admits solution

w2 (x; y) = pϕ (x) + qψ (x),

(A.8)

where ϕ and ψ are the two classical solutions to (A.7) with initial conditions ϕ (0) = 0, ϕ ′ (0) = 1, ψ (0) = 1, ψ ′ (0) = 0,6 p and q are constants to be determined. On the other hand, we can see from (a) that, for x ∈ [x0 , xS ], the ODE (17) admits solution (A.4). Thus, from w2 (x0 −; y) = w2 (x0 +; y) and w2′ (x0 −; y) = w2′ (x0 +; y) we have pϕ (x0 ) + qψ (x0 ) = B2 (xS − x0 )s + ℓ(xS − x0 ) + κ2 , pϕ ′ (x0 ) + qψ ′ (x0 ) = −sB2 (xS − x0 )s−1 − ℓ.

{

(

xS − x0 ) s

( xS − x0 ) + q ψ (x0 ) + ψ ′ (x0 ) s

Combining this equation with pϕ ′ (y) + qψ ′ (y) = 1, we have (p, q)⊤ =

( ×

1

ψ ′ (y) ψ (x0 ) + ψ ′ (x0 ) xS −s x0 )

s

(xS − x0 )1−s .

(A.10)

It is clear that p, q and B2 are continuous in y, thus w2 (y; y) is continuous in y ∈ [0, x0 ). (iii) Finally, when y → x0 −, we have pϕ ′ (x0 ) + qψ ′ (x0 ) = 1, B2 = −m(xS − x0 )1−s and

w2 (x0 −; x0 −) = (ℓ − m)(xS − x0 ) + κ2 = w2 (x0 +; x0 +),

(A.11)

where the second equality is due to (A.6). Thus, w2 (y; y) is continuous at y = x0 and hence continuous in [0, xs ]. Specially, when y = 0, since ϕ ′ (0) = 1 and ψ ′ (0) = 0, we have p = 1 and w2 (0; 0) = q = q0 , where q0 :=

(1 − 1s )ℓ(xS − x0 ) + κ2 − (ϕ (x0 ) + ϕ ′ (x0 )

ψ (x0 ) + ψ

′ (x ) xS −x0 0 s

xS −x0 ) s

.

(A.14)

and w (xw ) = (ℓ − m)(xS − xw ) + κ2 . Applying the conditions w′ (xw −) = 1 and w(xw −) = w(xw +) yields (A.15)

t

t

t1 B11 yw1 + t2 B12 yw2 = −smyw , t t B11 yw1 + B12 yw2 = −myw + n,

(A.16)

n : = (ℓ − m)(xS − xs ) + κ2 − κ1 (A.9)

(1 − 1s )ℓ(xS − x0 ) + κ2

pϕ ′ (x0 ) + qψ ′ (x0 ) + ℓ

B2 = −m(xS − xw )1−s < 0,

where

)−1

and B2 = −

(A.13)

with

{

= (1 − )ℓ(xS − x0 ) + κ2 .

ϕ ′ (y) ϕ (x0 ) + ϕ ′ (x0 ) xS −s x0

w(x) ⎧ B (x − x)t1 + B12 (xs − x)t2 + ℓ(xs − x) + κ1 , ⎪ ⎨ 11 s x ∈ [0, xw ), = B2 (xS − x)s + ℓ(xS − x) + κ2 , ⎪ ⎩ x ∈ [xw , xS ],

Let yw := xs − xw ∈ (0, xs − x0 ). Then (A.15) can be rewritten as

s

1

(

(i) First, consider the case xw ∈ [x0 , xs ). (a) We construct an explicit solution to (16)–(18). In this case, from (A.3) and (A.4) a solution to (16) and (17) is given by

⎧ ⎨−B11 t1 (xs − xw )t1 −1 − B12 t2 (xs − xw )t2 −1 − ℓ = 1, B11 (xs − xw )t1 + B12 (xs − xw )t2 + ℓ(xs − xw ) + κ1 ⎩ = (ℓ − m)(xS − xw ) + κ2 .

It follows that p ϕ (x0 ) + ϕ ′ (x0 )

227

(A.12)

This completes our proof. □

1+ℓ

1 1 )λσ0 (θ2 − θ1 ) + λσ0 (θ2 − θ1 ) s r ρ [( 1)1 1] 1 + = λσ0 (θ2 − θ1 ) ℓ(1 − ) − s s r ρ [ r −ρ 1 r −ρ ] = λσ0 (θ2 − θ1 ) + < 0. ρ (r + ρ ) s r(r + ρ )

= (ℓ −

(A.17)

Solving (A.16) leads to

⎧ ⎨B11 = ⎩B12 =

−1 t1 − t2

[

]

−t

1 t1 − t2

[

]

−t

t2 (n − myw ) + smyw yw 1 , smyw + t1 (n − myw ) yw 2 .

Since s > t1 > 1, it is clear that B11 < 0. With boundary condition w(0) = 0, we see that yw satisfies 𭟋(yw ) = 0, where 𭟋 is defined as xs 𭟋(y) := (t2 n + (s − t2 )my)( )t1 (A.18) y xs − (t1 n + (s − t1 )my)( )t2 + (ℓxs + κ1 )(t2 − t1 ), y y ∈ [0, xs − x0 ]. Direct calculation shows limy→0+ 𭟋(y) = +∞ and

A.5. Proof of Theorem 4.4 We prove the theorem by considering q0 > 0 and q0 ≤ 0 respectively. A.5.1. Proof of the case q0 > 0 Lemma A.1. If q0 > 0, P1 admits solutions (v, w ) which are also the solutions to HJB equations (7) and (8). Proof. Since w2 (xs ; xs ) < w1 (xs ; xs ) = κ1 and w2 (0; 0) = q0 > 0 = w1 (0; 0), due to the continuity of w1 (·; ·) and w2 (·; ·), there exists xw ∈ (0, xs ) such that w1 (xw ; xw ) = w2 (xw ; xw ). 6 The pair ϕ and ψ forms a set of linearly independent solutions to (A.7); see Borodin and Salminen (2002).

𭟋′ (y) = −t1 t2 nxts1 y−t1 −1 + (1 − t1 )(s − t2 )mxts1 y−t1

+ t1 t2 nxts2 y−t2 −1 − (s − t1 )(1 − t2 )mxts2 y−t2 ) xs 1 ( xs t2 +1 = t1 t2 n ( ) − ( )t1 +1 xs

y

+ (1 − t1 )(s −

y

t2 )mxts1 y−t1

− (s − t1 )(1 − t2 )mxts2 y−t2 < 0,

i.e. 𭟋 is strictly decreasing. Therefore, if and only if 𭟋(xs − x0 ) ≤ 0 Eq. (A.18) has a unique solution yw ∈ (0, xs − x0 ], with which xw ∈ [x0 , xs ) and B11 , B12 are also uniquely determined. (b) Once xw is determined, we construct a solution to (13)– (15). For x ∈ (0, xw ), with θ ∗ (x) = θ1 , (13) admits solution

v (x) = A11 − A12 (xs − x)γ1 +1 ,

(A.19)

where A11 and A12 are constants. v (0) = 1 indicates that A11 = γ +1 1 + A12 xs 1 .

228

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

For x ∈ (xw , xS ), with boundary condition v (xS ) = 0, Eq. (14) admits solution

where m > 0 is defined in (A.5). Solving the system of equations leads to

v (x) = −A2 (xS − x)γ2 +1 ,

⎧ ⎨B11 =

(A.20)

where A2 is a constant to be determined. Combining (A.19) and (A.20), we have

v (x) =

γ +1

1 + A12 xs 1 − A12 (xs − x)γ1 +1 , −A2 (xS − x)γ2 +1 ,

{

x ∈ [0, xw ), x ∈ [xw , xS ].

γ1 + 1

γ1 + 1

1 + A12 xs − A12 (xs − xw ) = −A2 (xS − xw ) (γ1 + 1)A11 (xs − xw )γ1 = A2 (γ2 + 1)(xS − xw )γ2 .

γ2 +1

,

By solving the system of equations, A11 and A2 are uniquely characterized as

⎧ A = ⎪ ]−1 ⎨ 12[ γ +1 +1 < 0, − xs 1 − (xs − xw )γ1 +1 + (xs − xw )γ1 (xS − xw ) γγ1 + 2 1 ⎪ ⎩ (xs −xw )γ1 γ1 +1 A2 = (x −x )γ2 γ +1 A12 < 0. w S 2 (A.22) (c) Finally, we prove the optimality of (α , θ ). It is clear that (α ∗ , θ ∗ ) are admissible, by verification Theorem 3.1 we just need to show that (v, w ) are the solutions to (7) and (8). By construction, v, w ∈ C1 [0, xS ]∩C2 (0, xS )\{xw }. From (A.22) we can see that ∗ ∗ ∗ v is convex, and hence 0 = Lα ,θ v (x) ≤ Lα,θ v (x), ∀α ∈ [0, 1]. ′ ′ Since w is concave on (xw , xS ), w (x) < w (xw ) = 1 for all x ∈ (xw , xS ). Once we prove that w ′ (x) > 1 for x ∈ (0, xw ), then ∗ ∗ ∗ 0 = Aα ,θ w (x) ≥ Aα ,θ w (x), ∀θ ∈ [θ1 , θ2 ], and thus prove our argument. In fact, we claim that w ′′ (x) < 0, x ∈ (0, xw ). Assume other∗

∗

(α ∗ (xw ))2

wise. Since w ′′ (xw −) = w ′′ (xw +) (α2∗ (x

< 0 (see Remark 4.1), let xh := sup{x ∈ (0, xw ) : w (x) = 0}. It is clear that w′ (xh )∗ > 1, w′′ (xh ) = 0 and w′′′ (xh ) ≤ 0. Thus, taking derivative on Aα1 ,θ1 w (x) = 0 shows 1

′′

r

γ1

2

w ))

(xs − xh )2 w ′′′ (xh ) = (ρ + r)w ′ (xh ) − 2r ≤ 0,

(A.23)

2r and hence w ′ (xh ) ≤ r +ρ < 1, which is a contradiction. The concavity of w indicates that w ′ (x) > 1 for x ∈ (0, xw ). (ii) If 𭟋(xs − x0 ) > 0, we are left with xw ∈ (0, x0 ). (a) We construct an explicit solution to (16)–(18). In this case, from Appendix A.4 a solution to (16) and (17) is given by

w(x) ⎧ ⎪B11 (xs − x)t1 + B12 (xs − x)t2 + ℓ(xs − x) + κ1 , ⎪ ⎪ ⎪ x ∈ [0, xw ), ⎪ ⎨ F (x; xw ) := pϕ (x) + qψ (x), = x ∈ [xw , x0 ), ⎪ ⎪ ⎪ ⎪ B2 (xS − x)s + ℓ(xS − x) + κ2 , ⎪ ⎩ x ∈ [x0 , xS ],

(A.24)

(A.25)

− B11 t1 (xs − xw )t1 −1 − B12 t2 (xs − xw )t2 −1 − ℓ = 1, (A.26) t1 t2 B11 (xs − xw ) + B12 (xs − xw ) + ℓ(xs − xw ) + κ1 = F (xw ; xw ), (A.27) Let yw := xs − xw ∈ (xs − x0 , xs ). Then (A.26) and (A.27) become

{

t

t

B11 t1 yw1 + B12 t2 yw2 = −smyw , t t B11 yw1 + B12 yw2 = F (xw ; xw ) − ℓyw − κ1 ,

]

−t

1 t1 − t2

[

]

−t

smyw + t2 (F (xw ; xw ) − ℓyw − κ1 ) yw 1 , t1 (F (xw ; xw ) − ℓyw − κ1 ) + smyw yw 2 .

(A.28)

𭟋1 (x) := F (x; x),

x ∈ [0, x0 ].

(A.29)

Thus, (A.25) becomes 𭟋2 (yw ) = 0 with xw = xs − yw , where

[

(

𭟋2 (y) := smy + t2 𭟋1 (xs − y) − ℓy − κ1

)] x s

) t1 y [ ( ) ] x s − t1 𭟋1 (xs − y) − ℓy − κ1 + smy ( )t2 y (

+ (ℓxs + κ1 )(t2 − t1 ).

(A.30)

We proceed to show that (A.30) admits a solution yw ∈ (xs − x0 , xs ). From (A.11), 𭟋1 (x0 ) = (ℓ − m)(xS − x0 ) + κ2 and

𭟋1 (x0 ) − ℓ(xs − x0 ) − κ1 = n − m(xs − x0 ), where n is defined in (A.17). Thus,

[

]

xs

) t1 xs − x0 [ ] x s − t1 n + (s − t1 )m(xs − x0 ) ( )t2 xs − x0 + (ℓxs + κ1 )(t2 − t1 ) = 𭟋(xs − x0 ) > 0,

𭟋2 (xs − x0 ) = t2 n + (s − t2 )m(xs − x0 ) (

where 𭟋 is defined in (A.18). On the other hand,

𭟋2 (xs ) = [smxs + t2 (𭟋1 (0) − ℓxs − κ1 )]

− [t1 (𭟋1 (0) − ℓxs − κ1 ) + smxs ] + (ℓxs + κ1 )(t2 − t1 ) = (t2 − t1 )𭟋1 (0) = (t2 − t1 )q0 < 0. Therefore, we may determine yw and hence xw by solving the non-linear equation 𭟋2 (y) = 0. Substituting the value of xw into (A.9), (A.10) and (A.28), we obtain p, q, B2 , B11 , and B12 . (b) We construct a solution to (13)–(15). For x ∈ (xw , x0 ), with θ ∗ (x) = θ2 and α2∗ (x) = 1 Eq. (7) becomes (rx + λσ0 η)v ′ (x) +

1 2

λσ02 v ′′ (x) = 0,

which admits solution

v (x) = v (xw ) + A0

∫

x

−

e

r

λσ02

(y+

λσ0 η 2 r )

dy.

(A.31)

xw

Combining (A.19), (A.20) and (A.31), a solution to (13)–(15) is given by

where p, q and B2 are given by (A.9) and (A.10), B11 , B12 and xw are determined by B11 xts1 + B12 xts2 + ℓxs + κ1 = 0,

[

Let us denote (A.21)

The value-matching and smooth-pasting conditions at point xw imply

{

⎩B12 =

−1 t1 − t2

⎧ γ1 +1 − A12 (xs − x)γ1 +1 , ⎪ ⎨1 + A12 xs λσ0 η 2 r v (x) = v (x ) + A ∫ x e− λσ02 (y+ r ) dy, w 0 xw ⎪ ⎩ −A2 (xS − x)γ2 +1 ,

x ∈ [0, xw ), x ∈ [xw , x0 ), x ∈ [x0 , xS ],

(A.32)

where A0 , A12 , and A2 are determined by the value-matching and smooth-pasting conditions: v ′ (xw −) = v ′ (xw +), v ′ (x0 +) = v ′ (x0 −) and v (x0 +) = v (x0 −). That is,

⎧ − r 2 (xw +λσ0 ηr )2 ⎪ ⎪ ⎪ A12 = A0 (γ1 + 1)−1 (xs − xw )−γ1 e λσ0 := A0 Γ1 , ⎪ ⎪ ⎪ − r (x +λσ η )2 ⎪ ⎨A = A (γ + 1)−1 (x − x )−γ2 e λσ02 2 0 r := A Γ , 2 0[ 2 S 0 0 2 γ +1 ⎪ A0 = − Γ1 (xs 1 − (xs − xw )γ1 +1 ) ⎪ ⎪ ⎪ ⎪ ] ⎪ ∫ x − r 2 (y+λσ0 ηr )2 ⎪ ⎩ + xw0 e λσ0 dy + Γ2 (xS − x0 )γ2 +1 −1 . It is clear that A0 < 0 and hence A1 < 0 and A2 < 0.

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

(c) We show that w and v given by (A.24) and (A.32) are the solutions to (7)–(8). By construction, we already know that v and w are C1 on [0, xS ] and C2 on (0, xS ) \{xw∗ }.∗Since A1 , A2∗ and A3 are negative, v is convex and hence 0 = Lα ,θ v (x) ≤ Lα,θ v (x), ∀α ∈ ∗ ∗ ∗ [0, 1]. We just need to show that Aα ,θ w(x) ≥ Aα ,θ w (x), which is proved as following:

• We claim that w′ (x0 ) < 1. Otherwise, if w′ (x0 ) ≥ 1, from (A.10) it is clear that B2 < 0 and w ′′ (x) < 0, x ∈ (x0 , xS ). Since w is twice continuously differentiable at x0 (see Remark 4.1 (ii)), we also have w ′′ (x0 −) = w ′′ (x0 +) < 0 and w′ (x) > 1, x ∈ (x0 −ϵ, x0 ) with ϵ > 0 being small. Moreover, since w ′ (xw ) = 1, there exists x¯ ∈ (xw , x0 ) such that w2′ (x¯ ) > 1, w ′′ (x¯ ) = 0, and w ′′′ (x¯ ) ≤ 0. On the other hand, by taking derivative on (A.7) and letting x = x¯ , we have λ 0 > (r − ρ )w ′ (x¯ ) = − σ02 w ′′′ (x¯ ), (A.33)

2 which is a contradiction. • We show that w′ (x) < 1, x ∈ (xw , xS ). If B2 < 0, then w is strictly concave in (x0 , xS ) and, for x ∈ (x0 , xS ), w ′ (x) < w′ (x0 ) < 1; if B2 ≥ 0, w2 is convex and hence w′ (x) ≤ w′ (xS ) = −ℓ < 1 for x ∈ (x0 , xS ). On the other hand, by performing exactly the same procedure as above, we are able to show that w ′ (x) < 1, x ∈ (xw , x0 ). • We show that w′ (x) > 1, x ∈ (0, xw ). Since w′ (x) < 1, x ∈ (xw , xS ) and w ′ (xw ) = 1, we have w ′′ (xw +) ≤ 0. If w ′′ (xw +) = 0, similar to (A.33) we have w ′′′ (xw +) > 0, and hence w ′′ (x) > 0, w ′ (x) > 1, x ∈ (xw , xw + ϵ ) with ϵ > 0 being small, which contradicts argument (ii). Thus, we have w ′′ (xw +) < 0. According to Remark 4.1 (i), we have w′′ (xw −) < 0. By a similar proof as in (c) in part (i), we see that w ′′ (x) < 0, x ∈ (0, xw ) and hence w ′ (x) > 1. With the above observations, we have ∗ ∗ ∗ Aα ,θ w (x) ≥ Aα ,θ w (x),

A

w (x) = (rx + ηλσ0 )w (x) + =A

α ∗ ,θ

2

w(x),

λ 2

,

ϕ ′ (x0 )

1

ℓ(xS − x0 )(1 − ) + κ2 = ℓ

λσ0 θ2

s

= λσ0 θ2 (

1

ρ

−

1

ρ+r

r

1

(1 −

s

1

(1 −

2

)) > λσ0 θ2 (

) + κ2 s 1 1

ρ

−

r +ρ

) > 0.

It is clear that v is strictly decreasing and convex. Thus, we just need to show that w ′ (x) < 1 for x ∈ (0, xS ). Then, (v, w ) are the equilibrium value functions. First, we claim that w ′ (0) ≤ 1. Otherwise, suppose w ′ (0) > 1. Define process X¯ t = x +

t

∫

[

r X¯ s + (θ2 (α2∗ (X¯ s ) ∧ 1) − (θ2 − η))λσ0 ds

]

0

∫ +

t

√

(α2∗ (X¯ s ) ∧ 1) λσ0 dBs + ζt ,

t ∈ [0, τS ],

0

where τS := inf{t ≥ 0 : X¯ t = xS } is a stopping time, ζt is adapted to Ft , nondecreasing, continuous and satisfies ζ0 = 0. For w2 ∈ C2 (0, xS ) being the solution of (17) with xw = 0, by Itô’s formula (see e.g. Shreve et al., 1984), τS

∫ [ w2 (0; 0) = E0 κ2 e−ρτS +

λσ0 θ2 (1 − α2∗ (X¯ t ))dt −

∫

τS

]

e−ρ t dζt .

0

0

+

τS

∫

τS

∫

−ρτS

−w′ (0)

α2∗ ,θ2

λσ0 θ2 (1 − α2∗ (X¯ t ))dt

0

]

e−ρ t dζt .

0

Since w ′ (0) > 1, by comparing the above two equations we see that 0 = w (0) < w2 (0; 0) = q0 ≤ 0, which is a contradiction. Second, since w ′ (xS ) = −ℓ < 1, similar to (c) of part (i) in Appendix A.5.1, we have w ′ (x) < 1, x ∈ (0, xS ]. This completes our proof. □ A.6. Proof of Lemma 5.2

dy,

x ∈ [0, x0 ), x ∈ [x0 , xS ],

(A.34) Proof. By first order condition,

where A0 and A2 are determined by the value-matching and smooth-pasting conditions at x0 : v (x0 −) = v (x0 +), v ′ (x0 −) = v ′ (x0 +). i.e.,

⎧ ⎪ ⎪ ⎪ A2 ⎪ ⎪ ⎪ ⎪ ⎨ −1 = r < 0, λσ η λσ η (x0 + r0 )2 − r 2 (y+ r0 )2 2 ∫ x0 λσ0 λσ0 ⎪ γ γ + 1 e (γ2 +1)(xS −x0 ) 2 0 e dy+(xS −x0 ) 2 ⎪ ⎪ ⎪ ⎪ ⎪ λσ0 η 2 r ⎪ (x + r ) ⎩ 2 0 (γ2 + 1)(xS − x0 )γ2 < 0. A0 = A2 e λσ0 For the ODE (20), similar to our analyses in Appendix A.5.1, we can see that

w(x) =

(xS − x0 )−s < 0,

where p > 0 because

′′

Proof. From (A.20), (A.31), and boundary condition v (0) = 1, a solution to the ODE (19) is given by

{

ϕ (x0 ) xS −x0 ϕ ′ (x0 ) s

xS −x0 s

ϕ (x0 ) +

θ ∈ [θ1 , θ2 ].

λσ0 η 2 r )

1+

ℓ(xS − x0 )(1 − 1s ) + κ2

σ w (x) − ρw (x) 2 0

0

w(0) = E0 κ2 e

A.5.2. Proof of the case q0 ≤ 0

∫ x − λσr 2 (y+ v (x) = 1 + A0 0 e 0 −A2 (xS − x)γ2 +1 ,

p=

ℓ ϕϕ′(x(x0 )) + ℓ(xS − x0 ) + κ2

[

This completes our proof. □

{

B2 = −

w(x) = 0 On the other hand, since w ∈ C (0, xS ) satisfies A with boundary condition w (xS ) = κ2 . By Itô’s formula,

x ∈ [0, xw ) ∪ [x0 , xS ].

′

Using boundary condition w (0) = 0, we have q = 0. p and B2 are characterized by w (x0 −) = w (x0 +), w ′ (x0 −) = w2′ (x0 +). That is,

2

Moreover, when x ∈ (xw , x0 ), with α ∗ (x) ≡ 1 we have α ∗ ,θ

229

pϕ (x) + qψ (x), B2 (xS − x)s + ℓ(xS − x) + κ2 ,

x ∈ [0, x0 ), x ∈ [x0 , xS ],

(A.35)

d Lα,θi dα

λθi F¯ (αi∗ )v ′ (x) + λαi∗ F¯ (αi∗ )v ′′ (x) = 0 ⇒

v (x) |α=αi∗ = 0, we have

α∗ v ′ (x) =− i . ′′ v (x) θi

Combining the above equation with Lαi ,θi v (x) = 0, we have Pi (αi∗ ) = 0. We now proceed to show that (25) has unique solution. Direct calculation shows Pi (0) = 0, and ∗

lim Pi (α ) = lim

α→+∞

Pi′ (0) = −

[ −α

α→+∞

r

θi

[x −

λµ0 r

θi

(rx + ληµ0 ) +

(θi − η)],

1 2

] λσ02 = −∞,

Pi′′ (α ) = −λF¯Z (α ) < 0.

Thus, P1 (α ) = 0 admits a unique solution α1∗ (x) ∈ (0, +∞) if x < xs and a trivial solution α1∗ (x) ≡ 0 if x ≥ xs ; P2 (α ) = 0 admits a unique solution α2∗ (x) ∈ (0, +∞) if x < xS and a trivial

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solution α2∗ (x) ≡ 0 if x ≥ xS . Specially, when r = 0, one can see that Pi (α ) = 0 admits a unique solution α¯ i > 0. Finally, observe that Pi (α ) is strictly decreasing in r , x and η; thus, if αi∗ (x) > 0, it is strictly decreasing in r , x and η with αi∗ (x) ≤ α¯ i . Similarly, we see that αi∗ (x) is strictly increasing in θ and hence α1∗ (x) < α2∗ (x). □ A.7. Proof of Lemma 5.3

) ( − ρw2 (y; y) + λθ2 µ0 − µ(α2∗ (y)) = ry + ληµ0 − ρw2 (y; y) ⇒ w2 (y; y) > λθ µ

ry + ληµ0

ρ

.

ry+ληµ

0 = ρr (xS − y) < xS − y, Thus, w2 (xS ; y) − w2 (y; y) < 2ρ 0 − ρ which is contradiction. On the other hand, assume w2′′ (y+; y) = 0. With w2′ (y+; y) = 1, direct calculation shows

σ 2 (α2∗ (y))

Proof. We prove the arguments of this lemma in three steps. (i) First, we show that ODEs (16) and (17) admit solutions. To this end, given y ∈ (0, xs ], consider two stochastic processes {Xˆ t }t ≥0 and {Xˇ t }t ≥0 . The former has initial state Xˆ 0 = xˆ ∈ (0, y), and is absorbed at 0 and reflected at y; whereas the latter has initial state Xˇ 0 = xˇ ∈ (y, xS ), and is reflected at y and absorbed at xS . The dynamics of Xˆ and Xˇ are defined by

0=λ

⎧ [ ] ⎪ dXˆ t = r Xˆ t + λ[θ1 µ(α1∗ (Xˆ t )) + (η − θ1 )µ0 ] dt ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ + λσ (α1∗ (Xˆ t ))dBt − dζˆty , ⎨ [ ] ˇ t = r Xˇ t + λ[θ2 µ(α ∗ (Xˇ t )) + (η − θ2 )µ0 ] dt d X 2 ⎪ ⎪ ⎪ √ ⎪ ∗ ˇ ⎪ ˇy ⎪ + ( X ))dB λσ ( α t t + dζt , ⎪ 2 ⎪ ⎩ Xˆ 0 = xˆ ∈ [0, y], Xˇ 0 = xˇ ∈ [y, xS ].

Thus, we have w2′′ (x; y) > 0, x ∈ (y, y + ϵ ) with ϵ > 0 being small. By performing exactly the same procedure as above, we see that contradiction exists, and thus the assumption w2′′ (y+; y) = 0 does not hold. Argument (ii) is proved. ∗ (iii) From w2′ (y+; y) = 1 and Aα2 ,θ2 w2 (y; y) = 0, we have

y

(A.36)

1

0

where τˆ0 := inf{t ≥ 0 : Xˆ t = 0} and τˇS := inf{t ≥ 0 : Xˇ t = xS } are times. According to Lemma 5.2, the volatilities √ F -stopping √ λσ (α1∗ (x)) and λσ (α2∗ (x)) are bounded away from 0 in (0, y) and (y, xS ). Moreover, since µ′ (αi∗ (·)) is bounded, w1 and w2 are twice continuously differentiable in (0, y) and (y, xS ) and satisfy ODEs (16) and (17) with xw = y (see Øksendal and Sulem, 2005). Also, since w1 (·; y) and w2 (·; y) are continuously dependent on y, we see that w1 (y; y) and w2 (y; y) are continuous in [0, xs ]. (ii) We prove w2′′ (y+; y) < 0, y ∈ (0, xs ), by contradiction. Assume w2′′ (y+; y) > 0. Since w2′ (y+; y) = 1, either of the following two cases hold: (a) w2′ (x; y) has a local maximum at xh ∈ (y, xS ) such that w2′ (xh ; y) > 1, w2′′ (xh ; y) = 0 and w2′′′ (xh ; y) ≤ 0; (b) w2′ (x; y) has a global maximum at xS such that w2′ (xS −; y) > 1 and w2′′ (xS −; y) ≥ 0. For case (a), by evaluating dα2∗ (xh )

∗ d Aα2 ,θ2 dx

w2 (x; y) |x=xh = 0 and the fact

< 0 (see Lemma 5.2), we have ) dα ∗ (xh ) 0 ≤ r + λθ2 µ′ (α2∗ (xh )) 2 − ρ w2′ (xh ; y) dx

− λθ2 µ (α2 (xh )) ∗

dα2∗ (xh )

dx

dx

= (r − ρ )w2 (xh ; y) + (w2′ (xh ; y) − 1)λθ2 µ′ (α2∗ (xh )) ′

dα2∗ (xh ) dx

< 0,

which is a contradiction. For case (b), we have w2′ (x; y) > 1, x ∈ (y, xS ]. However, 0 = Aα2 ,θ2 w2 (y; y) ∗

[

<

] > ry + λ[θ2 µ(α2∗ (y)) + (η − θ2 )µ0 ]

ry + ληµ0 + λ2 σ 2 (α2∗ (y))w2′′ (y+; y) ry + ληµ0

ρ

ρ = κ1 .

Specially, by letting y = xs we have w2 (xs ; xs ) < Our proof is completed. □

rxs +ληµ0

ρ

= κ1 .

Proof of Theorem 5.4. (i) For the case w2 (0; 0) > 0, we prove this theorem in three steps: (a) Characterizing xw and showing that (v, w ) are the solutions to P1; (b) Proving that w ′ (x) ≤ 1, x ∈ (xw , xS ); (c) Proving that w ′ (x) > 1, x ∈ (0, xw ). Then, since v given in (27) is strictly convex and that (α ∗ , θ ∗ ) are admissible, similar to the proof of Theorem 4.4, we see that V = v, W = w , and (θ ∗ , α ∗ ) are the equilibrium strategies. (a) Since w1 (0; 0) = 0 < w2 (0; 0) and w1 (xs ; xs ) = κ1 > w2 (xs ; xs ), by continuity there exists xw ∈ (0, xs ) such that w1 (xw ; xw ) = w2 (xw ; xw ). From Lemma 5.3, we see that w is the solution to (16)–(18). Also, it is clear that v satisfies (13)–(15). (b) Direct calculation shows

σ 2 (αi∗ (x))

[ w′′′ (x) + λαi∗ (x)µ′ (αi∗ (x))(αi∗ (x))′ 2 ] + rx + λ[θi µ(αi∗ (x)) + (η − θi )µ0 ] w′′ (x) ( ) + r + λθi µ′ (αi∗ (x))(αi∗ )′ (x) − ρ w′ (x)

λ

− λθi µ′ (αi∗ (x))(αi∗ )′ (x) = 0,

(

′

2

A.8. Proof of Theorem 5.4

⎪ w2 (xˇ ;[y) := ⎪ ⎪ ] ⎪ ∫ ∫ ) ( ⎪ ⎩ Exˇ e−ρ τˇS κ2 + τˇS e−ρ t λθ2 µ0 − µ(α ∗ (Xˇ t )) dt − τˇS e−ρ t dζˇ y , t 2 0 0

that

− λθ2 µ′ (α2∗ (y))(α2∗ )′ (y) σ 2 (α2∗ (y)) ′′′ =λ w2 (y+; y) + r − ρ ⇒ w2′′′ (y+; y) > 0.

y

⎧ ⎪ := ⎪ ⎪w1 (xˆ ;[y) ] ⎪ ∫ ∫ ) ( ⎪ ⎨ Exˆ τˆ0 e−ρ t λθ1 µ0 − µ(α ∗ (Xˆ t )) dt + τˆ0 e−ρ t dζˆty ,

w2′′′ (y+; y)

) + r + λθ2 µ′ (α2∗ (y))(α2∗ )′ (y) − ρ w2′ (y+; y)

w2 (y; y) =

Here, {ζˆt }t ≥0 and {ζˇt }t ≥0 are F -adapted, nondecreasing, lefty y continuous processes with ζˆ0 = ζˇ0 = 0. Define

0

2

(

i = 1, 2.

(A.37)

We prove by contradiction and assume that supx∈(xw ,xS ] w ′ (x) > 1. Assume there exists local maximum point x¯¯ ∈ (xw , xS ) such that ′′ ¯¯ w′ (x¯¯ ) > 1, w (x) = 0 and w ′′′ (x¯¯ ) ≤ 0. Combining (A.37) and the dα2∗ (x) fact that dx |x=x¯¯ < 0 (see Lemma 5.2), we have 0≥λ

σ 2 (α2∗ (x)) 2

w′′′ (x¯¯ )

= λθ2 µ′ (α2∗ (x¯¯ ))

d dx

α2∗ (x¯¯ )(1 − w′ (x¯¯ )) + (ρ − r)w′ (x¯¯ )

≥ (ρ − r)w′ (x¯¯ ) > 0,

S. Chen, Y. Liu and C. Weng / Insurance: Mathematics and Economics 86 (2019) 216–231

which is a contradiction. Otherwise, assume w ′ (x) has global maximum point at xS such that w ′ (xS ) > 1. Let x˜ := sup{x ∈ (xw , xS ) : w ′ (x) = 1}. Then, w ′ (x˜ ) = 1, w ′′ (x˜ ) ≥ 0, and

[

]

0 = Aα2 ,θ2 w (x˜ ) ≥ r x˜ + λ[θ2 µ(α2∗ (x˜ )) + (η − θ2 )µ0 ] ∗

− ρw (x˜ ) + λ(µ0 − µ(α2∗ (x˜ )))θ2 = r x˜ + ληµ0 − ρw (x˜ ) = r(x˜ − xS ) + λµ0 θ2 − ρw (x˜ ) = r(x˜ − xS ) + ρw (xS ) − ρw (x˜ ) w(xS ) − w(x˜ ) r ⇒ ≤ < 1, xS − x˜ ρ which contradicts the fact that w ′ (x) > 1, x ∈ (x˜ , xS ). (c) Since w ′ (xw ) = 1 and w ′′ (xw +) < 0 (see Lemma 5.3), we have w ′′ (xw −) < 0 and hence w ′ (x) > 1, x ∈ (xw − ϵ, xw ) with ϵ > 0 being small. We claim that w is strictly concave on (0, xw ) and thus our argument holds. Otherwise, let xˆˆ := sup{x ∈ (0, xw ) : w′′ (x) = 0}. Then w′′′ (xˆˆ ) ≤ 0, w′′ (xˆˆ ) = 0 and w′ (xˆˆ ) > 1. Thus, 0≥λ

σ 2 (α1∗ (xˆˆ )) 2

w′′′ (xˆˆ )

d = λθ1 µ′ (α1∗ (xˆˆ )) α1∗ (xˆˆ )(1 − w′ (xˆˆ )) + (ρ − r)w′ (xˆˆ ) > 0,

dx which is a contradiction. (ii) For the case w2 (0; 0) ≤ 0, since v given in (28) is strictly convex, we just need to show that w ′ (x) < 1, x ∈ (0, xS ). First, we show that w ′ (0) ≤ 1. Otherwise, if w ′ (0) > 1, according to Shreve et al. (1984), we have

∫ τS [ w(x) = Ex κ2 e−ρτS + e−ρ t λ(µ0 − µ(α2∗ (Xˇ t )))θ2 dt 0 ∫ τS ] ′ e−ρ t dζˇt − w (0) 0 ∫ τS [ −ρτS < Ex κ2 e + e−ρ t λ(µ0 − µ(α2∗ (Xˇ t )))θ2 dt 0 ∫ τS ] − e−ρ t dζˇt = w2 (x; 0), 0

where Xˇ is defined in (A.36) with y = 0. Thus, 0 = w (0) < w2 (0; 0) ≤ 0, which is a contradiction. Similar to the proof in step (b) of case (i), we can also see that

w′ (x) < 1, x ∈ (0, xS ). This completes our proof. □ References

Bai, L., Cai, J., Zhou, M., 2013. Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting. Insurance Math. Econom. 53, 664–670. Bai, L., Hunting, M., Paulsen, J., 2012. Optimal dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints. Finance Stoch. 16, 477–511. Bi, J., Liang, Z., Xu, F., 2016. Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence. Insurance Math. Econom. 70, 245–258. Borodin, A.N., Salminen, P., 2002. Stochastic processes in general. In: Handbook of Brownian Motion-Facts and Formulae. Birkhäuser, Basel.

231

Chen, S., Li, Z., Zeng, Y., 2018. Optimal dividend strategy for a general diffusion process with time-inconsistent preferences and ruin penalty. SIAM J. Financial Math. 9, 274–314. Cheng, G., Wang, R., Fan, K., 2016. Optimal risk and dividend control of an insurance company with exponential premium principle and liquidation value. Stochastics 88, 904–926. Chiu, M.C., Wong, H.Y., 2014. Mean–variance asset–liability management with asset correlation risk and insurance liabilities. Insurance Math. Econom. 59, 300–310. Deng, C., Zeng, X., Zhu, H., 2018. Non-zero-sum stochastic differential reinsurance and investment games with default risk. European J. Oper. Res. 264, 1144–1158. Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, Princeton. Evstigneev, I.V., Schürger, K., Taksar, M.I., 2004. On the fundamental theorem of asset pricing: Random constraints and bang–bang no-arbitrage criteria. Math. Finance 14, 201–221. Golubin, A.Y., 2008. Optimal insurance and reinsurance policies in the risk process. Astin Bull. 38, 383–397. Grandell, J., 1991. Aspects of Risk Theory. Springer, New York. Grenadier, S.R., Wang, N., 2007. Investment under uncertainty and timeinconsistent preferences. J. Financ. Econ. 84, 2–39. Hipp, C., Vogt, M., 2003. Optimal dynamic XL reinsurance. Astin Bull. 33, 193–207. Højgaard, B., Taksar, M., 2001. Optimal risk control for a large corporation in the presence of returns on investments. Finance Stoch. 5, 527–547. Hu, D., Chen, S., Wang, H., 2018a. Robust reinsurance contracts in continuous time. Scand. Actuar. J. 2018, 1–22. Hu, D., Chen, S., Wang, H., 2018b. Robust reinsurance contracts with uncertainty about jump risk. European J. Oper. Res. 266, 1175–1188, 2018. Jang, B.G., Kim, K.T., 2015. Optimal reinsurance and asset allocation under regime switching. J. Bank. Financ. 56, 37–47. Krylov, N.V., 2008. Controlled Diffusion Processes. Springer, New York. Li, D., Rong, X., Zhao, H., 2014. Optimal reinsurance-investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model. J. Comput. Appl. Math. 255, 671–683. Li, D., Rong, X., Zhao, H., 2016. The optimal investment problem for an insurer and a reinsurer under the constant elasticity of variance model. IMA J. Manag. Math. 27, 255–280. Liang, Z., Yuen, K.C., 2016. Optimal dynamic reinsurance with dependent risks: Variance premium principle. Scand. Actuar. J. 2016, 18–36. Luo, S., Taksar, M., Tsoi, A., 2008. On reinsurance and investment for large insurance portfolios. Insurance Math. Econom. 42, 434–444. Øksendal, B.K., Sulem, A., 2005. Applied Stochastic Control of Jump Diffusions. Springer, Berlin. Schmidli, H., 2001. Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuar. J. 2001, 55–68. Schmidli, H., 2008. Stochastic Control in Insurance. Springer, New York. Shreve, S.E., Lehoczky, J.P., Gaver, D.P., 1984. Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optim. 22, 55–75. Siu, C., Yam, S.C.P., Yang, H., Zhao, H., 2016. A class of nonzero-sum investment and reinsurance games subject to systematic risks. Scand. Actuar. J. 2016, 1–38. Taksar, M., Zeng, X., 2011. Optimal non-proportional reinsurance control and stochastic differential games. Insurance Math. Econom. 48, 64–71. Zeng, X., Luo, S., 2013. Stochastic Pareto-optimal reinsurance policies. Insurance Math. Econom. 53, 671–677. Zhang, X., Meng, H., Zeng, Y., 2016. Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling. Insurance Math. Econom. 67, 125–132. Zhao, H., Weng, C., Shen, Y., Zeng, Y., 2017. Time-consistent investmentreinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models. Sci. China Math. 60, 317–344.