Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps

Insurance: Mathematics and Economics 66 (2016) 138–152

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

Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps Yan Zeng a , Danping Li b,∗ , Ailing Gu c a

Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, PR China

b

School of Science, Tianjin University, Tianjin 300072, PR China

c

School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, PR China

highlights • • • • •

A new robust mean–variance reinsurance-investment model with jumps is established. Different ambiguity-averse levels towards diffusion and jump risks are adopted. The robust equilibrium strategy and corresponding value function are derived. Some special cases and utility losses from model uncertainty are illustrated. Some interesting results and phenomena are presented.

article

info

Article history: Received August 2015 Received in revised form October 2015 Accepted 28 October 2015 Available online 18 November 2015 JEL classification: C61 G11 G22 Keywords: Robust optimal control Reinsurance and investment Jump-diffusion model Mean–variance criterion Equilibrium strategy

abstract This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean–variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér–Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The application of stochastic control theory to the optimal reinsurance-investment problem has been the focus of a good part of actuarial research, and the interest in this problem continues to grow. In recent years, this problem has been studied in terms of a variety of objectives, such as minimizing the probability of ruin (see Promislow and Young, 2005; Azcue and Muler, 2013), maximizing the expected utility from terminal wealth (see Bai and

∗

Corresponding author. E-mail addresses: [email protected] (Y. Zeng), [email protected] (D. Li), [email protected] (A. Gu). http://dx.doi.org/10.1016/j.insmatheco.2015.10.012 0167-6687/© 2015 Elsevier B.V. All rights reserved.

Guo, 2010; Liang and Yuen, 2016), and the mean–variance criterion (see Pressacco et al., 2011; Bi et al., 2013). Although the optimal reinsurance-investment problem has been widely investigated by many scholars, only a few have incorporated the model uncertainty into it. However, it is a notorious fact that the return of risky assets is difficult to be estimated with precision. Thus, some scholars have advocated and investigated the effect of model uncertainty on portfolio selection. Robust decision making in the portfolio context is introduced by Maenhout (2004). Maenhout (2004, 2006) investigates the effect of ambiguity on the intertemporal portfolio choice in a setting with constant investment opportunities and in a setting with a mean-reverting equity risk premium, respectively. A number of other papers are built on Maenhout (2004) to

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

address the implications of ambiguity on portfolio choice. Liu (2010) examines the robust consumption and portfolio choice for time-varying investment opportunities. Flor and Larsen (2014) determine the optimal investment strategy for an ambiguityaverse investor with a stochastic interest rate. Munk and Rubtsov (2014) introduce a stochastic interest rate and inflation into a portfolio management problem for an ambiguity-averse investor. Yi et al. (2015b) focus on an optimal portfolio selection problem with model uncertainty in a financial market that contains a pair of stocks. From these papers, we find that compared with making ad-hoc decisions about how many errors are contained in the estimates for the parameters of risky assets, investors may consider alternative models that are close to the estimated model. This method has also been accepted in the robust optimal reinsurance-investment problem. Lin et al. (2012) and Korn et al. (2012) investigate the optimal reinsurance problem or the optimal reinsurance-investment problem with model uncertainty by using a stochastic differential game approach. Yi et al. (2013) and Yi et al. (2015a) study the problem of robust optimal reinsurance-investment for an ambiguity-averse insurer (AAI) under the expected exponential utility maximization and mean–variance criteria, respectively. Pun and Wong (2015) consider the problem of robust optimal reinsurance-investment with multi-scale stochastic volatility using a general concave utility function. However, most of the literature on the robust optimal reinsurance and investment problem assumes that the AAI’s surplus process and the risky asset’s price process follow the diffusion model, which ignores the significant effect that jumps have on the optimal strategy. As is mentioned in Branger and Larsen (2013) and Aït-Sahalia and Matthys (2014), there are pronounced differences between ambiguity aversion with respect to (w.r.t.) diffusion and jump risks. Therefore, in the portfolio selection problem, ignoring ambiguity w.r.t. the jump risk may result in large losses in the financial market. In this paper, we consider the optimal reinsurance-investment problem for an AAI who faces uncertainties regarding models in the financial and insurance markets with jumps. To the best of our knowledge, no published work addresses the robust optimal reinsurance-investment problem with jumps under the mean–variance criterion for an AAI. Traditional dynamic mean–variance optimization problem is a time-inconsistent problem, and most of the literature derives an optimal strategy that is only optimal at the initial time. However, time consistency of strategies is a basic requirement for rational decision making in many situations. A decision maker sitting at time t would consider that, starting from t + ∆t, she will follow the strategy that is optimal sitting at time t + ∆t. Namely, the optimal strategy derived at time t should agree with the optimal strategy derived at time t + ∆t. Because the time-consistency of strategies is important for a rational decision-maker, recently many scholars have developed a time-consistent strategy for the dynamic mean–variance asset allocation problem. The main approach is to formulate the problem within a non-cooperate game theoretic framework, with one player for each time t, where player t can be regarded as the future incarnation of the insurer at time t. Then we aim to derive the equilibrium strategy of the game. For more details, we refer the reader to Björk and Murgoci (2010), Zeng et al. (2013), Li and Li (2013), Björk et al. (2014) and references therein. In our model, the insurer’s surplus process is assumed to follow the classical Cramér–Lundberg (C–L) model, and the insurer is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. Given that the market (true model) may deviate from the estimated model (reference model) in reality, we incorporate model uncertainty into

139

our model and assume that the insurer is ambiguity-averse about diffusion and jump risks. On the basis of the above setup and by applying stochastic control theory, we formulate a robust optimization problem with alternative models and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function. Some special cases of our model are also provided, which show that our model and results extend some ones in the existing literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples. The main contributions of this paper are as follows: (i) a new optimal reinsurance-investment model incorporating model uncertainty and jumps under the mean–variance criterion is established; (ii) the robust equilibrium strategy and the corresponding equilibrium value function are derived explicitly, and our model and results extend some ones in the existing literature; and (iii) utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance for the AAI are analyzed, and some new findings are provided. The remainder of this paper is organized as follows. Section 2 describes the formulation of the model. Section 3 derives the explicit expressions of the robust equilibrium reinsuranceinvestment strategy and the corresponding equilibrium value function, and provides some special cases of our model. Section 4 presents some numerical examples to illustrate our results and sensitivity analysis of utility losses. Section 5 concludes this paper. 2. General formulation Let (Ω , F , {Ft }t ∈[0,T ] , P) be a filtered complete probability space satisfying the usual condition, where T > 0 is a finite constant representing the investment time horizon; Ft stands for the information available until time t; and P is a reference measure. Any decision made at time t is based on Ft , and all stochastic processes below are supposed to be well-defined and adapted to this probability space. In addition, suppose that there are no transaction costs or taxes in the financial and insurance markets, and trading can be continuous. Suppose that an insurer’s surplus process follows the classical C–L model. In this model, without reinsurance and investment, her surplus process is described by N1 (t )

R(t ) = x0 + ct −



Zi ,

i=1

where x0 ≥ 0 is the initial surplus; c is the premium rate; and i=1 Zi is a compound Poisson process, representing the cumulative claims up to time t , {N1 (t )}t ∈[0,T ] is a homogeneous Poisson process with intensity λ1 > 0, and the claim sizes Z1 , Z2 , . . . , independent of N1 (t ), are assumed to be independent and identically distributed (i.i.d.) positive random variables with common distribution F (z ), finite first moment E[Zi ] = µZ and second moment E[Zi2 ] = σZ2 . Furthermore, we assume that the premium rate c is assumed to be calculated according to the expected value principle, i.e., c = (1 + θ )λ1 µZ , where θ > 0 is the safety loading of the insurer. In addition, we assume that the insurer can control her insurance risk by purchasing proportional reinsurance or acquiring new business, such as acting as a reinsurer of other insurers (see Bäuerle, 2005). For each t ∈ [0, T ], the proportional reinsurance/new business level is denoted by the value of risk exposure p(t ) ∈ [0, +∞). When p(t ) ∈ [0, 1], it corresponds to a proportional reinsurance cover. In this case, the insurer diverts parts of the premium to the reinsurer at the rate of (1 + η)(1 − p(t ))λ1 µZ ,

N1 (t )

140

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

where η is the safety loading of the reinsurer with η > θ . Meanwhile, the insurer pays 100p(t )% and the reinsurer pays the remaining 100(1 − p(t ))% for each claim occurring at time t. When p(t ) ∈ [1, +∞), it corresponds to acquiring new business. For simplicity, {p(t )}t ∈[0,T ] is called a reinsurance strategy hereafter. With such a reinsurance strategy, the surplus process of the insurer follows

Denote π (t ) as the money amount invested in the risky asset at time t. The remainder X u (t )−π (t ) is invested in the risk-free asset, where X u (t ) is the wealth process at time t associated with strategy u := {(p(t ), π (t ))}t ∈[0,T ] . Thus, the wealth process {X u (t )}t ∈[0,T ] follows dX u (t ) = r0 X u (t ) + (µ − r0 )π (t ) + (θ − η)λ1 µZ



+ (1 + η)p(t )λ1 µZ ] dt  ∞ p(t )zN1 (dt , dz ) + π (t )σ dW (t ) −

N1 (t )

dR(t ) = [(θ − η)λ1 µZ + (1 + η)p(t )λ1 µZ ]dt − p(t )d



Zi .

i=1

0

Moreover, we assume that the insurer is allowed to invest in a financial market consisting of a risk-free asset and a risky asset. The price process of the risk-free asset evolves according to dS0 (t ) = r0 S0 (t )dt ,

(1)

where r0 > 0 represents the risk-free interest rate. The price process of the risky asset follows a jump-diffusion process as N2 (t )

 dS (t ) = S (t −) µdt + σ dW (t ) + d



Yi ,

(2)

where µ and σ are positive constants; {W (t )}t ∈[0,T ] is a standard Brownian motion; {N2 (t )}t ∈[0,T ] , representing the number of the price jumps that occur in the risky asset during time interval [0, t ], is a homogeneous Poisson process with an intensity λ2 ; Yi is the ith jump amplitude of the risky asset price; and Yi , i = 1, 2, . . . are i.i.d. random variables with distribution function G(y), finite first-order moment E[Yi ] = µY and second-order moment E[Yi2 ] = σY2 . Similar to Branger and Larsen (2013), we assume that

N1 (t )

N

(t )

Zi and i=21 Yi are independent. Besides, we assume that P{Yi ≥ −1 for all i ≥ 1} = 1 to ensure that the risky asset price remains positive. Generally, the expected return of the risky asset is larger than the risk-free interest rate, so we assume that µ + λ2 µY > r0 . Next, we use Poisson random measures N1 (·, ·) on Ω × [0, T ] × [0, ∞) and N2 (·, ·) on Ω × [0, T ] × [−1, ∞) to denote the N (t ) N (t ) compound Poisson processes i=11 Zi and i=21 Yi , respectively, as N1 (t )



∞

 t Zi = 0

i =1 N2 (t )



i =1

 t 0

∀t ∈ [0, T ],

yN2 (ds, dy),

∀t ∈ [0, T ].

0

∞

Yi =

i =1

zN1 (ds, dz ),

−1

If we denote by ν1 (dt , dz ) = λ1 dtdF (z ) and ν2 (dt , dy) = λ2 dtdG(y), then N (t )    t ∞ 1  E Zi = z ν1 (ds, dz ), ∀t ∈ [0, T ], 0

i=1

E

N (t )  2  Yi

yν2 (ds, dy),

∀t ∈ [0, T ],

−1

and ν1 (·, ·) and ν2 (·, ·) are the compensators of the random measures N1 (·, ·) and N2 (·, ·), respectively. Thus, the compensated measures N˜ 1 (·, ·) = N1 (·, ·) − ν1 (·, ·) and N˜ 2 (·, ·) = N2 (·, ·) − N (t ) ν2 (·, ·) are related to the compound Poisson processes i=11 Zi and

N2 (t ) i =1

0

N1 (t )

z N˜ 1 (ds, dz ) = 0

 t 0

Yi as follows

∞

 t



Zi − E

i=1

Most of the existing literature on the optimal investment problem under the mean–variance criterion considers the optimality of the solution at the initial time, which is given by



γ

u∈Π

2



Var0,x0 [X u (T )] ,

(4)

where Et ,x [·] = E[·|X u (t ) = x], Vart ,x [·] = Var[·|X u (t ) = x], Π is the corresponding admissible set and γ > 0 is the riskaverse coefficient of the insurer. In Kryger and Steffensen (2010), problem (4) is called the mean–variance optimization problem with precommitment due to E0,x0 [X u (T )] in the variance term not being updated at subsequent dates. The corresponding optimal strategy is called the precommitment strategy, which is timeinconsistent and only optimal at the initial time. However, time-consistency of strategies is a basic requirement for rational investors in many situations. Similar to Björk and Murgoci (2010), we formulate a reinsurance-investment problem with the objective changing over time: for any (t , x) ∈ [0, T ] × R, the optimization problem is as follows:



sup Et ,x [X u (T )] − u∈Π

γ 2



Vart ,x [X u (T )] ,

(5)

and the target is to develop the corresponding equilibrium reinsurance-investment strategy,1 which is time-consistent. The above-mentioned framework is the traditional reinsuranceinvestment model, where the insurer is assumed to be ambiguityneutral. However, large numbers of insurers are ambiguity-averse in reality, and want to guard themselves against worst-case scenarios. To incorporate ambiguity-aversion into the mean–variance problem, we assume that the reference model regarding the knowledge of the AAI’s ambiguity is described by the probability measure P, but she is skeptical about this reference model, and hopes to consider some alternative models, which are defined as a class of probability measures equivalent to P as follows (cf. Anderson et al., 1999):

yN˜ 2 (ds, dy) =

 i =1

N (t )  1 

Zi ,

∀t ∈ [0, T ],

Definition 2.1 (Admissible Strategy). A strategy u π (t ))}t ∈[0,T ] is said to be admissible if (1) ∀t ∈ [0, T ], p(t ) ∈ [0, +∞);

Q∗

=

{(p(t ),

T

(2) u is predictable w.r.t. {Ft }t ∈[0,T ] , and Et ,x { 0 [(p(t ))2 + (π (t ))2 ]dt } < ∞, where Q∗ is the chosen model to describe Q∗

∗

the worst case and Et ,x [·] = EQ [·|X u (t ) = x]; (3) ∀(t , x) ∈ [0, T ] × R, Eq. (3) has a pathwise unique solution {X u (t )}t ∈[0,T ] .

i=1

N2 (t )

∞

−1

(3)

−1

∞

= 0

i=1

π (t )yN2 (dt , dy).

+

Q := {Q|Q ∼ P}.

0

 t

∞

sup E0,x0 [X u (T )] −



i=1

{W (t )}t ∈[0,T ] ,



Yi − E

N (t )  2 

Yi ,

i=1

∀t ∈ [0, T ].

1 The properties of the equilibrium strategy can be referred to in Björk and Murgoci (2010), Chen et al. (2014), Björk et al. (2014), and so on.

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

1

Denote by Π the set of all admissible strategies. Suppose that φ(t ) := (φ1 (t ), φ2 (t ), φ3 (t )), t ∈ [0, T ] satisfies the three conditions: (i) φ1 (t ), φ2 (t ) and φ3 (t ) are Ft -measurable, for each t ∈ [0,T ]; (ii) φ2 (t ), φ3 (t ) > 0, for a.a. (t , ω) ∈ [0, T ]× Ω ; and (iii) E exp

T 1 0

2  

φi+1 (t ) + 1 dt

(φ1 (t ))2 dt +

 T i=1 λi 0 φi+1 (t ) ln φi+1 (t ) −

2

< ∞. For each φ(t ) = (φ1 (t ), φ2 (t ), φ3 (t )), t ∈

[0, T ], define a real-valued process {Λφ (t )|t ∈ [0, T ]} as  t  1 t φ φ1 (s)dW (s) − Λ (t ) = exp (φ1 (s))2 ds 2 0 0  t ∞ ln φ2 (ω)N1 (ds, dz ) + 0 0  t ∞ (1 − φ2 (s))ν1 (ds, dz ) + 0 0  t ∞ ln φ3 (s)N2 (ds, dy) + +

sup J u (t , x) = sup inf Jˆu,Q (t , x), u∈Π Q∈Q

Q∈Q

Definition 2.2. For an admissible strategy u∗ = {(p∗ (t ), ∗ π (t ))}t ∈[0,T ] with any fixed chosen initial state (t , x) ∈ [0, T ] × R, we define the following strategy



uε (s) =

 ∞ (1 − φ3 (s))ν2 (ds, dy) .

(6)

= Λφ (T ).

(˜p, π˜ ), u∗ (s),

t ≤ s < t + ε, t + ε ≤ s ≤ T,

where p˜ ∈ R+ , π˜ ∈ R and ε ∈ R+ . If ∀(˜p, π˜ ) ∈ R+ × R, ∗

J u (t , x) − J uε (t , x)

lim inf

ε

ε→0

W (t , x) = J u (t , x).

dW Q (t ) = dW (t ) − φ1 (t )dt ,

C 1,2 ([0, T ] × R)

and Poisson processes N1 (t ) and N2 (t ) become N1 (t ) and N2 (t ) with intensities λ1 φ2 (t ) and λ2 φ3 (t ). As is shown in Branger and Larsen (2013), for tractability and ease of interpretation, the distributions of jumps Zi and Yi are assumed to be known, and are restricted to be identical under P and Q. Furthermore, the dynamics of the wealth process under Q are Q

dX u (t ) = [r0 X u (t ) + (µ − r0 )π (t ) + (θ − η)λ1 µZ

∗

For convenience, we first provide some notations. Let

= {ψ(t , x)|ψ(t , ·) is once continuously differentiable on [0, T ] and ψ(·, x) is twice continuously differentiable on R} , D1p,2 ([0, T ] × R)  = ψ(t , x)|ψ(t , x) ∈ C 1,2 ([0, T ] × R) and all once partial derivatives satisfy the polynomial growth condition on R} . Before giving the verification theorem, we define a variational operator Au,φ : for ∀(t , x) ∈ [0, T ]× R, ∀ψ(t , x) ∈ C 1,2 ([0, T ]× R), denote

+ (1 + η)p(t )λ1 µZ + π (t )σ φ1 (t )]dt  ∞ Q Q + π (t )σ dW (t ) − p(t )zN1 (dt , dz )

Au,φ ψ(t , x) = ψt (t , x) + [r0 x + (µ − r0 )π + (θ − η)λ1 µZ

0

+ (1 + η)pλ1 µZ + π σ φ1 ]ψx (t , x)

∞



π (t )yN2Q (dt , dy),

+

(7)

1

+ π 2 σ 2 ψxx (t , x) + λ1 φ2 EQ [ψ(t , x − pz ) − ψ(t , x)]

−1

where N1 (dt , dz ) and N2 (dt , dy) are Poisson random measures under measure Q. Denote that Q

2

Q

Jˆu,Q (t , x)

(φ1 (s))2 λ1 (φ2 (s) ln φ2 (s) − φ2 (s) + 1) + 2ϕ1 (s) ϕ2 (s) t   λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) + ds ϕ3 (s) γ + EQt ,x [X u (T )] − VarQt ,x [X u (T )],

≥ 0,

then u∗ is called an equilibrium strategy and the equilibrium value function W (t , x) is defined as

dP FT According to Girsanov’s Theorem, under the alternative measure Q, the stochastic process {W Q (t )} is a standard Brownian motion, where Q

(9)

and the following definition gives the statements of equilibrium strategy and equilibrium value function.

According to the definition of φ(t ), we know that Λ (t ) is a Pmartingale. For each φ(t ), a new alternative measure Q absolutely continuous to P on FT is defined by putting

 

Following Anderson et al. (2003), Maenhout (2004) and Yi et al. (2015a), we assume that the insurer seeks a robust optimal control, which is the best choice among some worst-case models, i.e.,

J u (t , x) = inf Jˆu,Q (t , x),

φ

dQ 

(8)

where

−1

0

(φ1 (t ))2 + λ1 (φ2 (t ) ln φ2 (t ) − φ2 (t ) + 1)  + λ2 (φ3 (t ) ln φ3 (t ) − φ3 (t ) + 1) dt .

u∈Π

−1

0

 t

2

141

 T 

= EQt ,x

2 where ϕ1 (t ), ϕ2 (t ), ϕ3 (t ) are nonnegative and capture the AAI’s ambiguity aversions, that is, the larger ϕ1 (t ), ϕ2 (t ) and ϕ3 (t ), the more ambiguity-averse the AAI. The deviations from the reference measure are penalized by the first three terms in the expectation, which depends on the relative entropy arising from the diffusion and jump risks. In Appendix A, we show that the increase in relative entropy from t to t + dt equals

+ λ2 φ3 EQ [ψ(t , x + π y) − ψ(t , x)]. Theorem 2.3 (Verification Theorem). For problem (9), if there exist real value functions V (t , x) and g (t , x) ∈ D1p,2 ([0, T ] × R) satisfying the following conditions: ∀(t , x) ∈ [0, T ] × R, sup inf

u∈Π Q∈Q



Au,φ V (t , x) − Au,φ

γ 2

(g (t , x))2

(φ1 (t ))2 2ϕ1 (t ) λ1 (φ2 (t ) ln φ2 (t ) − φ2 (t ) + 1) + ϕ2 (t )  λ2 (φ3 (t ) ln φ3 (t ) − φ3 (t ) + 1) + = 0, ϕ3 (t ) V (T , x) = x, + γ g (t , x)Au,φ g (t , x) +

Au

∗ ,φ ∗

g (t , x) = 0,

g ( T , x) = x,

(10)

(11)

142

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152



and

× (1 + η)µZ e

 γ (u∗ , φ ∗ ) := arg sup inf Au V (t , x) − Au (g (t , x))2 2

u∈Π Q∈Q

π (t ) = ∗

 λ2 (φ3 (t ) ln φ3 (t ) − φ3 (t ) + 1) + , ϕ3 (t ) u∗

(12)

then W (t , x) = V (t , x), E [X (T )] = g (t , x) and u is the robust equilibrium reinsurance-investment strategy. Q

γ σZ2 (p∗ (t ))2 e2r0 (T −t ) 2

 )

− µZ , (14)

µ − r0 λ2 (−µY + γ σY2 π ∗ (t )er0 (T −t ) ) − β1 + γ σ2 σ2  2r ( T − t ) 2 ∗ 2 0 r (T −t ) γ σY (π (t )) e ∗ + ) 2 × eβ3 (−µY π (t )e 0 , (15)

(φ1 (t ))2 + γ g (t , x)Au g (t , x) + 2ϕ1 (t ) λ1 (φ2 (t ) ln φ2 (t ) − φ2 (t ) + 1) + ϕ2 (t )

∗

The proof of the verification theorem can be adapted from Theorem 4.1 of Björk and Murgoci (2010) and Theorem 1 of Kryger and Steffensen (2010). We omit it here.

e−r0 (T −t )

−β2 (µZ p∗ (t )er0 (T −t ) +



and the corresponding equilibrium value function is

(η − θ )λ1 µZ

V (t , x) = xer0 (T −t ) +

r0

T



l1 (s)ds +

+



(1 − er0 (T −t ) )

T

l2 (s)ds.

(16)

t

t

The worst-case measure is given by

φ1∗ (t ) = −β1 π ∗ (t )σ er0 (T −t ) ,

3. Solution to the model This section is devoted to deriving the robust equilibrium reinsurance-investment strategy under the mean–variance criterion. Following the idea suggested by Maenhout (2004, 2006), we have to impose ambiguity-aversion preference parameters ϕ1 (t ), ϕ2 (t ) and ϕ3 (t ), which render problem (9) analytically tractable, and ensure that the penalty in problem (9) is reasonable. We assume that ϕ1 (t ), ϕ2 (t ) and ϕ3 (t ) are fixed and stateindependent functions, i.e., ϕ1 (t ) = β1 , ϕ2 (t ) = β2 and ϕ3 (t ) = β3 , where β1 , β2 and β3 are nonnegative (cf. Maenhout, 2004). Then, we can rewrite the HJB equation (10) as



sup inf Vt (t , x) + [r0 x + (µ − r0 )π + (θ − η)λ1 µZ

φ2∗ (t ) = e φ3∗ (t ) = e

(17)

γ σ 2 (p∗ (t ))2 e2r0 (T −t ) β2 (µZ p∗ (t )er0 (T −t ) + Z 2

)

,

γ σ 2 (π ∗ (t ))2 e2r0 (T −t ) β3 (−µY π ∗ (t )er0 (T −t ) + Y 2

(18) )

,

(19)

where l1 (s) = (1 + η)p∗ (s)λ1 µZ er0 (T −s)

λ1 + β2

 1−e

β2 (µZ p∗ (s)er0 (T −s) +

l2 (s) = (µ − r0 )π ∗ (s)er0 (T −s) −

λ2 + β3

u∈Π Q∈Q

+ (1 + η)pλ1 µZ + π σ φ1 ]Vx (t , x)

 1−e

γ σZ2 (p∗ (s))2 e2r0 (T −s) 2

 )

,

(20)

(π ∗ (s))2 σ 2 (β1 + γ )e2r0 (T −s) 2

β3 (−µY π ∗ (s)er0 (T −s) +

γ σY2 (π ∗ (s))2 e2r0 (T −s) 2

 )

.

(21)

1

+ π 2 σ 2 [Vxx (t , x) − γ (gx (t , x))2 ] 2  γ + λ1 φ2 E V (t , x − pz ) − (g (t , x − pz ))2 2  + γ g (t , x)g (t , x − pz )   γ − λ1 φ2 V (t , x) + (g (t , x))2 2  γ + λ2 φ3 E V (t , x + π y) − (g (t , x + π y))2 2  + γ g (t , x)g (t , x + π y)

Proof. See Appendix C.

Remark 3.3 (Ambiguity-Neutral Insurer (ANI) Case). If all of the ambiguity-aversion coefficients β1 , β2 and β3 equal 0, i.e., β1 = β2 = β3 = 0, our model reduces to an optimization problem for an ANI. The wealth process of the ANI under measure P is described by



dX u1 (t ) = r0 X u1 (t ) + (µ − r0 )π1 (t )



+ (θ − η)λ1 µZ + (1 + η)p1 (t )λ1 µZ ] dt  ∞ + π1 (t )σ dW (t ) − p1 (t )zN1 (dt , dz )

The following theorem provides the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function.

0

 Theorem 3.1. For the optimal reinsurance-investment problem (9) with the wealth process (7), the robust equilibrium strategy is given by e

∞

π1 (t )yN2 (dt , dy),

+

(22)

−1

and the optimization problem is

−r 0 ( T −t )

γ σZ2



Next, we discuss some special cases of our model, i.e., the ambiguity-neutral insurer (ANI) case, the no jump case and the investment-only case, and provide the corresponding equilibrium strategies and equilibrium value functions.

(13)

p∗ (t ) =



Proposition 3.2. Both Eqs. (14) and (15) have unique positive roots, i.e., there exist unique p∗ (t ) ∈ [0, +∞) and π ∗ (t ) ∈ [0, +∞) that satisfy Eqs. (14) and (15), respectively.

φ2 − λ2 φ3 V (t , x) + (g (t , x))2 + 1 2 2β1  λ1 (φ2 ln φ2 − φ2 + 1) λ2 (φ3 ln φ3 − φ3 + 1) + + = 0. β2 β3 

γ

Proof. See Appendix B.

sup

u1 ∈Π



Et ,x [X u1 (T )] −

γ 2



Vart ,x [X u1 (T )] .

(23)

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

The equilibrium reinsurance-investment strategy u∗1 = {(p∗1 (t ), π1∗ (t ))}t ∈[0,T ] and the corresponding equilibrium value function V1 (t , x) for the ANI who ignores model uncertainty are given by p∗1 (t ) =

π1∗ (t ) =

ηµZ e−r0 (T −t ) , γ σZ2

(24)

(µ − r0 + λ2 µY )e−r0 (T −t ) , γ (σ 2 + λ2 σY2 )

(25)

V1 (t , x) = xer0 (T −t ) +

+

(η − θ )λ1 µZ r0

143

and the corresponding equilibrium value function V2 (t , x) is V2 (t , x) = xer0 (T −t ) +

+

(η − θ )λ1 µZ r0

λ1 η2 µ2Z (T − t ) + 2σZ2 (β2 + γ )

λ1 η2 µ2Z (T − t ) (µ − r0 + λ2 µY )2 (T − t ) + . (26) 2γ σZ2 2γ (σ 2 + λ2 σY2 )

We find that the equilibrium reinsurance-investment strategy for the ANI in Eqs. (24) and (25) can reduce to the result in Zeng et al. (2013), if C–L model is considered in Zeng et al. (2013), which means that our model extends the equilibrium strategy in Zeng et al. (2013) to the case of robust optimal formulation.

T



l3 (s)ds,

(32)

t

where l3 (s) = (µ − r0 )π2∗ (s)er0 (T −s) −

(1 − er0 (T −t ) )

(1 − er0 (T −t ) )

λ2 + β3

 1−e

(π2∗ (s))2 σ 2 (β1 + γ )e2r0 (T −s) 2

γ σ 2 (π ∗ (s))2 e2r0 (T −s) β3 (−µY π2∗ (s)er0 (T −s) + Y 2 2 )

 T 

 Q

sup inf Et ,x Q∈Q

u3 ∈Π

t

  λ1 (φ2 (s) ln φ2 (s) − φ2 (s) + 1) ds + ϕ2 (s)  γ + EQt ,x [X u3 (T )] − VarQt ,x [X u3 (T )]

dR(t ) = [(θ − η)λ1 µZ + ηp2 (t )λ1 µZ ]dt

dX u3 (t ) = [r0 X u3 (t ) + (µ − r0 )π3 (t ) + (θ − η)λ1 µZ

+ (1 + η)p3 (t )λ1 µZ + π3 (t )σ φ1 (t )]dt  ∞ Q + π3 (t )σ dW Q (t ) − p3 (t )zN1 (dt , dz ).

and with consideration of investment, the wealth process of the AAI under the measure Q becomes

e−r0 (T −t )

p∗3 (t ) = (28)

−1

where {W0 (t )}t ∈[0,T ] is a standard Brownian motion and is

γ σZ2

 × (1 + η)µZ e

Q

i=1

(φ1 (s)) (φ2 (s)) + Q ∈ Q 2 ϕ ( s ) 2ϕ2 (s) u2 ∈Π 1 t   λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) + ds ϕ3 (s)  γ + EQt ,x [X u2 (T )] − VarQt ,x [X u2 (T )] .  T 



2

Yi . The corresponding

(t ))2 e2r0 (T −t ) γ σZ2 (p∗ 3 2

 )

− µZ , (36)

(µ − r0 )e−r0 (T −t ) , σ 2 (β1 + γ )

V3 (t , x) = xer0 (T −t ) +

Q

+

(37)

(η − θ )λ1 µZ r0

(µ − r0 )2 (T − t ) + 2σ 2 (β1 + γ )

(1 − er0 (T −t ) ) T



l4 (s)ds,

(38)

t

where (29)

Then, by some similar calculations, the robust equilibrium reinsurance-investment strategy u∗2 = {(p∗2 (t ), π2∗ (t ))}t ∈[0,T ] is given by

ηµZ e−r0 (T −t ) p2 (t ) = 2 , (30) σZ (β2 + γ )  e−r0 (T −t ) µ − r0 λ2 (−µY + γ σY2 π2∗ (t )er0 (T −t ) ) π2∗ (t ) = − 2 β1 + γ σ σ2  2 ∗ 2 2r0 (T −t ) r (T −t ) γ σY (π2 (t )) e ∗ + ) 2 × eβ3 (−µY π2 (t )e 0 , (31) ∗

π3∗ (t ) =

−β2 (µZ p∗3 (t )er0 (T −t ) +

and the corresponding equilibrium value function V3 (t , x) is

2

sup inf Et ,x

2

(35)

0

Then the robust equilibrium reinsurance-investment strategy u∗3 = {(p∗3 (t ), π3∗ (t ))}t ∈[0,T ] is given by

dX u2 (t ) = [r0 X u2 (t ) + (µ − r0 )π2 (t ) + (θ − η)λ1 µZ

independent of {W Q (t )}t ∈[0,T ] and optimization problem becomes

(34)

2

(27)

N2Q (t )

(33)

(φ1 (s))2 2ϕ1 (s)

and

 + ηp2 (t )λ1 µZ + π2 (t )σ φ1 (t ) + λ1 σZ φ2 (t )]dt  + π2 (t )σ dW Q (t ) + λ1 p2 (t )σZ dW0Q (t )  ∞ + π2 (t )yN2Q (dt , dy),

.

(ii) Second, we consider the optimal reinsurance-investment problem for an AAI in which there is no jump in the risky asset’s price process. If no such jump occurs, i.e., λ2 = 0 in our model, the optimization problem and the wealth process become

Remark 3.4 (No Jump Case). (i) First, we consider the optimal reinsurance-investment problem for an AAI in which there is no jump in the surplus process. If no such jump occurs, following Grandell (1991) and Promislow and Young (2005), we can approximate the surplus process by the following diffusion model

 + λ1 p2 (t )σZ dW0Q (t ),



l4 (s) = (1 + η)p∗3 (s)λ1 µZ er0 (T −s)

λ1 + β2

 1−e

β2 (µZ p∗3 (s)er0 (T −s) +

γ σZ2 (p∗ (s))2 e2r0 (T −s) 3 2

 )

.

(39)

(iii) Third, we consider the optimal reinsurance-investment problem for an AAI in a model without jumps. If there is no jump in both the surplus process of the insurer and the price process of the risky asset, the wealth process and optimization problem become dX u4 (t ) = [r0 X u4 (t ) + (µ − r0 )π4 (t ) + (θ − η)λ1 µZ

 + ηp4 (t )λ1 µZ + π4 (t )σ φ1 (t ) + λ1 σZ φ2 (t )]dt  + π4 (t )σ dW Q (t ) + λ1 p4 (t )σZ dW0Q (t )

(40)

144

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

and

+ π6 (t )σ dW (t ) −

(φ1 (s))2 (φ2 (s)) + 2 ϕ ( s ) 2ϕ2 (s) u4 ∈Π 1 t  γ + EQt ,x [X u4 (T )] − VarQt ,x [X u4 (T )] . 



 T

 2

Q

sup inf Et ,x Q∈Q

ds

Q

(41)

(42)

π4∗ (t ) =

(µ − r0 )e−r0 (T −t ) , σ 2 (β1 + γ )

(43)

r0

(µ − r0 )2 (T − t ) λ1 η2 µ2Z (T − t ) + + . 2σ 2 (β1 + γ ) 2σZ2 (β2 + γ )

(44)

(iv) Finally, we consider the optimal reinsurance-investment problem for an ANI in a model without jumps. If there is no jump in the model for the ANI, the wealth process reduces to dX u5 (t ) = r0 X u5 (t ) + (µ − r0 )π5 (t ) + (θ − η)λ1 µZ



 λ1 p5 (t )σZ dW0 (t ). (45)

For optimization problem



γ

Et ,x [X u5 (T )] −

2

ηµZ e γ σZ2

π5∗ (t ) =

(µ − r0 )e−r0 (T −t ) , γσ2

,

+

2 Z

sup inf

T



(47)

π6 (t ) = ∗



e−r0 (T −t )

β1 + γ

λ2 (−µY + γ σY2 π6∗ (t )er0 (T −t ) ) µ − r0 − σ2 σ2 

r (T −t )

× eβ3 (−µY π6 (t )e 0 ∗

+

γ σY2 (π6∗ (t ))2 e2r0 (T −t ) 2

r0

(1 − er0 (T −t ) ) 2

(49)

We find that the robust equilibrium investment strategies given by Eqs. (37) and (43) are similar to that in Maenhout (2004), which considers the robust portfolio selection with the power utility maximization. Moreover, the equilibrium reinsurance-investment strategy for the ANI given by Eqs. (47) and (48) reduces to that in Zeng and Li (2011). In other words, the model in Zeng and Li (2011) can be taken as a special case of our model.

V6 (t , x) = xer0 (T −t ) −

(1 + θ )λ1 µZ r0

T

l5 (s)ds +

+

dX u6 (t ) = [r0 X u6 (t ) + (µ − r0 )π6 (t ) + (θ − η)λ1 µZ

+ (1 + η)λ1 µZ + π6 (t )σ φ1 (t )]dt

,

(52)

(1 − er0 (T −t ) )

T



l6 (s)ds,

(53)

t

where



λ1 l5 (s) = β2

1−e

β2 (µZ er0 (T −s) +

γ σZ2 e2r0 (T −s) 2

l6 (s) = (µ − r0 )π6∗ (s)er0 (T −s) −

λ2 + β3

 1−e

 )

,

(54)

(π6∗ (s))2 σ 2 (β1 + γ )e2r0 (T −s) 2

β3 (−µY π6∗ (s)er0 (T −s) +

γ σY2 (π6∗ (s))2 e2r0 (T −s) 2

 )

.

(55)

(ii) On the other hand, we consider the optimal investment problem in a model without an insurance business. If we do not consider the insurance business, our model reduces to an optimization problem for an ambiguity-averse investor, the wealth process of the ambiguity-averse investor and the corresponding optimization problem become dX u7 (t ) = [r0 X u7 (t ) + (µ − r0 )π7 (t ) + π7 (t )σ φ1 (t )]dt

+ π7 (t )σ dW (t ) +



∞

π7 (t )yN2Q (dt , dy)

Q

(56)

−1

and

 Remark 3.5 (Investment-only Case). (i) On the one hand, we consider the optimal investment problem for an AAI in a model without reinsurance. If there is no reinsurance in our model, i.e., all of the claim risks are retained by the AAI, then the wealth process becomes

)

and the corresponding equilibrium value function V6 (t , x) is

(48)

(η − θ )λ1 µZ

(51)

2

(46)

λ1 η µ (T − t ) (µ − r0 ) (T − t ) + . 2γ σ 2 2γ σZ2 2



(φ1 (s))2 (φ2 (s))2 + 2ϕ1 (s) 2ϕ2 (s) u6 ∈Π Q∈Q t   λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) + ds ϕ3 (s)  γ Q Q u6 u6 + Et ,x [X (T )] − Vart ,x [X (T )] , Q E t ,x

t

p∗5 (t ) =

V5 (t , x) = xer0 (T −t ) +







Vart ,x [X u5 (T )] ,

the equilibrium reinsurance-investment strategy u∗5 = {(p∗5 (t ), π5∗ (t ))}t ∈[0,T ] and the corresponding equilibrium value function V5 (t , x) are respectively given by −r0 (T −t )

For optimization problem

the robust equilibrium investment strategy u∗6 = {π6∗ (t )}t ∈[0,T ] is given by

(1 − er0 (T −t ) )

+ ηp5 (t )λ1 µZ ] dt + π5 (t )σ dW (t ) +

(50)

−1

ηµZ e−r0 (T −t ) , σZ2 (β2 + γ )

(η − θ )λ1 µZ

π6 (t )yN2Q (dt , dy).

+

p∗4 (t ) =

V4 (t , x) = xer0 (T −t ) +

∞



Furthermore, the robust equilibrium reinsurance-investment strategy u∗4 = {(p∗4 (t ), π4∗ (t ))}t ∈[0,T ] and the corresponding equilibrium value function V4 (t , x) are respectively given by

u5 ∈Π

zN1 (dt , dz ) 0



2

sup

∞



Q

sup inf

u7 ∈Π Q∈Q

Q E t ,x

 t

T



(φ1 (s))2 2ϕ1 (s)

  λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) + ds ϕ3 (s)  γ Q Q u7 u7 + Et ,x [X (T )] − Vart ,x [X (T )] , 2

(57)

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

145

Table 1 Values of model parameters in the numerical examples. r0

µ

σ

θ

η

γ

β1

β2

β3

λ1

λ2

λZ

λY

x

T

t

0.03

0.08

0.25

0.10

0.20

1

0.5

0.5

0.7

1

1

1

1

1

10

0

Fig. 1. Effects of parameters β2 , γ , λZ and η on p∗ (t ).

respectively. Then, the robust equilibrium investment strategy u∗7 = {π7∗ (t )}t ∈[0,T ] is given by

π7 (t ) = ∗

e−r0 (T −t )

β1 + γ



µ − r0 λ2 (−µY + γ σY2 π7∗ (t )er0 (T −t ) ) − 2 σ σ2 

r (T −t )

× eβ3 (−µY π7 (t )e 0 ∗

+

γ σY2 (π7∗ (t ))2 e2r0 (T −t ) 2

)

,

Fig. 1 shows the effects of the ambiguity aversion coefficient

(58)

and the corresponding equilibrium value function V7 (t , x) is V7 (t , x) = xer0 (T −t ) +

T



l7 (s)ds,

(59)

t

where l7 (s) = (µ − r0 )π7∗ (s)er0 (T −s) −

λ2 + β3

 1−e

(π7∗ (s))2 σ 2 (β1 + γ )e2r0 (T −s)

β3 (−µY π7∗ (s)er0 (T −s) +

2 γ σY2 (π7∗ (s))2 e2r0 (T −s) 2

 )

.

4.1. Effects of model parameters on the robust equilibrium reinsuranceinvestment strategy

(60)

We find that the robust equilibrium investment strategies given in Eqs. (37) and (43) are similar to that in Branger and Larsen (2013), which considers the robust optimal investment strategy to maximize the expected utility from terminal wealth with jumps. 4. Numerical simulations This section provides some numerical examples to illustrate the effects of model parameters on the robust equilibrium reinsurance-investment strategy and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance. Suppose that both the claim size Zi and the jump size Yi follow exponential distributions with parameters λZ and λY , respectively, i.e., the density functions of Zi and Yi are given by f (z ) = λZ e−λZ z , z ≥ 0 and g (y) = λY e−λY (y+1) , y ≥ −1. Throughout the numerical analyses, unless otherwise stated, the basic parameters are given in Table 1 (cf. Liang et al., 2012; Branger and Larsen, 2013; Yi et al., 2013).

β2 , the risk aversion coefficient γ , the parameter of the jump’s distribution function in the surplus process λZ and the reinsurer’s safety loading η on the robust equilibrium reinsurance strategy p∗ (t ). From Fig. 1, we find that p∗ (t ) decreases w.r.t. β2 and γ , whereas it increases w.r.t. λZ and η, because the AAI with higher ambiguity aversion level β2 is prone to purchasing more reinsurance to disperse the underlying insurance business risks. As the AAI becomes more risk averse, i.e., as γ increases, she would like to cede more risks to the reinsurer to spread the risks between both parties. When λZ increases, the mean and variance of each claim Zi become smaller, and hence, with the same risk tolerance, the AAI can retain more insurance business. As η increases, to decrease the expensive payment for reinsurance, the AAI prefers to take more insurance business and raise the retention level of reinsurance. Fig. 2 illustrates the effects of the ambiguity aversion coefficients β1 and β3 , jump intensity and the parameter of the jump’s distribution function of the risky asset’s price process λ2 and λY on the robust equilibrium investment strategy π ∗ (t ). From Fig. 2, we find that the effects of β3 and λ2 on π ∗ (t ) are not obvious and are difficult to determine. So we further provide Table 2 to show the slight changes. As is shown in Fig. 2 and Table 2, π ∗ (t ) decreases w.r.t. β1 and β3 , because due to the model uncertainty, if the AAI has higher ambiguity aversion coefficients, she will reduce the investment in the risky asset. Besides, π ∗ (t ) decreases w.r.t. λ2 and increases w.r.t. λY . This can be attributed to the fact that the larger the λ2 , the stronger the intensity of jumps in the risky asset’s price process and the higher the risky asset, and therefore, the less money is invested in the risky asset. Besides, a larger λY implies that the mean and variance of Yi become smaller, and then with the same risk tolerance, the AAI will invest more in the risky asset. 4.2. Effects of model parameters on the utility loss functions This subsection uses numerical examples to discuss the utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance for the AAI.

146

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

Fig. 2. Effects of parameters β1 , β3 , λY and λ2 on π ∗ (t ). Table 2 Values of π ∗ (t ) with different β3 and λ2 .

β3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 0.3713

π (t )

0.3722

0.3721

0.3720

0.3719

0.3718

0.3717

0.3715

λ2

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

π ∗ (t )

0.3719

0.3718

0.3717

0.3716

0.3715

0.3714

0.3713

0.3712

∗

First, we study the utility loss from ignoring model uncertainty. Suppose that the AAI does not take the optimal strategy u∗ = {(p∗ (t ), π ∗ (t ))}t ∈[0,T ] given in Theorem 3.1, but makes decision as if she were an ANI, i.e., the AAI follows the strategy u∗1 = {(p∗1 (t ), π1∗ (t ))}t ∈[0,T ] given in Remark 3.3. The equilibrium value function for the AAI following the given strategy u∗1 is defined by

 V8 (t , x) = inf

Q∈Q

+

Q E t ,x



T t



(φ1 (s))2 2ϕ1 (s)

2

+

(η − θ )λ1 µZ r0

(61)

(1 − er0 (T −t ) )

(µ − r0 )(µ − r0 + λ2 µY )(T − t ) γ (σ 2 + λ2 σY2 )

σ 2 (µ − r0 + λ2 µY )2 (β1 + γ )(T − t ) 2γ 2 (σ 2 + λ2 σY2 )2   ηµ2 η2 µ2 Z ) β2 ( Z2 + λ1 (T − t )  η(1 + η)λ1 µ2Z (T − t ) γ σZ 2γ σ 2  Z + 1−e + β2 γ σZ2

−

On the basis of the above derivations, we derive the equilibrium value functions under different cases. To make a fair comparison of these equilibrium value functions, we assume that the appreciation and volatility rates of the risky asset with no jump are the same 2 as those with jumps, i.e., µno jump = µ + λ2 E[Yi ] and σno jump =

Furthermore, we defined the utility loss from ignoring model uncertainty as follows:

Note that φ1 (t ), φ2 (t ) and φ3 (t ), which describe the alternative model, are still determined endogenously and depend on the reinsurance-investment strategy. Different from the optimal case, the reinsurance-investment strategy is now prespecified. The functions ϕ1 (t ), ϕ2 (t ) and ϕ3 (t ) are also defined as ϕ1 (t ) = β1 , ϕ2 (t ) = β2 and ϕ3 (t ) = β3 . Similar to the above derivations, we have the equilibrium value function under the suboptimal strategy V8 (t , x) = xer0 (T −t ) +

(62)

σ 2 + λ2 E[Yi2 ]. From Table 3, we find that the equilibrium value functions under the case without jumps (V4 (t , x)), the case without reinsurance (V6 (t , x)) and the case with a suboptimal strategy (V8 (t , x)) are less than that of our model (V (t , x)).

λ1 (φ2 (s) ln φ2 (s) − φ2 (s) + 1) ϕ2 (s)

  λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) + ds ϕ3 (s)  γ Q Q u∗ u∗ 1 1 + Et ,x [X (T )] − Vart ,x [X (T )] .

  2 2 −µ (µ−r +λ µ ) σ (µ−r0 +λ2 µY ) ) β3 ( Y 2 0 22 Y + Y λ2 (T − t )  2 2 2 γ (σ +λ2 σY ) 2γ (σ +λ2 σ ) . Y + 1−e β3

L1 (t ) := 1 −

V8 (t , x) V ( t , x)

,

(63)

where V (t , x) and V8 (t , x) are given by Eqs. (16) and (62), respectively. Fig. 3 discloses the utility loss from ignoring model uncertainty for the AAI as an increasing function of the remaining time T − t, which indicates that the utility loss decreases as the remaining time T − t diminishes. Comparing the two figures in Fig. 3, we find that the difference of L1 (t ) is rather large, so the effect of the remaining time T − t on the utility loss L1 is significant. Moreover, from Fig. 3 and Table 4, we can see that ambiguity aversion coefficients β1 , β2 and β3 have positive effects on L1 (t ), implying that the utility loss is higher for the AAI with less information about model P (larger β1 , β2 and β3 ) than for the AAI with more information about model P (smaller β1 , β2 and β3 ). Moreover, the utility loss L1 (t ) is more sensitive to the ambiguity-aversion coefficients for the risky asset’s price process β1 and β3 than that for the surplus process β2 . Fig. 4 reveals that the utility loss L1 (t ) increases w.r.t. λ1 and λ2 , whereas it decreases w.r.t. λZ and λY . Larger λ1 and λ2 imply that the intensities of the jumps in the surplus process and the risky asset’s price process become larger; therefore, the AAI will suffer from more utility losses. In addition, the variance of Zi and

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

147

Table 3 Equilibrium value functions under different cases. t

0

1

2

3

4

5

6

7

8

V (t , x ) V 4 (t , x ) V 6 (t , x ) V 8 (t , x )

2.9585 0.9763 0.1538 2.6660

2.7741 0.9901 0.2422 2.5109

2.5869 1.0012 0.3353 2.3530

2.3971 1.0096 0.4257 2.1924

2.2047 1.0154 0.5135 2.0293

2.0098 1.0187 0.6088 1.8636

1.8124 1.0195 0.6816 1.6955

1.6127 1.0203 0.7521 1.5250

1.4107 1.0210 0.8103 1.3522

Table 4 Values of L1 (t ) with different β2 .

β2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L 1 (t )

NaN

0.0941

0.0945

0.0951

0.0958

0.0967

0.0977

0.0988

Fig. 3. Effects of parameters T − t , β1 , β2 and β3 on L1 (t ).

Fig. 4. Effects of parameters λ1 , λZ , λ2 and λY on L1 (t ).

Yi increase with the decrease in λZ and λY ; thus, there is more volatility uncertainty regarding the surplus process and the risky asset’s price process, and furthermore, the utility loss function becomes larger. Second, we investigate the utility loss from ignoring jump risks, which is defined as L2 (t ) := 1 −

V4 (t , x) V (t , x)

,

(64)

where V (t , x) and V4 (t , x) are given by Eqs. (16) and (44), respectively. Comparing Figs. 3–4 with Figs. 5–6, we find that the effects of the model parameters on the utility loss L2 (t ) become heavier than those on L1 (t ). The utility loss from ignoring jump risks L2 (t ) is clearly larger than that from ignoring model uncertainty L1 (t ),

which implies that in portfolio choice problems, jump risks should not be ignored. In reality, from the time of the financial crisis in 2008, frequent jumps have occurred, especially cojumping in the financial market (cf. Bollerslev et al., 2008; Dungey and Hvozdyk, 2012). Therefore, jump risks are universal in the financial market, and they should always be taken into account in our model. In addition, we also find that the effect of λY on L2 (t ) is not as obvious as that on L1 (t ), and according to Table 5, we clearly know that L2 (t ) decreases w.r.t. λY as well. Finally, we introduce the utility loss to measure the effect of reinsurance on the equilibrium value function, which is defined as L3 (t ) := 1 −

V6 ( t , x )

, (65) V (t , x) where V (t , x) and V6 (t , x) are given by Eqs. (16) and (53), respectively.

148

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

Fig. 5. Effects of parameters T − t , β1 , β2 and β3 on L2 (t ).

Fig. 6. Effects of parameters λ1 , λZ , λ2 and λY on L2 (t ).

Fig. 7. Effects of parameters t , β1 , β2 and β3 on L3 (t ). Table 5 Effects of β2 and λY on L2 (t ) and L3 (t ).

β2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L2 (t ) L3 (t )

NaN NaN

0.6648 0.2223

0.6667 0.2805

0.6683 0.3506

0.6696 0.4354

0.6708 0.5385

0.6718 0.6642

0.6727 0.8182

λY

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

L2 (t ) L3 (t )

0.7727 0.8189

0.7727 0.8188

0.7727 0.8187

0.7726 0.8186

0.7726 0.8185

0.7725 0.8184

0.7725 0.8183

0.7724 0.8182

Comparing Figs. 7–8 with Figs. 3–6, and according to Tables 4–5, we find that the effects of some of the model parameters on

the utility loss L3 (t ) are also similar to those on L1 (t ) and L2 (t ). Different from the utility losses that occur from ignoring model

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

149

Fig. 8. Effects of parameters λ1 , λZ , λ2 and λY on L3 (t ).

uncertainty and jump risks, the utility loss L3 (t ) is more sensitive to the ambiguity aversion coefficients for jumps in the surplus process β2 and the risky asset’s price process β3 than to that for diffusion β1 , which implies that the jump risks also have an important effect on the utility loss from ignoring reinsurance. From Figs. 1–8, we find that both model uncertainty and jump risks have significant effects on the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function. Moreover, the utility losses under the three cases are obvious, and the utility losses from ignoring jump risks and prohibiting reinsurance are larger than that from ignoring model uncertainty, which shows the importance of the consideration of jump risks and reinsurance in the robust equilibrium reinsurance-investment problem. 5. Conclusion In this paper, we consider a robust equilibrium reinsuranceinvestment problem in a model with jumps for an AAI who worries about model uncertainty and aims to develop robust equilibrium strategies. The surplus process of the insurer is assumed to follow the classical C–L process, and the insurer is allowed to purchase proportional reinsurance or acquire new business and invest her surplus in a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. Meanwhile, the insurer may lack confidence in the model describing the financial and reinsurance markets, in which case we formulate a systematic analysis of the robust reinsurance-investment problem. By applying a stochastic dynamic programming approach, explicit expressions for the robust equilibrium reinsurance-investment strategy under the mean–variance criterion and the corresponding equilibrium value function are obtained. Some special cases of our model and the associated results are also discussed, and the economic implications of our findings and numerical examples are presented. We also numerically analyze the utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance for the AAI. We find that: (i) the robust equilibrium reinsurance-investment strategy for the AAI is affected by her attitude towards ambiguity, such that the AAI facing model uncertainty has a smaller optimal strategy than an ANI; (ii) the effects of ambiguity-aversion coefficients for the surplus process on the utility losses from ignoring model uncertainty and jump risks are different from those for the risky asset’s price process; (iii) the effects of ambiguity-aversion coefficients on the utility loss from prohibiting reinsurance are different w.r.t. diffusion and jump risks; (iv) the utility loss from ignoring jump risks is larger than that from ignoring model uncertainty, which means that jump risks are very important in the optimization problem and they should not be

ignored; and (v) the AAI suffers from significant utility loss if she ignores reinsurance, which implies that reinsurance is important in risk management. In future research, more complicated models, such as the mean–variance criterion with state-dependent risk aversion and the stochastic volatility model with jumps, can be taken into account, although doing so may make it difficult to obtain a closedform solution. Thus, other methods, such as asymptotic or other practical methods, may be introduced to deal with the robust optimal reinsurance-investment problem. Acknowledgments The authors would be very grateful to referees for their suggestions and this research was supported by the National Natural Science Foundation of China under Grant (Nos. 11301376, 71571195, 71231008, 71201173, 71501050, 71573110), Guangdong Natural Science Funds for Distinguished Young Scholars (No. 2015A030306040) and Natural Science Foundation of Guangdong Province of China (Nos. 2014A030310195, S2013010011959). Appendix A Derivation of relative entropy. The relative entropy is defined as the expectation under the alternative measure of the log Radon–Nikodym derivative defined in Eq. (6). By using Itô’s formula, we have d ln Λφ = φ1 (t )dW (t ) + λ1 (1 − φ2 (t ))dt + λ2 (1 − φ3 (t ))dt 1



∞

− (φ1 (t )) dt + ln φ2 (t )N1 (dt , dz ) 2 0  ∞ + ln φ3 (t )N2 (dt , dy). 2

−1

The relative entropy over the interval from t to t + ε is then given by

 Λφ (t + ε) Λφ (t )  t +ε = EQ φ1 (s)(dW Q (s) + φ1 (s)ds) 

EQ ln

t t +ε

 +

   1 λ1 (1 − φ2 (s)) + λ2 (1 − φ3 (s)) − (φ2 (s))2 ds 2

t t +ε



∞



ln φ2 (s)N˜ 1 (ds, dz )

+ t

0

150

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152 t +ε



Differentiating Eq. (73) w.r.t. u implies

λ1 φ2 (s) ln φ2 (s)ds  t +ε  ∞ ln φ3 (s)N˜ 2 (ds, dy) +

+

t

∗

−1

t t +ε



λ2 φ3 (s) ln φ3 (s)ds

+ t

 =E

t +ε



Q

1 2

t

(φ1 (s)) + λ1 (φ2 (s) ln φ2 (s) − φ2 (s) + 1) 2

Let ε → 0 and we have the continuous-time limit of the relative entropy which is given by Eq. (8). Appendix B Proof of Theorem 3.1. To solve Eqs. (11) and (13), we try to conjecture the solutions in the following forms

g (t , x) = A¯ (t )x + B¯ (t ) A¯ (T ) = 1, B¯ (T ) = 0.

(66) (67)

Vx = A, Vxx = 0,

gt = A¯ t x + B¯ t ,

gx = A¯ , gxx = 0,

(68)



sup inf At x + Bt + [r0 x + (µ − r0 )π + (θ − η)λ1 µZ u∈Π Q∈Q

+ (1 + η)pλ1 µZ + π σ φ1 − λ1 µZ pφ2 + λ2 µY π φ3 ]A γ p2 λ1 σZ2 φ2 A¯ 2 γ π 2 (σ 2 + λ2 σY2 φ3 )A¯ 2 φ2 − − + 1 2 2 2β1  λ1 (φ2 ln φ2 − φ2 + 1) λ2 (φ3 ln φ3 − φ3 + 1) + + = 0. (69) β2 β3 According to the first-order optimality conditions, the functions φ1∗ (t ), φ2∗ (t ) and φ3∗ (t ), which realize the infimum part of Eq. (69) are given by

φ2 = e ∗

β2 (µZ pA+

(70)

γ σZ2 p2 A¯ 2

φ3∗ = eβ3 (−µY π A+

2

)

,

γ σY2 π 2 A¯ 2 2

(71) )

.

2 ∗ 2 ¯2 ∗ A+ γ σZ (p ) A )

¯ β2 (µZ p − λ1 µZ p∗ Ae

)

= 0.

A¯ t + r0 A¯ = 0,

(π ∗ )2 σ 2 (β1 A2 + γ A¯ 2 ) + (1 + η)p∗ λ1 µZ A − 2   γ σZ2 (p∗ )2 A¯ 2 λ1 ∗ ) 2 1 − eβ2 (µZ p A+ + β2   γ σ 2 (π ∗ )2 A¯ 2 λ2 ) β3 (−µY π ∗ A+ Y 2 = 0, + 1−e β3 B¯ t + (µ − r0 )π ∗ A¯ + (θ − η)λ1 µZ A¯ + (1 + η)p∗ λ1 µZ A¯ − (π ∗ )2 σ 2 β1 AA¯ 2 ∗ 2 ¯2 ∗ A+ γ σZ (p ) A )

¯ β2 (µZ p − λ1 µZ p∗ Ae

¯ β3 (−µY π + λ2 µY π ∗ Ae

2

∗ A+

γ σY2 (π ∗ )2 A¯ 2 2

)

= 0.

Considering the boundary conditions, we have A(t ) = er0 (T −t ) , B(t ) =

A¯ (t ) = er0 (T −t ) ,

B¯ (t ) =

(76)

(η − θ )λ1 µZ

(1 − er0 (T −t ) )  T  T + l1 (s)ds + l2 (s)ds, r0

t

u∈Π

π 2 σ 2 (β1 A2 + γ A¯ 2 ) + (1 + η)pλ1 µZ ]A − 2   γ σ 2 p2 A¯ 2 λ1 β2 (µZ pA+ Z 2 ) + 1−e β2   γ σ 2 π 2 A¯ 2 λ2 β3 (−µY π A+ Y 2 ) + 1−e = 0. β3

2

Bt + (µ − r0 )π A + (θ − η)λ1 µZ A

Inserting Eqs. (70)–(72) into Eq. (69) yields



∗ A+

∗

(72)

sup At x + Bt + [r0 x + (µ − r0 )π + (θ − η)λ1 µZ

2

γ σY2 (π ∗ )2 A¯ 2

By separating the variables with and without x, we can derive the following equations At + r0 A = 0,

where V , g , A, B, A¯ and B¯ are short for V (t , x), g (t , x), A(t ), B(t ), A¯ (t ) and B¯ (t ). Substituting Eqs. (66)–(68) into Eq. (13), we have

φ1∗ = −β1 π σ A,

(π ∗ )2 σ 2 (β1 A2 + γ A¯ 2 ) + (1 + η)p∗ λ1 µZ A − 2   γ σ 2 (p∗ )2 A¯ 2 λ1 β2 (µZ p∗ A+ Z 2 ) + 1−e β2   γ σY2 (π ∗ )2 A¯ 2 λ2 ∗ ) 2 1 − eβ3 (−µY π A+ + = 0, β3 x[A¯ t + r0 A¯ ] + B¯ t + (µ − r0 )π ∗ A¯ + (θ − η)λ1 µZ A¯ + (1 + η)p∗ λ1 µZ A¯ − (π ∗ )2 σ 2 β1 AA¯

¯ β3 (−µY π + λ2 µY π ∗ Ae

The partial derivatives are Vt = A t x + B t ,

(75)

x[At + r0 A] + Bt + (µ − r0 )π ∗ A + (θ − η)λ1 µZ A

+ λ2 (φ3 (s) ln φ3 (s) − φ3 (s) + 1) ds .

A(T ) = 1, B(T ) = 0,

(74)

Plugging Eqs. (74)–(75) into Eqs. (11) and (73) implies

 

V (t , x) = A(t )x + B(t ),

γ σZ2 (p∗ )2 A¯ 2

) 2 (µZ A + γ σZ2 p∗ A¯ 2 )eβ2 (µZ p A+ = (1 + η)µZ A,  (µ − r0 )A 1 π∗ = 2 2 ¯ σ2 β1 A + γ A  λ2 (−µY A + γ σY2 π ∗ A¯ 2 ) β3 (−µY π ∗ A+ γ σY2 (π ∗ )2 A¯ 2 ) 2 − e . σ2

(77)

t

(η − θ )λ1 µZ

(1 − er0 (T −t ) )  T  T ¯l1 (s)ds + ¯l2 (s)ds, + r0

t

(78)

t

where l1 (s) = (1 + η)p∗ (s)λ1 µZ er0 (T −s) (73)

λ1 + β2

 1−e

β2 (µZ p∗ (s)er0 (T −s) +

γ σZ2 (p∗ (s))2 e2r0 (T −s) 2

 )

,

(79)

Y. Zeng et al. / Insurance: Mathematics and Economics 66 (2016) 138–152

l2 (s) = (µ − r0 )π ∗ (s)er0 (T −s) −

λ2 + β3

 1−

(π ∗ (s))2 σ 2 (β1 + γ )e2r0 (T −s) 2

2 ∗ 2 2r0 (T −s) r0 (T −s) γ σY (π (s)) e ∗ + ) 2 eβ3 (−µY π (s)e

We obtain

 ,

(80)

¯l1 (s) = (1 + η)p∗ (s)λ1 µZ er0 (T −s) r (T −s)

) + 2 , − λ1 µZ p∗ (s)er0 (T −s) eβ2 (µZ p (s)e 0 ¯l2 (s) = (µ − r0 )π ∗ (s)er0 (T −s) − (π ∗ (s))2 σ 2 β1 e2r0 (T −s) ∗

+ λ2 µY π ∗ (s)er0 (T −s) eβ3 (−µY π

2 ∗ 2 2r0 (T −s) ∗ (s)er0 (T −s) + γ σY (π (s)) e ) 2

(81)

.

(82)

Therefore, Eqs. (74) and (75) become e−r0 (T −t )

p∗ (t ) =

×e

γσ



2 Z

γ σ 2 (p∗ (t ))2 e2r0 (T −t ) −β2 (µZ p∗ (t )er0 (T −t ) + Z ) 2

e−r0 (T −t )

π ∗ (t ) =

β1 + γ

× eβ3 (−µY π



− µZ ,

(83)

λ2 (−µY + γ σY2 π ∗ (t )er0 (T −t ) ) µ − r0 − 2 σ σ2 

2 ∗ 2 2r0 (T −t ) ∗ (t )er0 (T −t ) + γ σY (π (t )) e ) 2

. 

(84)

Appendix C Proof of Proposition 3.2. (1) Proof of the existence–uniqueness of p∗ (t ). After simplification, Eq. (14) becomes 2 ∗ 2 2r0 (T −t ) ∗ er0 (T −t ) + γ σZ (p ) e )

(µZ + γ σZ2 p∗ er0 (T −t ) )eβ2 (µZ p

2

= (1 + η)µZ .

Let f (p) = (µZ + γ σZ2 per0 (T −t ) )eβ2 (µZ pe

2 2 2r0 (T −t ) r0 (T −t ) γ σZ p e + 2

)

− (1 + η)µZ . We have f ′ (p) = (µZ + γ σZ2 per0 (T −t ) )2 β2 er0 (T −t ) eβ2 (µZ pe

+ γ σZ2 er0 (T −t ) e

2 2 2r0 (T −t ) r0 (T −t ) γ σZ p e + 2

γ σ 2 p2 e2r0 (T −t ) β2 (µZ per0 (T −t ) + Z 2

)

)

> 0,

which means that f (p) is an increasing function w.r.t. p. Moreover, we can easily obtain that there exists a positive parameter κ such r0 (T −t )

γ σZ2 κ 2 e2r0 (T −t )

+ ) 2 that (µZ + γ σZ2 κ er0 (T −t ) )eβ2 (µZ κ e > (1 + η)µZ , i.e., f (κ) > 0. Since f (0) = −ηµZ er0 (T −t ) < 0, Eq. (14) has a unique positive root. (2) Proof of the existence–uniqueness of π ∗ (t ). Eq. (15) can be transformed into

(β1 + γ )π ∗ (t )σ 2 er0 (T −t ) = µ − r0 − λ2 (−µY + γ σY2 π ∗ (t ) × er0 (T −t ) )eβ3 (−µY π

2 ∗ 2 2r0 (T −t ) ∗ (t )er0 (T −t ) + γ σY (π (t )) e ) 2

Suppose that h(π) = µ − r0 − λ2 (−µY + γ σY2 π ∗ (t )er0 (T −t ) ) r (T −t )

γ σY2 (π ∗ (t ))2 e2r0 (T −t )

+ × eβ3 (−µY π (t )e 0 − (β1 + γ )π ∗ (t )σ 2 er0 (T −t ) . ∗

2 2 2r0 (T −t ) r0 (T −t ) γ σY π e + 2

)

− λ2 β3 er0 (t −T ) (−µY + γ σY2 π er0 (T −t ) )2 γ σY2 π 2 e2r0 (T −t )

+ 2 × eβ3 (−µY π e 0 2 r0 (T −t ) − (β1 + γ )σ e < 0,

)

which implies that h(π ) is a decreasing function w.r.t. π . Furthermore, we have h(0) = µ − r0 + λ2 µY > 0. Also, we can µ−r0 µ find that if π > max{ 2 r0Y(T −t ) , } > 0, we have 2 r0 (T −t ) γ σY e

(β1 +γ )σ e

h(π ) < 0. Therefore, Eq. (15) also has a unique positive root.



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151

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)

.

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