Stochastic differential reinsurance games with capital injections

Insurance: Mathematics and Economics 88 (2019) 7–18

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

Stochastic differential reinsurance games with capital injections Nan Zhang a , Zhuo Jin b , Linyi Qian a , Kun Fan a ,

∗

a

Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, 3663 North Zhongshan Road, Shanghai, 200062, China b Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

article

info

Article history: Received September 2018 Received in revised form March 2019 Accepted 16 May 2019 Available online 27 May 2019 Keywords: Stochastic differential game Reflected process Nash equilibrium Quasi-variational inequality Singular control

a b s t r a c t This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Reinsurance is the insurance among two insurance companies of contractual liabilities incurred under contracts of reinsurance. It is an effective risk management tool that could indemnify the direct insurer for part of its underwriting losses and improve its solvency. The design of optimal reinsurance is a risk sharing problem between an insurer and a reinsurer. Since Borch (1960) investigates the optimal proportional and stop-loss retentions that maximize the product of expected utility functions of the two parties’ wealth at a specific time in the future, the optimal reinsurance design has boasted an extensive literature and remains as an active subject for several decades. In general, optimal reinsurance problems are analyzed under both static models and the dynamic ones. Regarding the dynamic models, Højgaard and Taksar (1998) find a closed-form expression for a proportional reinsurance policy by solving the Hamilton–Jacobi–Bellman equation derived by means of standard dynamic programming techniques. Thereafter, stochastic control theory becomes a popular tool in solving optimal reinsurance problems. See Asmussen et al. (2000), Choulli et al. (2003), Azcue and Muler (2005), Zhang et al. (2007), Bai and Guo (2008), Wei et al. (2010), Liang and Guo (2011), Liang et al. (2012), Li et al. (2015), Meng et al. (2016), Zhang et al. (2016), and references therein. The aforementioned literature investigates the optimal reinsurance strategies of a single insurer. However, the insurance ∗ Corresponding author. E-mail addresses: [email protected] (N. Zhang), [email protected] (Z. Jin), [email protected] (L. Qian), [email protected] (K. Fan). https://doi.org/10.1016/j.insmatheco.2019.05.002 0167-6687/© 2019 Elsevier B.V. All rights reserved.

industry is a complex unit involving plenty of insurance institutions that are inevitably competing with others for higher profits and market shares. To reflect the competition and cope with the strategic interactions among several insurance companies, researchers proceed to investigate reinsurance strategies under the framework of stochastic differential games, which may be used to analyze oligopoly, i.e., a market with so few sellers that they must take account of their actions on the market price or each other. Most insurance markets can be considered as an oligopoly as there are relatively small numbers of firms who collectively influence the price and supply of insurance policies. To simplify the modeling, researchers tend to constrain the market to be duopoly, that is the market of two participants, insurer A and insurer B. The earliest work about game theory from the mathematical viewpoint can be traced back to Neumann and Morgenstern (1953). Since then game theory has drawn great attention of both practitioners and academia. In the contexts of insurance and finance, there emerge some works that solve the optimal reinsurance problem under the stochastic differential game framework. Zeng (2010) develops a zero-sum differential game between two insurance companies and derives the Nash equilibrium for the dynamic proportional reinsurance. Bensoussan et al. (2014) formulate a non-zero-sum stochastic differential game between two insurers by applying the concept of relative performance and obtain explicit solutions for optimal reinsurance and investment strategies under some special cases. Meng et al. (2015) consider a stochastic differential reinsurance game between two insurers with quadratic risk control processes, and derive an explicit Nash equilibrium strategy by solving a pair of

8

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

HJB equations. Wang et al. (2019) formulate a class of non-zerosum reinsurance and investment games between two AAIs who faced default risk and explicit expressions for Nash equilibrium investment and reinsurance strategies are established. In this paper, we consider a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. We assume that both insurers have the freedom to purchase proportional reinsurance contracts from reinsurance markets. The reinsurance premium is determined by the generalized variance premium principle. To guarantee the solvency of an insurance institution, regulators require the insurance company to hold sufficient risk capital to absorb contingent claim payments and unexpected losses. In the case that an insurer is not adequately capitalized, capital injection strategies might be conducted to alleviate the negative impacts of insolvency. Therefore, we also incorporate capital injections to prevent insolvency and ruin in our reinsurance game formulation. In general, the behavior of capital injections is formulated in two ways, depending on whether fixed cost is generated by the conduct of each capital injection. When the fixed transaction cost exists, times and amounts of capital injections are formulated by impulse controls. For example, in a dual risk model, Yao et al. (2011) consider the dividend payments and capital injections control problem under the objective of maximizing the expected present value of the difference between dividends income and the cost of capital injections. In the regime-switching jump diffusion model, Jin et al. (2013) study the optimal dividend payment and investment strategies with capital injections by focusing on numerical methods. Aïd et al. (2018) investigate the non-zero-sum stochastic differential games between two competing institutions with one pair of impulse controls. In the case of proportional penalties for capital injections, academia choose to describe the cumulative amount of capital injections by a non-decreasing stochastic process. Then, capital injections make the institutions’ surplus processes reflected at a lower bound, and the optimal capital injections strategies can be described by local times or singular controls. Detailed techniques applied in such problem are similar with analyzing the dividend strategies, which are firstly solved by Shreve et al. (1984) in a diffusion model. The reader may also refer to Asmussen and Taksar (1997), Choulli et al. (2003), Yin and Wen (2013) and references therein. In the context of capital injections, Eisenberg and Schmidli (2009) solve the problem of minimizing the expected discounted capital injections. The optimal strategy for capital injections, that are penalized proportionally, is described by a local time that makes surplus process reflected in zero. Yin and Yuen (2015) analyze the optimal control problem in the presence of proportional transaction costs for the dual model with random return on investment under a uniform framework. Lindensjö and Lindskog (2019) investigate an insurer’s dividends and capital injections strategies under a singular stochastic control formulation. Under the assumption of existing proportional cost in raising equity capitals and certain parameter constraints, it is optimal to pay dividends and inject capital to reflect the surplus process at an upper and a lower barrier, respectively. Further investigations on capital injection optimizations with proportional penalty include Løkka and Zervos (2008), Zhou and Yuen (2012), Li and Yin (2017), etc. Like these existing literature, we depict the two insurance companies’ accumulative amount of injected capitals by two stochastic processes that are non-negative, non-decreasing and right-continuous with left limits. We assume that the objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor’s. The penalties and gains of

interventions to both insurers are proportional to the amount of corresponding capital injections. Using the dynamic programming principle, the value functions are a pair of solutions to the quasi-variational inequalities. We obtain the Nash equilibrium reinsurance strategies by standard techniques in stochastic control theory. Regarding strategies for capital injections, points of reflection are determined by first derivative conditions of value functions at places where corresponding insurance companies intervene. Expressions for value functions in the region that insurers’ competitors inject capitals can be obtained from the intuitional analysis. Our results indicate that both insurers’ capital injection strategies in equilibrium are in threshold type and their corresponding value functions are piecewise functions. Specifically, insurer 1 needs to inject capitals as long as the surplus difference falls below a threshold (denoted by x), and the amount of his capital injections elevates the surplus difference back to the target state that equals this threshold. The capital injections of insurer 2 are triggered once the surplus difference is higher than another threshold (denoted by x), which is also the level of surplus difference after insurer 2’s intervention. Namely, capital injections of insurer 1 and insurer 2 make the surplus difference process reflect at x and x, respectively. In addition, the equilibrium value function of both insurers is convex when neither insurer injects capitals, that is, the surplus difference takes values between the two thresholds. In the remaining region, that is when the surplus difference is lower than insurer 1’s capital injections threshold x, or higher than insurer 2’s threshold for capital injections x, both equilibrium value functions are linear in the surplus difference between the two insurance companies. In our formulation, the major contribution is considering reinsurance strategies (regular controls) together with capital injections (singular controls) simultaneously in the game formulation. Comparing to the work in Aïd et al. (2018), capital injections in our model is in singular type instead of impulse controls, and we have one more pair of regular controls in the dynamic processes. The reason we cannot adopt impulse controls in our case of proportional penalties/gains of capital injection is that capitals could be continuously injected in the absence of fixed costs. In addition, the extra pair of reinsurance regular controls adds significant difficulties to find the smooth pasting conditions on the boundaries of regions. By solving the optimal controls in the continuation region and comparing the boundary conditions with other regions, we obtain the explicit expressions of optimal controls and value functions in presence of proportional transaction cost. Verification theorem is provided accordingly. The remainder of this paper is organized as follows. In Section 2, we provide a general formulation of the proportional reinsurance and capital injections game problem between two insurance companies. Section 3 derives the Nash equilibrium reinsurance and capital injections strategies and the corresponding value functions under the generalized formulation. The verification theorem is proved in Section 4. Section 5 presents the results under several simplified situations. To illustrate the results, several numerical examples are presented in Section 6. Finally, some general remarks are provided in Section 7. 2. Formulation Following the classical Cramér–Lundberg process, we assume that ˆ Xk (t), k ∈ {1, 2}, the surplus of insurance company k without reinsurance satisfies dˆ Xk (t) = ck dt − dYk (t), t ≥ 0,

(2.1)

where ˆ Xk (0) := xk is the initial surplus, ck is the rate of premium, ∑ˆ N(t) i and Yk (t) = i=1 yk is a compound Poisson process with claim

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

size yik , with {yik : i ≥ 1} being a sequence of positive, independent and identically distributed random variables. For k =∑ 1, 2, let ηn be the inter-arrival time of the nth claim, n then νn = j=1 ηj represents the time of the nth claim. In general, we consider a Poisson measure in lieu of the traditionally used Poisson process. Suppose Θ ⊂ R+ is a compact set, N(t , H) = number of claims on [0, t ] with claim size taking values in H ∈ Θ

(2.2)

counts the number of claims up to time t, which is a Poisson counting process. For k = 1, 2, Yk (t) is jump processes representing claims for each company with arrival rate λk . Let yk be the magnitude of the claim sizes, and follows the distribution Πk (·). Then the Poisson measure N(·) has intensity λk dt ×Πk (dyk ), where Πk (dyk ) = f (yk )dyk . Let mk = E[yk ], and lk = E[y2k ]. 2.1. Reinsurance arrangement Let uk (t) be an exogenous retention level, which is a control chosen by the insurance company representing the reinsurance policy. Let q(yk , uk ) be the fraction of each claim paid by the primary insurance company. Let the reinsurance premium rate be g(uk ). Considering the reinsurance strategies, the surplus process u of the insurance company k, denoted by ˜ Xk k (t), follows u u d˜ Xk k (t) = (ck − g(uk ))dt − dYk k (t), uk ˜ Xk (0) = x˜ k ,

{

(2.3)

where u

Yk k (t) =

∫ t∫ 0

q(yk , uk )N(dt , dyk ) R+

is the aggregate claim amount paid by the primary insurance company. Similar to the work in Bensoussan et al. (2014), by using the techniques of diffusion approximation applied to the Cramér– Lundberg model, the surplus process of the insurance company k satisfies

{

u

u

d˜ Xk k (t) = (ck − g(uk ) − λk mk k )dt +

√

u ˜ Xk k (0) = x˜ k , u mk k

u

˜

= x˜ k .

We are now working on a filtered probability space (Ω , F , {Ft }t ≥0 , P), where Ft is the σ -algebra generated by {M1 (s), M2 (s), N(s) : 0 ≤ s ≤ t }. Definition 2.1. A strategy π = (π1 , π2 ), where πk = (uk , Zk ), k = 1, 2, is said to be admissible if i. the retained proportional reinsurance process {uk (t)} is {Ft }t ≥0 -adapted and uk (t) ∈ [0, 1] for any t ≥ 0; ii. the capital injections process Zk (t) is a non-decreasing, right continuous, {Ft }t ≥0 -adapted cádlág process. We denote the set of all admissible controls of insurer k (k = 1, 2) by Uk . 2.3. Objectives of insurance companies In this work, we model the competition of two insurance companies with reinsurance and capital injections schemes. The relative performance of the two companies is measured by the difference of their surpluses X1 (t) − X2 (t). Thus, the competition between the two companies formulates a game with two players, each of which can adjust his reinsurance and capital injection strategies based on his competitor’s scheme. Let the surplus difference performance for the two insurance companies is X (t) := X1 (t) − X2 (t). Hence, X (t) is governed by the following dynamics t

∫

[c1 − c2 − g(u1 ) + g(u2 ) − λ1 u1 m1 + λ2 u2 m2 ] ds

X (t) = x + 0

∫

t

+

u1

√

λ1 l1 dM1 (s)

u2

√

λ2 l2 dM2 (s) + Z1 (t) − Z2 (t),

∫0 t 0

X (0) = x = x1 − x2 . u lk k

d˜ Xk k (t) = (ck − g(uk ) − λk uk mk )dt + uk

u Xk k (0)

⎧ ∫ t ⎪ ⎪ X (t) = x + (ck − g(uk ) − λk uk mk )ds ⎪ k k ⎪ ⎨ ∫ t 0√ + uk λk lk dMk (s) + Zk (t), ⎪ ⎪ ⎪ 0 ⎪ ⎩ Xk (0) = xk .

(2.4)

where = E[q(yk , uk )], = E[q2 (yk , uk )], and Mk (t) is a standard Brownian motion, where k = 1, 2. Let ρ be the correlation coefficient between M1 (t) and M2 (t). In the case of u u u proportional reinsurance, Yk k = uk Yk , mk k = uk mk , lk k = u2k lk . Thus, following (2.4), the surplus is given by

{

injections and reinsurance is driven by the following stochastic differential equation

−

u

λk lk k dMk (t),

9

√

λk lk dMk (t),

(2.5)

2.2. Capital injections We define the process of capital injections as Eisenberg and Schmidli (2009). Let the non-decreasing process Zk (t) describe the accumulated amount of capitals injected to insurer k (k = 1, 2) up to time t, such that Zk (t) is a right-continuous stochastic process with left limits, and Zk (0−) = 0. Then, capital injection occurs at points where Zk has jumps, that is where Zk (s) ̸ = Zk (s−). In the meantime, the amount injected equals the size of the corresponding jump Zk (s) − Zk (s−). The evolution of surplus process, subject to capital injections, follows a one-dimensional process on an unbounded domain G′ = (−∞, ∞). For insurance company k, k ∈ {1, 2}, the surplus process Xk (t) considers capital

(2.6)

Let Ak and Bk be the collection of all admissible reinsurance and capital injections strategies, respectively. Let πk = (uk , Zk ) and πk ∈ Ak × Bk = Uk . Moreover, we assume that the effects of capital injections on the insurer itself and on his competitor are both proportional. Specifically, when insurer k has the capital injection of ζ , he suffers from the intervention penalties of severity βζ , β > 0, and simultaneously, the corresponding gain for his competitor is ˜ β ζ , where β and ˜ β are fixed non-negative constants. If we have β < ˜ β , then, for a suitable capital injection ζ , the two insurance companies could realize a mutual gain by an instantaneous double intervention. Therefore, we assume that

β >˜ β≥0 to avoid arbitrage. We can regard βζ as the proportional transaction costs. Denote by r > 0 the discount factor. For an arbitrary pair of admissible control (π1 , π2 ), we define the objective function of insurer 1 as J1 (x, π1 ) := Ex

∞

[∫

e−rt U1 (X (t) − κ1 ) dt 0

−β

∞

∫

−rt

e 0

dZ1 (t) + ˜ β

∞

∫

e 0

−rt

]

dZ2 (t) ,

(2.7)

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N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

and the objective function of insurer 2 follows J2 (x, π2 ) := Ex

∞

[∫

e−rt U2 (κ2 − X (t)) dt

0

−β

∞

∫

e

−rt

dZ2 (t) + ˜ β

0

∞

∫

e

−rt

]

dZ1 (t) ,

(2.8)

0

where Uk , k = 1, 2 are the utility functions adopted by insurer k; κ1 and κ2 are reference points the insurance companies 1 and 2 track by adopting a pair of reinsurance and capital injections strategy πk = (uk , Zk ). The game problem can be described as follows. Problem 2.2. Find a Nash equilibrium (π1∗ , π2∗ ) = (u∗1 , Z1∗ ; u∗2 , Z2∗ )

∈ U1 × U2 such that J1 (x; π1∗ , π2∗ ) ≥ J1 (x; π1 , π2∗ ),

∀ π1 ∈ U1 , J2 (x; π1∗ , π2∗ ) ≥ J2 (x; π1∗ , π2 ), ∀ π2 ∈ U2 .

For notational convenience, we denote f1 (x) := U1 (x − κ1 ) and f2 (x) := U2 (κ2 − x). We also define an operator Lπ by L(u1 ,u2 ) V (x) = (c1 − c2 − g(u1 ) + g(u2 ) − λ1 u1 m1 + λ2 u2 m2 ) Vx

+

(2.9)

If a Nash equilibrium exists, the value functions of the game problem are defined as V k (x) := Jk (x; π1∗ , π2∗ ),

inject, say, raising ∆ capitals to lift surplus differences from level x to x + ∆, he suffers from the penalty at proportion β , i.e., the severity of β ∆. In the meantime, his competitor, insurer 2, has no choice but to passively accept the new surplus difference and enjoy an opportunity gain at proportion ˜ β , that is ˜ β ∆ in quantity. In this way, the rate of changing in value functions for insurer 1 and insurer 2 with respect to surplus differences are β and −˜ β , respectively. Therefore, we have Vx1 (x) = β if and only if 2 Vx (x) = −˜ β . By the same reasoning, we have Vx2 (x) = −β is equivalent to Vx1 (x) = ˜ β.

k = 1, 2.

3. Nash equilibrium In this section, we first prove a lemma which illustrates some properties of value functions. Afterwards, we derive both insurers’ Nash equilibrium reinsurance and capital injections strategies by analyzing the corresponding quasi-variational inequalities. Lemma 3.1. Value function V 1 (x) is increasing on R and value function V 2 (x) is decreasing on R. In addition, for any y ≤ x, we have

˜ β (x − y) ≤ V 1 (x) − V 1 (y) ≤ β (x − y), −β (x − y) ≤ V 2 (x) − V 2 (y) ≤ −˜ β (x − y).

where Vx and Vxx represent the first and second derivatives of V with respect to x. Following standard dynamic programming techniques, value function V k (k = 1, 2) (assuming V k is sufficiently smooth) satisfies the following quasi-variational inequalities (QVIs)

{ max

V 1 (y) ≥ V 1 (x) − β (x − y),

V 2 (y) ≥ V 2 (x) + ˜ β (x − y).

Regarding admissible strategy π2 we constructed, i.e., insurer 2 conducts capital injections with amount (x − y) at the time that the surplus difference equals x, which will be reduced to level y. Then, we obtain V 2 (x) ≥ V 2 (y) − β (x − y),

V 1 (x) ≥ V 1 (y) + ˜ β (x − y).

Therefore, we obtain the inequality satisfied by value function V k . The properties of first order derivatives of value functions follow from the two inequalities. □ Remark 3.2. If first derivatives of value functions exist, the equivalence between Vx1 (x) = β and Vx2 (x) = −˜ β can also be explained in the intuitive sense. Derivative condition Vx1 (x) = β corresponds to the situation that insurer 1 actively injects capitals. When insurer 1 chooses the amount of capitals he will

[

sup

uk ∈[0,1]

]

L(uk ,um ) V k (x) − rV k (x) + fk (x) , ∗

} Vx1 (x)

− β, −β −

Vx2 (x)

= 0,

(3.2)

for k, m ∈ {1, 2}, with k ̸ = m. We divide the set of the surplus difference into three regions: (i) Continuation region

{ sup

C :=

If the right hand derivatives exist, we have ˜ β ≤ Vx1 (x+) ≤ β and −β ≤ Vx2 (x+) ≤ −˜ β , with Vxk (x+) = (−1)m β ⇔ Vxm (x+) = (−1)k˜ β , k, m ∈ {1, 2} and k ̸= m. Proof. We construct two new admissible strategies πk ∈ Uk , k ∈ {1, 2} as follows: insurer k injects capitals with the amount of (x − y) at the initial time and then conducts the optimal strategy πk∗ is followed. We assume the two insurers will not inject capitals simultaneously. If insurer 1 takes actions at the initial time, that is, when the surplus difference is at level y, he injects (x − y) capitals to elevate the difference up to level x. By the definition of value functions and objective functions (2.7)–(2.8), we have

) √ 1( 2 u1 λ1 l1 − 2ρ u1 u2 λ1 λ2 l1 l2 + u22 λ2 l2 Vxx , 2 (3.1)

[

uk ∈[0,1]

]

L(uk ,um ) V k (x) − rV k (x) + fk (x) = 0, ∗

} Vx1 (x)

< β,

Vx2 (x)

> −β .

(ii) Insurer 1’s capital injections region

{ I1 :=

[

sup

L

(uk ,u∗ m)

uk ∈[0,1]

}

]

V (x) − rV (x) + fk (x) < 0, k

k

Vx1 (x)

=β .

(iii) Insurer 2’s capital injections region

{ I2 :=

sup

[

uk ∈[0,1]

]

L(uk ,um ) V k (x) − rV k (x) + fk (x) < 0, ∗

} Vx2 (x)

= −β .

3.1. Reinsurance strategies in equilibrium We assume the reinsurance premium is calculated under the generalized variance premium principle, i.e., gk (uk ) = (1 + [ ] η) (1 − uk )λk mk + ν (1 − uk )2 λk lk . We require η ≥ 0, ν ≥ 0 and (1 + η)λk (mk + ν lk ) ≥ ck to avoid cheap reinsurance. Then the controlled operator becomes Lu1 ,u2 V (x) = C + A1 u1 − A2 u2 − B1 (1 − u1 )2 + B2 (1 − u2 )2 Vx

[

+

]

) √ 1( D1 u21 − 2ρ D1 D2 u1 u2 + D2 u22 Vxx 2 (3.3)

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

11

where C := c1 − c2 − (1 + η)λ1 m1 + (1 + η)λ2 m2 , Ak := ηλk mk , Bk := (1 + η)νλk lk , Dk := λk lk . The optimal reinsurance strategy u∗k of insurer k, given the optimal reinsurance strategy of his competitor, u∗m , m ̸ = k ∈ {1, 2}, can be solved as follows

Proof. We assume that the two insurers’ value functions take the form

⎧ ⎪ ⎪ u∗1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

The first and the second derivatives of W k (x) with respect to x are

⎪ ⎪ u∗2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

{ = arg max

A1 u1 − B1 (1 − u1 )

u1 ∈[0,1]

+

[

1( 2

D1 u21 − 2ρ

√

2

]

Vx1

1 (x) = γ12 K1 (u∗2 ) exp (−γ1 (x − κ1 )) , Wxx

Wx2 (x) = γ2 K2 (u∗1 ) exp (−γ2 (κ2 − x)) ,

{

] [ = arg max −A2 u2 + B2 (1 − u2 )2 Vx2 x u2 ∈[0,1] } ) √ 1( + −2ρ D1 D2 u∗1 u2 + D2 u22 Vxx2 .

2 (x) = γ22 K2 (u∗1 ) exp (−γ2 (κ2 − x)) . Wxx

2

k Assume that 2(1 + η)ν Wxk + (−1)k Wxx ̸= 0. Let the first order derivatives equal to 0 give

⎧ [ ]+ √ (A +2B )W 1 −ρ D1 D2 u∗ W1 ⎪ 2 xx ⎨ u∗1 = 1 1 x 1 ∧ 1 , 1 2B1 Wx −D1 Wxx [ ]+ √ ∗ 2 2 ⎪ (A + 2B 2 2 )Wx +ρ D1 D2 u1 Wxx ⎩ u∗ = ∧ 1 , 2 2 2 2B W +D W 2

x

2

W 2 (x) = K2 (u∗1 ) exp (−γ2 (κ2 − x)) + k2 . Wx1 (x) = −γ1 K1 (u∗2 ) exp (−γ1 (x − κ1 )) ,

}

1 D1 D2 u1 u2 Vxx ,

) ∗

W 1 (x) = K1 (u∗2 ) exp (−γ1 (x − κ1 )) + k1 ,

(3.4)

Plugging these derivatives into (3.4), it is easy to verify that the optimal retained proportion of claims risks for the two insurers becomes (3.7). Expression for value functions of the two insurers can be derived by similar procedures. Therefore, without loss of generality, we focus on deriving the value function of insurer 1. Since the optimized retained proportion of risks might be obtained at the inner point of interval (0, 1) or at either sides of this interval, we conduct the derivation in the following three cases.

√

xx

CASE 1. ργ1 D1 D2 u∗2 ≤ −(A1 + 2B1 ); √ CASE 2. −(A1 + 2B1 ) < ργ1 D1 D2 u∗2 < γ1 D1 − A1 ; √ CASE 3. ργ1 D1 D2 u∗2 ≥ γ1 D1 − A1 .

k

where W are the solutions of the following system of coupled ordinary differential equations:

[ ] ⎧ 0 = C + A1 u∗1 − A2 u∗2 − B1 (1 − u∗1 )2 + B2 (1 − u∗2 )2 Wx1 ⎪ ⎪ ⎪ ] √ 1[ ⎪ ⎪ + D1 (u∗1 )2 − 2ρ D1 D2 u∗1 u∗2 + D2 (u∗2 )2 Wxx1 ⎪ ⎪ ⎪ 2 ⎨ 1 − ] [ rW (x) ∗+ f1 (x),∗ 0 = C + A1 u1 − A2 u2 − B1 (1 − u∗1 )2 + B2 (1 − u∗2 )2 Wx2 ⎪ ⎪ ⎪ ] √ ⎪ 1[ ⎪ ⎪ + D1 (u∗1 )2 − 2ρ D1 D2 u∗1 u∗2 + D2 (u∗2 )2 Wxx2 ⎪ ⎪ 2 ⎩ −rW 2 (x) + f2 (x).

CASE 1. √ When ργ1

(A1 +2B1 )+ργ1

2B1 +γ1 D1

When both insurers have constant absolute risk aversion (CARA) utility functions, i.e., they both adopt the exponential utility function the QVIs for insurer 1

γk

exp(−γk x),

(3.6)

and fk in (3.5) have the expression f1 (x) = U1 (x − κ1 ) = η1 − f2 (x) = U2 (κ2 − x) = η2 −

1

γ1 1

γ2

exp (−γ1 (x − κ1 )) ,

Under such formulation, we obtain the solution of QVIs (3.2) as follows. Theorem 3.3. Expressions for the equilibrium reinsurance strategies of the two insurers are

(3.7)

× exp (−γ1 (x − κ1 )) + η1 − rk1 = 0.

(3.9)

To ensure that (3.9) holds for all x ∈ R, parameter k1 and function K1 (u∗2 ) should satisfy

⎧ η1 − rk1 = 0, ⎪ [ ] ⎪ ⎪ ] 1 [ ⎨ ∗ 2 ∗ ∗ 2 2 −γ1 C − A2 u2 − B1 + B2 (1 − u2 ) + D2 (u2 ) γ1 − r 2 ⎪ ⎪ 1 ⎪ ∗ ⎩ × K1 (u2 ) − = 0, γ1 = η1 / r , 1 [ K1 (u2 ) = −r − (C + B2 − B1 )γ1 + (A2 + 2B2 )γ1 u∗2 γ 1 ⎪ ( ) ]−1 ⎪ ⎪ ⎩ + γ12D2 − B2 γ1 (u∗2 )2 . √ CASE 2. When −(A1 + 2B1 ) < ργ1 D1 D2 u∗2 < γ1 D1 − A1 ,

⎧ k1 ⎪ ⎪ ⎪ ⎨

∗

u∗1 =

(A1 + 2B1 ) + ργ1

(a)

(3.8)

(b)

D1 D2 u∗2

2B1 + γ1 D1

[ C + A1

(A1 + 2B1 ) + ργ1

γk D m 2

− Bm ,

( − B1

√

∈ (0, 1).

D1 D2 u∗2

2B1 + γ1 D1

∗

where Kk (um ) is given in Box I , with Pk,m := (−1)k C − Bm , Qk := Ak + 2Bk , Rk,m := k, m ∈ {1, 2}, and k ̸ = m.

√

1 Plugging the expression for u∗1 , W 1 , Wx1 , Wxx , and f1 (x) into the first equation of (3.5), we obtain

The corresponding value functions are W 2 (x) = K2 (u∗1 ) exp (−γ2 (κ2 − x)) + η2 /r ,

[ [ ] −γ1 K1 (u∗2 ) C − A2 u∗2 − B1 + B2 (1 − u∗2 )2 ] 1 1 + D2 (u∗2 )2 γ12 K1 (u∗2 ) − rK1 (u∗2 ) − 2 γ1

the optimal retained proportion of claims for insurer 1 after purchasing the optimal reinsurance contract is

⎧ ]+ [ √ (A +2B1 )+ργ1 D1 D2 u∗ ⎪ 2 ⎨ u∗1 = 1 2B ∧ 1 , 1 +γ1 D1 [ ]+ √ ∗ ⎪ ⎩ u∗ = (A2 +2B2 )+ργ2 D1 D2 u1 ∧ 1 . 2 2B2 +γ2 D2 W 1 (x) = K1 (u∗2 ) exp (−γ1 (x − κ1 )) + η1 /r ,

≤ 0. Therefore, insurer 1’s optimal retained

which gives

exp (−γ2 (κ2 − x)) .

{

D1 D2 u∗2 ≤ −(A1 + 2B1 ), we always have

proportion of risks u∗1 = 0. Plugging the expression of W 1 , Wx1 , 1 Wxx , f1 (x), and u∗1 into the first ODE of (3.5) and rearrange, insurer 1’s QVI in the continuous region becomes

(3.5)

Uk (x) = ηk −

D1 D2 u∗ 2

√

(γ1 D1 − A1 ) − ργ1

√

2B1 + γ1 D1

− A2 u∗2

D1 D2 u∗2

)2

12

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

⎧ √ 1 ⎪ ⎪ [ ] , for ργk Dk Dm u∗m ≤ −(Ak + 2Bk ), ⎪ ∗ ∗ 2 ⎪ −r + (Pk,m + Bk )γk + Qm γk um + Rk,m γk (um ) γk ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ [ ( ) ( ) ( ) ] , √ ⎪ ⎪ 2 ργ Q Dk Dm ⎪ Q ρ 2 γk2 Dk Dm k k k ⎪ ∗ + R ∗ )2 γ γ + Q − γ u − γ (u ⎨ −r + Pk,m + Bk − 2(2Bk +γ k m k k , m k k m m 2Bk +γk Dk 2(2Bk +γk Dk ) k Dk ) Kk (u∗m ) = √ ⎪ ⎪ for − (Ak + 2Bk ) < ργk Dk Dm u∗m < γk Dk − Ak , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ [ ( ] , ) ⎪ √ ⎪ γk D k ⎪ ⎪ −r + Pk,m − Ak + 2 γk + (Qm − ργk Dk Dm )γk u∗m + Rk,m γk (u∗m )2 γk ⎪ ⎪ ⎪ √ ⎩ for ργk Dk Dm u∗m ≥ γk Dk − Ak ,

Box I.

]

of (3.5), becomes

] [ + B2 (1 − u∗2 )2 × −γ1 K1 (u∗2 ) exp (−γ1 (x − κ1 )) +

1

[

( D1

2

(A1 + 2B1 ) + ργ1

√

2B1 + γ1 D1 ∗ (A1 + 2B1 ) + ργ1

√

D1 D2 u∗2

√

)2

D 1 D 2 u∗

1

γ1

]

−

Q12

− −P1,2 − B1 +

{

2(2B1 + γ1 D1 )

)

+ (A2 + 2B2 − ργ1 D1 D2 )γ1 u∗2 K1 (u∗2 ) } ) ( 1 γ1 D2 ∗ 2 ∗ + − B2 γ1 (u2 ) − rK1 (u2 ) − 2 γ1

Q2 −

− rK1 (u∗2 ) −

1

γ1

( ) γ1 D1 − C + B2 + A1 − γ1 K1 (u∗2 ) 2

γ1 K1 (u∗2 )

) ργ1 Q1 D1 D2 γ1 u∗2 K1 (u∗2 ) 2B1 + γ1 D1 ( ) ρ 2 γ12 D1 D2 + R1,2 − γ1 (u∗2 )2 K1 (u∗2 ) 2(2B1 + γ1 D1 ) }

+

exp (−γ1 (x − κ1 )) = 0.

√

√

(

1

γ1

Rearranging and simplifying such equation gives

exp (−γ1 (x − κ1 )) = 0.

(

] −γ1 K1 (u∗2 ) exp (−γ1 (x − κ1 )) ]

] [ − r K1 (u∗2 ) exp (−γ1 (x − κ1 )) + k1 + η1

Rearranging and simplifying this equation, we obtain

{

][

√ 1[ D1 − 2ρ D1 D2 u∗2 + D2 (u∗2 )2 2 × γ12 K1 (u∗2 ) exp (−γ1 (x − κ1 )) +

2 + D2 (u∗2 )2 − 2ρ D1 D2 u2 2B1 + γ1 D1 ] [ × γ12 K1 (u∗2 ) exp (−γ1 (x − κ1 )) ] [ − r K1 (u∗2 ) exp (−γ1 (x − κ1 )) + k1 + η1

−

C + A1 − A2 u∗2 + B2 (1 − u∗2 )2

[

× exp (−γ1 (x − κ1 )) + η1 − rk1 = 0. The previous equation holds for all real numbers iff

⎧ ⎪ ) [ 1 −( rk1 = 0, ⎨ η √ − −P1,2 + A1 − γ12D1 γ1 + (Q2 − ργ1 D1 D2 )γ1 u∗2 ] ⎪ ⎩ + R γ (u∗ )2 − r K (u∗ ) − 1 = 0,

exp (−γ1 (x − κ1 ))

1,2 1

+ η1 − rk1 = 0,

1

2

γ1

which gives

which will always hold if and only if

⎧ η ⎪ [1 − rk1 = 0, ⎪ ⎪ ( ) ⎪ ⎪ Q12 ⎪ ⎪ − − P1,2 − B1 + 2(2B +γ γ1 ⎪ D ) ⎪ 1 1 1 ⎨ ( ( ) √ + Q2 − ργ2B1 Q1+γ DD1 D2 γ1 u∗2 + R1,2 − ⎪ ⎪ 1 ⎪ ]1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ×γ1 (u∗2 )2 − r K1 (u∗2 ) − γ1 = 0. ⎪ ⎩ 1

2

⎧ k1 ⎪ ⎪ ⎨

∗

K1 (u2 )

⎪ ⎪ ⎩ ρ 2 γ12 D1 D2

)

2(2B1 +γ1 D1 )

Therefore, we have ⎧ ⎪ k1 = η1 /r , ⎪ ⎪ ⎨ K (u∗ ) = 1 2 1 [ ( ) ( ) ] . ( ) ⎪ √ 2 ⎪ ρ2 γ 2 D D ργ1 Q1 D1 D2 ⎪ 1 ⎩ −r − P1,2 −B1 + 2(2B1Q+γ γ1 u∗2 + R1,2 − 2(2B 1+γ1 D2 ) γ1 (u∗2 )2 γ1 γ + Q − 1 2 2B1 +γ1 D1 1 D1 ) 1 1 1

= η1 /r , ( ) 1 [ = −r + P1,2 − A1 + γ12D1 γ1 γ1 ]−1 √ + (Q2 − ργ1 D1 D2 )γ1 u∗2 + R1,2 γ1 (u∗2 )2 .

Combining the three different cases, we obtain that parameter k1 is always a constant η1 /r, and K1 is a function of insurer 2’s equilibrium reinsurance strategy u∗2 and employing the following form K1 (u∗2 ) =

≥ γ1 D1 − A1 , inequality ≥ 1 always holds, and hence u∗1 = 1. Then, the

√ ⎧ 1 , for ργ1 D1 D2 u∗2 ≤ −(A1 + 2B1 ), ⎪ [−r +(P1,2 +B1 )γ1 +Q2 γ1 u∗2 +R1,2 γ1 (u∗2 )2 ]γ1 ⎪ ⎪ ⎪ 1 [ ( ) ( ) ] , ⎪ ( ) √ ⎪ ⎪ Q2 ρ2 γ 2 D D ργ1 Q1 D1 D2 ⎪ 1 γ1 + Q2 − 2B γ1 u∗2 + R1,2 − 2(2B 1+γ1 D2 ) γ1 (u∗2 )2 γ1 ⎨ −r + P1,2 +B1 − 2(2B1 +γ 1 D1 ) 1 +γ1 D1 1 1 1 √ for − (A1 + 2B1 ) < ργ1 D1 D2 u∗2 < γ1 D1 − A1 , ⎪ ⎪ ⎪ [ ( 1 ⎪ ) ] ⎪ , √ γ D ⎪ ∗ ∗ 2 ⎪ 2 )γ1 u2 +R1,2 γ1 (u2 ) γ1 ⎪ −r + P1,2 −A1 + 12 1 γ1 +(Q2 −ργ1 D1 D√ ⎩ ∗ for ργ1 D1 D2 u2 ≥ γ1 D1 − A1 .

QVIs for insurer 1 in the continuous region, i.e. the first equation

(3.10)

CASE 3.√ When ργ1

(A1 +2B1 )+ργ1

D1 D2 u∗ 2

2B1 +γ1 D1

√

D1 D2 u∗2

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

By using the same procedure, we obtain k2 = η2 /r and ∗

K2 (u1 ) =

√ ⎧ 1 , for ργ2 D1 D2 u∗1 ≤ −(A2 + 2B2 ), ⎪ [−r +(P2,1 +B2 )γ2 +Q1 γ2 u∗1 +R2,1 γ2 (u∗1 )2 ]γ2 ⎪ ⎪ ⎪ 1 [ ( ) ( ) ] , ⎪ ) ( √ ⎪ ⎪ Q2 ρ2 γ 2 D D ργ2 Q2 D1 D2 ⎪ 2 γ2 u∗1 + R2,1 − 2(2B 2+γ1 D2 ) γ2 (u∗1 )2 γ2 γ2 + Q1 − 2B ⎨ −r + P2,1 +B2 − 2(2B2 +γ D ) +γ D 2 2 2 2 2 2 2 2 √ for − (A2 + 2B2 ) < ργ2 D1 D2 u∗1 < γ2 D2 − A2 , ⎪ ⎪ ⎪ [ ( 1 ⎪ ) ] , ⎪ √ γ D ⎪ ∗ ∗ 2 ⎪ 2 )γ2 u1 +R2,1 γ2 (u1 ) γ2 ⎪ −r + P2,1 + 22 2 −A2 γ2 +(Q1 −ργ2 D1 D√ ⎩ ∗ for ργ2 D1 D2 u1 ≥ γ2 D2 − A2 . □

(3.11)

Proof. Insurer 1 will inject capital once the value of dynamic ∗ process X π (t) drops below x, while insurer 2 will intervene when the dynamic surpasses x. Referring to Shreve et al. (1984), we consider the equilibrium capital injection strategies { } in the local time {type, that is, Z1∗}(t) = x − min inf0≤s≤t X (s), x and Z2∗ (t) = max sup0≤s≤t X (s), x −x. Then, in order to determine the optimal capital injection strategies Zk∗ (x), we only need to find x and x. The capital injection region of insurer 1 is represented as Vx1 (x) = β. Then, the differentiability of V 1 (x) at x requires that

Remark 3.4. From (3.7) in Theorem 3.3, we find that the equilibrium reinsurance strategies do not depend on the level of surplus difference X (t). Therefore, we can determine the reinsurance strategies in equilibrium of both insurers before analyzing their interventions of capital injections.

W 1 (x) = β.

3.2. Equilibrium capital injections strategies

we require

From the intuitive interpretation, insurer 1 would like to keep a high value of the surplus difference X (t) = X1 (t) − X2 (t), and hence we assume that he will intervene by capital injections when the dynamic process takes values below a threshold x. In the meantime, insurer 2 prefers a low value of the dynamic process. Therefore, he will choose to intervene when the value is higher than another threshold x. We assume x < x. Then, the partition of surplus takes the form as follows I1 = Vx1 (x) = β = (−∞, x),

{

13

(3.15)

For the capital injection region of insurer 2, which is represented as Vx2 (x) = −β,

W 2 (x) = −β

(3.16)

in order to make sure V 2 (x) is differentiable at x. (3.15) and (3.16) imply that

⎧ ( ) 1 β ⎪ ⎪ ln − , ⎨ x = κ1 − γ1 ( γ1 K1 (u∗2 ) ) β 1 ⎪ ⎪ , ln − ⎩ x = κ2 + γ2 γ2 K2 (u∗1 )

□

}

3.3. A solution to the QVIs

C = Vx1 (x) < β, Vx2 (x) > −β = [x, x],

{

}

I2 = Vx2 (x) = −β = (x, ∞).

Theorem 3.7. A couple of solutions to the QVIs (3.2) V k , k = 1, 2, is

Remark 3.5. This intuitive explanation coincides with the result derived by Shreve et al. (1984). That is, under equilibrium strategies, the surplus difference process X (t) described by (2.6) is reflected at a lower barrier x and an upper barrier x. To be specific, the equilibrium capital injection process for insurer 1 makes X (t) immediately jump up to x if the surplus difference is smaller than the lower barrier; in the meantime, the equilibrium capital injection process for insurer 2 narrows the surplus difference to x when X (t) exceeds the upper barrier. Local times Z1∗ and Z2∗ (defined in the following Theorem) guarantee that X (t) is a doubly reflected process.

⎧ ⎨ W 1 (x) − β (x − x), 1 W 1 (x), V (x) = ⎩ 1 β (x − x), W (x) + ˜ ⎧ 2 ⎨ W (x) + ˜ β (x − x), V 2 (x) = W 2 (x), ⎩ 2 W (x) − β (x − x),

{

}

The equilibrium capital injection strategies are described as the following theorem. Theorem 3.6. For insurer k, k = 1, 2, the equilibrium capital injections processes follow the local times Z1∗ (t) = x − min

{

∗

Z2 (t) = max

{

inf X (s), x

}

1

sup X (s), x − x,

(

β

∗

V 1 (x) = W 1 (x) + ˜ β (x − x)

1

γ2

ln − γ

β

∗

2 K2 (u1 )

.

and V 2 (x) = W 2 (x) − β (x − x). □

∗

with expressions for K1 (u2 ) and K2 (u1 ) are given by (3.10) ( and (3.11) )

)

V 1 (x) = V 1 (x) − β (x − x) = W 1 (x) − β (x − x)

(3.13) (3.14)

respectively, and satisfying x < x when κ1 −κ2 <

(

Proof. It is easy to verify that smooth functions W 1 (x) and W 2 (x) given by (3.8)(a) and (3.8)(b) solve QVIs (3.2) in the continuous region C = [x, x]. For x ∈ I1 , we have Vx1 (x) = β and Vx2 (x) = −˜ β . Therefore, we obtain

from the continuous condition of V k , k = 1, 2 at x. In insurer 2’s capital injections region I2 , Vx2 (x) = −β , Vx1 (x) = ˜ β and the continuous condition of V k , k = 1, 2 at x give

)

, γ1 K1 (u∗2 ) ( ) 1 β x = κ2 + ln − , γ2 γ2 K2 (u∗1 ) γ1

where x, x are given by (3.13) and (3.14), respectively.

(3.12)

0≤s≤t

ln −

(3.18)

V 2 (x) = V 2 (x) + ˜ β (x − x) = W 2 (x) + ˜ β (x − x)

}

where x = κ1 −

x ∈ (−∞, x), x ∈ [x, x], x ∈ (x, ∞),

(3.17)

and

and

0≤s≤t

x ∈ (−∞, x), x ∈ [x, x], x ∈ (x, ∞),

1

γ1

ln − γ

β

∗

1 K1 (u2 )

+

Remark 3.8. Solutions V k (x), k = 1, 2, given by (3.17) and (3.18) satisfy V 1 ∈ C 2 (x, x) ∩ C 1 (−∞, x) ∩ C (−∞, +∞) and V 2 ∈ C 2 (x, x) ∩ C 1 (x, +∞) ∩ C (−∞, +∞), respectively. Therefore,

14

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

V k (x) are technically viscosity solutions, rather than smooth ones, of QVIs (3.2). More specifically, V k (x) are not twice continuous differentiable at x and x. But at all the other points, V k (x) are smooth enough, hence, they are viscosity solutions for (3.2). By the continuity of both viscosity solutions and V k (x), together with the uniqueness of viscosity solutions to QVIs, we obtain the identity between V k (x) and viscosity solutions to (3.2) everywhere, including the un-smooth points x and x.

Plugging (4.2), (4.3) and (4.4) into Eq. (4.1), we obtain (π1 ,π2∗ )

e−rt V 1 (Xt t

∫

) − V 1 (x) (π1 ,π2∗ )

[

e−rs −f1 (Xs

≤ 0

+β

∫

−˜ β

∫

t

π

e−rs d˜ Z1 1 (s) + β 0

4. Verification theorem

π∗

e−rs d˜ Z2 2 (s) − ˜ β (π1 ,π2∗ )

e−rs f1 (Xs

=−

∑

)ds + β

Proof. It is easy to check that V (x) (k = 1, 2), the couple of solutions to QVIs (3.2) are smooth, i.e., V k ∈ C 2 (C ) ∩ C 1 (C ∪ Im ) ∩ C (C ∪ Im ∪ Ik ). Besides, we can bound V k by a function with polynomial growth because V k is linear in capital injections regions Ik and Im , and is convex/concave in the continuous region C , which is a closed interval. In the following, we prove V 1 (x) ≥ J1 (x; π1 , π2∗ ) for all π1 ∈ U1 , and V 1 (x) = J1 (x; π1∗ , π2∗ ). The results for V 2 and J2 will be drawn by following the same procedure. Step I. V 1 (x) ≥ J1 (x; π1 , π2∗ ). For k = 1, 2, fix a strategy ∑ πk ∈ Uk , define Λk := {s : π π π πk πk Zk k (s−) ̸ = Zk k (s)}. Let ˆ Zk k (t) := s∈Λk ,s≤t [Zk (s) − Zk (s−)] be the discontinuous part of insurer k’s cumulative capital injections π π π π process Zk k (t), and ˜ Zk k (t) := Zk k (t)−ˆ Zk k (t) be the continuous part πk of Zk (t). From Itô formula, we have (π1 ,π2∗ )

t

∫

) − V 1 (x) (π1 ,π2∗ )

[

e−rs −rV 1 (Xs

=

(π1 ,π2∗ )

) + L(u1 ,u2 ) V 1 (Xs ∗

]

) ds

0 t

∫

(π1 ,π2∗ )

e−rs Vx1 (Xs

+

π

)d˜ Z1 1 (s)

0 (π1 ,π2∗ )

[

∑

+

e−rs V 1 (Xs

(π1 ,π2∗ )

) − V 1 (Xs−

)

]

(4.1)

s∈Λ1 ,s≤t t

∫

(π1 ,π2∗ )

e−rs Vx1 (Xs

−

π∗

)d˜ Z2 2 (s)

π

[

π∗

]

π∗

]

π

e−rs dZ1 1 (s) − ˜ β 0

t

∫

π∗

e−rs dZ2 2 (s) 0

Taking expectations on both sides and rearranging, we obtain 1

{∫

t

e

V (x) ≥ Ex

−rt

(

f1 X

(π1 ,π2∗ )

)

(t) dt − β

0

t

∫

π

e−rs dZ1 1 (s) 0

+˜ β

k

e−rt V 1 (Xt

π

e−rs Z2 2 (s) − Z2 2 (s−) t

∫

0

Theorem 4.1. Assume that (π1∗ , π2∗ ) = (u∗1 , Z1∗ ; u∗2 , Z2∗ ), with expressions of u∗k , and Zk∗ (k = 1, 2) are as in (3.4) and (3.12), respectively. Functions V k (k = 1, 2) are described by (3.17) and (3.18). Then, (π1∗ , π2∗ ) is a Nash Equilibrium of Problem 2.2 and value functions are V k (x) = Jk (x; π1∗ , π2∗ ).

[

e−rs Z1 1 (s) − Z1 1 (s−)

s∈Λ2 ,s≤t

t

∫

∑ s∈Λ1 ,s≤t

t 0

In this section, we prove a verification theorem for the Game Problem 2.2.

]

) ds

t

∫

π∗

e−rs dZ2 2 (s) 0

}

−rt

+e

V

1

(π1 ,π2∗ ) ) (Xt

= J1 (x; π1 , π2∗ )

Step II. V 1 (x) = J1 (x; π1∗ , π2∗ ). By the properties of π1∗ , all the inequalities in previous derivations become equalities. □ 5. Special Cases We assume the reinsurance premium is calculated under the generalized variance premium principle in our model. If we let the loading on the variance of the risks ν or the extra loading on variance premium principle η to be zero, the reinsurance premium reduces be the expected premium principle or the variance premium principle. This will simplify expressions of equilibrium strategies and value functions. Additionally, the result can also be simplified by introducing the independence assumption of the two insurers’ claim processes. In these special cases, Nash equilibrium strategies of the cumulative capital injections processes are given by local times described in (3.12), while Theorem 3.3, i.e., expressions of the equilibrium reinsurance purchase and value functions in the continuous region, will be greatly simplified. In this section, we present the simplified results under three different situations.

0 (π1 ,π2∗ )

[

∑

−

e−rs V 1 (Xs

(π1 ,π2∗ )

) − V 1 (Xs−

)

]

5.1. Variance premium principle

s∈Λ2 ,s≤t

The quasi-variational inequality implies that

( ) ( ) ( ) ∗ ∗ ∗ ∗ − rV 1 X (π1 ,π2 ) (t) + L(u1 ,u2 ) V 1 X (π1 ,π2 ) (t) ≤ −f1 X (π1 ,π2 ) (t) . (4.2) Moreover, we have ˜ β ≤ Vx1 (x) ≤ β . Therefore, for s ∈ Λ1 and s ≤ t,

(

(π1 ,π2∗ )

V 1 Xs

)

( (π ,π ∗ ) ) ] [ π π − V 1 Xs−1 2 ≤ β Z1 1 (s) − Z1 1 (s−)

(4.3)

V

1

(π1 ,π2∗ ) Xs

)

−V

1

(

Corollary 5.1.

The equilibrium reinsurance strategies are

⎧ [ ]+ √ 2B1 +ργ1 D1 D2 u∗ ⎪ 2 ⎨ u∗1 = ∧ 1 , 2B1 +γ1 D1 [ ]+ √ ∗ ⎪ ⎩ u∗ = 2B2 +ργ2 D1 D2 u1 ∧ 1 . 2 2B2 +γ2 D2 Corresponding value functions in the continuous region are

For s ∈ Λ2 , we have

(

When η = 0, i.e., reinsurance premium is determined under the variance premium principle, we have Ak = 0 and Bk = ν Dk . Then, Theorem 3.3 reduces to the following corollary.

(π1 ,π ∗ ) Xs− 2

)

[

π2∗

π2∗

{ ]

=˜ β Z2 (s) − Z2 (s−)

(4.4)

W 1 (x) = K1 (u∗2 ) exp (−γ1 (x − κ1 )) + η1 /r , W 2 (x) = K2 (u∗1 ) exp (−γ2 (κ2 − x)) + η2 /r ,

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18 Table 6.1 Model parameters.

where ∗

Kk (um ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 ( [ ) ] , γ D −r +(Pk,m +Bk )γk +2Bm γk u∗m + k 2 m −Bm γk (u∗m )2 γk

[

for ργk

√

Dk Dm um ≤ −2Bk , ∗

k

k

√

Dk Dm u∗m < γk Dk ,

for ργk

√

Corollary 5.2.

Dk Dm um ≥ γk Dk ,

Kk (u∗m ) =

The equilibrium reinsurance strategies are

where Kk (u∗m ) = 1 ] , for −r +(−1)k C γk +Am γk u∗m + D2m γk2 (u∗m )2 γk

√ ργk Dk Dm u∗m ≤ −Ak ,

1 [ ( ) ) ( ] , √ 2 ρA D D (A )2 −r + (−1)k C − 2γ k D γk + Am − k D k m γk u∗m + (1−ρ2 )Dm γk2 (u∗m )2 γk k

k k

for − Ak < ργk

√

Dk Dm u∗m < γk Dk − Ak ,

) ] , ( √ γ D −r + (−1)k C −Ak + k2 k γk +(Am −ργk Dk Dm )γk u∗m + D2m γk2 (u∗m )2 γk 1

for ργk

√

Dk Dm u∗m ≥ γk Dk − Ak ,

with k, m ∈ {1, 2}, and k ̸ = m. 5.3. Independent claim processes When the claim processes of the two insurance companies are independent, we have ρ = 0. The Nash equilibrium strategies and the corresponding value functions are given by the following corollary. Corollary 5.3. Expressions for the Nash equilibrium strategies of the cumulative capital injections processes are given by local times described in (3.12). The equilibrium reinsurance strategies are u∗1 = ∗

u2 =

ηm1 +2(1+η)ν l1 2(1+η)ν l1 +γ1 l1 ηm2 +2(1+η)ν l2 2(1+η)ν l2 +γ2 l2

∧ 1, ∧ 1.

The corresponding value functions V k (x) are described by (3.17) and (3.18), respectively, with

{

Insurer 2 l1 80

γ1 0.2

λ2 0.5

m2 2

γ2

l2 50

0.25

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

[

(

Q2

1 )

]

,

k γk +Qm γk u∗m +Rk,m γk (u∗m )2 γk −r + Pk,m +Bk − 2(2B +γ k k Dk )

for [

mk lk

<

γk , η

( ) 1 ] , γ D −r + Pk,m −Ak + k2 k γk +Qm γk u∗m +Rk,m γk (u∗m )2 γk m γ for l k ≥ ηk , k

with k, m ∈ {1, 2}, and k ̸ = m. Remark 5.4. When ρ = 0, the equilibrium reinsurance strategies of either insurance company do not rely on his competitor. In fact, when the two insurers’ claim processes are independent, the insurance business of these two companies is totally different. Therefore, there is no real competition relationship between them. 6. Numerical examples

W 1 (x) = K1 (u∗2 ) exp (−γ1 (x − κ1 )) + η1 /r , W 2 (x) = K2 (u∗1 ) exp (−γ2 (κ2 − x)) + η2 /r ,

[

0.8

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

The value functions in the continuous region are

[

m1 2.5

where

∗

⎧ [( ) ]+ √ A1 ∗ ⎨ u∗ = + ρ D / D u ∧ 1 , 2 1 1 2 [( γ1 D1 ) ]+ √ A ⎩ u∗ = 2 + ρ D1 /D2 u∗1 ∧ 1 . 2 γ2 D2

{

λ1

( ] , ) [ ( ) √1 γ γ D −r + Pk,m + k2 k γk +(2Bm −ργk Dk Dm )γk u∗m + 2k −ν Dm γk (u∗m )2 γk

When ν = 0, the generalized variance premium principle reduces to the commonly studied expected premium principle. Then we have Bk = 0. Therefore, the Nash equilibrium reinsurance strategies and the corresponding value functions reduce to the following corollary.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Insurer 1

r 0.06

k

for − 2Bk < ργk

5.2. Expected premium principle

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Discount rate

1 ] , ( ) ( ) ( ) √ ρ 2 γk2 2ρνγk Dk Dm 2ν 2 D γ 2 γ −r + Pk,m +Bk − 2ν+γk γk + 2Bm − Dm γk (u∗ γk u∗m + 2k −ν− 2(2ν+γ k m) 2ν+γ )

with k, m ∈ {1, 2}, and k ̸ = m.

{

15

W 1 (x) = K1 (u∗2 ) exp (−γ1 (x − κ1 )) + η1 /r , W 2 (x) = K2 (u∗1 ) exp (−γ2 (κ2 − x)) + η2 /r ,

In this section, we study some numerical examples on the game problem we have described in order to better understand the effects of certain model parameters on the equilibrium proportional reinsurance and capital injections strategies. Unless otherwise stated, we assume that common model parameters are as shown in Table 6.1. We first examine how the correlation between the two insurance companies’ claim processes, represented by correlation coefficient ρ between standard Brownian motions M1 (t) and M2 (t), influences the equilibrium reinsurance strategies u∗1 and u∗2 , respectively. Suppose the reinsurance institution charges 10% premium loading on the second moment of claim risks, that is ν = 0.1. Theoretically, correlation coefficient ρ takes values within interval [−1, 1]. But in practice, it is impossible to have two companies whose businesses are completely positively/negatively linear correlated. Therefore, we assume ρ ∈ [−0.9, 0.9], which is sufficient to reflect the linear relationship between two arbitrary insurers in the real insurance market. Fig. 6.1 depicts the change of equilibrium reinsurance strategies with respect to correlation ρ , where Fig. 6.1(a) plots insurer 1’s retained proportion of claims in equilibrium, and Fig. 6.1(b) describes that of insurer 2. In order to avoid chances of cheap insurance and reinsurance, the rate of insurance premium ck , k = 1, 2, should satisfy that λk mk < ck < (1 + η)λk (mk + ν lk ). Therefore, we choose ck as the average of its lower bound and upper bound. In both subfigures, the solid blue line, the dash green line and the dot-and-dash red line correspond to the cases when premium loading η equals 0, 0.2, and 0.5, respectively. Each line is part of a hyperbola curve. Since we assume that both insurance companies can only purchase strict proportional reinsurance contracts, all the curves are floored by 0 if u∗k is negative, and are capped by 1 when u∗k ≥ 1. The latter property is testified by Fig. 6.1(b) under our parameter setting. In addition, comparing with the three curves in each subfigure in Fig. 6.1, we find that both insurance companies tend to retain a higher proportion of claim risks when η is increasing. This phenomenon is consistent with the practical experience, i.e., with

16

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Fig. 6.1. The reinsurance equilibrium strategies v.s. correlation coefficient ρ , with ν = 0.1 and η = 0, 0.2, 0.5, respectively.

Fig. 6.2. The reinsurance equilibrium strategies v.s. correlation coefficient ρ , with η = 0.5 and ν = 0, 0.1, 0.2, respectively.

the growing up of reinsurance premium loading, insurance companies would like to retain more risks in order to reduce the cost spent on purchasing reinsurance contracts. When the reinsurance premium loading η is fixed as 0.5, Fig. 6.2 describes the variations of equilibrium reinsurance strategies with respect to the correlation coefficient ρ . Figs. 6.2(a) and 6.2(b) present retained proportions of claims in equilibrium for insurer 1 and insurer 2, respectively. When the premium loading on the second moment of claim risks ν takes values 0, 0.1, and 0.2, u∗k are depicted by the solid blue line, the dash green line and the dot-and-dash red line. Like previous analyses, each of those lines is part of a hyperbola curve, with a floor 0 and a cap 1. Besides, with the increase of ν , the reinsurance company charges more for the fluctuation of claim experiences. Therefore, both insurance companies choose to purchase less reinsurance protections to alleviate pressures of the high premium cost. Next, we analyze the influence of risk-aversion parameters γk , k = 1, 2, on capital injections interventions. To simplify the calculation, we consider a special case by adopting the expected premium principle and assuming the two insurers have uncorrelated claim processes. That is ν = 0 and ρ = 0. We set the parameter regarding intervention penalties as β = 0.1, the proportional intervention gain from competitor’s capital injections as ˜ β = 0.05, and choose the reference points for the two insurance companies as κ1 = −50 and κ2 = 40, respectively.

Fig. 6.3 shows the effect of risk-aversion parameters on target states, or thresholds of capital injections, of the equilibrium capital injections strategies. In Fig. 6.3(a), we plot the evolution of target states x and x with respect to the risk-aversion parameter of insurer 1, γ1 . The solid blue line corresponds to the value of x, that is the state below which insurer 1 will inject capitals and elevate the surplus differences between himself and his competitor back up to such level. In other word, when the surplus difference X (t) is smaller than x, insurer 1 will inject capitals of amount x − x to raise the difference up to the level x. If the surplus difference is greater than x, insurer 2 need to inject x − x capitals in order to make the difference descend to target state x. Fig. 6.3(b) displays the changing of target states x and x v.s. insurer 2’s risk-aversion parameter γ2 . Insurer 1 will conduct capital injections as long as the surplus difference is below x (the solid blue curve). While insurance company 2 will inject capital once the surplus difference surpasses x, which is plotted by the dash red line. Both subfigures indicate that the higher the coefficient of absolute risk aversion parameter, which means that an insurance company is more risk averse, the more prudent on actively taking capital injections insurance companies are. Besides, both insurers are more sensitive to their own risk attitudes than that of their competitors. Fig. 6.4 plots the value functions for Problem 2.2. Specifically, Fig. 6.4(a) depicts the evolution of insurer 1’s value function

N. Zhang, Z. Jin, L. Qian et al. / Insurance: Mathematics and Economics 88 (2019) 7–18

17

Fig. 6.3. Target states of capital injections strategies in equilibrium with respect to risk aversion parameter γk , k = 1, 2, under the case of η = 0.5 and ν = 0.

Fig. 6.4. Value functions of the two insurance companies in the case of η = 0.5 and ν = 0.

x ↦ → V 1 (x). We notice that V 1 is an increasing function of x. Additionally, V 1 grows linearly with the surplus difference x in capital injections regions I1 = (−∞, x] and I2 = [x, ∞), and is convex in the continuation region C = [x, x]. The slope of V 1 in insurer 1’s capital injections region I1 is the fixed proportion of intervention penalties β . In I2 , the capital injections region of insurer 2, the slope of insurer 1’s value function V 1 is the fixed proportional intervention gain ˜ β . Fig. 6.4(b) corresponds to the value function of insurer 2, i.e., x ↦ → V 2 (x). Since we take X (t) as the surplus of insurer 1 minuses that of insurer 2, it is not surprising to detect that V 2 decreases with the increasing of x. Symmetrical to V 1 , insurer 2’s value function shrinks linearly in β, insurer 1’s capital injections region I1 = (−∞, x] at rate −˜ then V 2 follows a convex function in continuous region C = [x, x], and monotonically decreases with slope −β in his own capital injections region I2 = [x, ∞). 7. Conclusion We investigate the proportional reinsurance and capital injections game problem between two insurance companies under the framework of non-zero-sum stochastic differential games. Each insurer’s objective function includes the expectation of discounted values of his utility of his surplus relative to a reference point, the penalties caused by his own capital injection

interventions, and the gains brought by capital injections of his competitors. We derive explicit expressions of the Nash equilibrium strategies and value functions by solving the corresponding quasi-variational inequalities, and then prove the verification theorem. In the formulation of this paper, we assume that both insurers’ intervention penalties and the corresponding gains for their competitor are proportional to the amount of capital injections. If we add a fixed penalty/gain to capital injections and formulating them by impulse controls, and then solve the problem by following similar steps of our derivation, we find that the equilibrium reinsurance strategies are the same as our current formulation, while thresholds and target states for capital injections will no longer be equal. Specifically, insurer 1 will inject capitals once the surplus difference is lower than a threshold, and raise the surplus difference to target state that is higher than this threshold. While for insurer 2, capital injections are triggered when the surplus difference is higher than a threshold and the difference in surpluses will be reduced to target state lower than the threshold for his intervention. However, the fixed penalty/gain terms make the value function of insurer i discontinuous at the point where his competitor conducts capital injections, that is when he passively accepts the surplus difference caused by competitor’s intervention. But with this discontinuity point in value functions, it is hard to prove the verification theorem. Therefore, we leave this as an open problem for further study.

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