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Optimal investment of an insurer withregime-switching and risk constraintJingzhen Liu a b , Ka-Fai Cedric Yiu b & Tak Kuen Siu c da School of Insurance , Central University Of Finance andEconomics , Beijing , 100081 , P.R. Chinab Department of Applied Mathematics , The Hong KongPolytechnic University , Hunghom, Kowloon, Hong Kong , P.R.Chinac Cass Business School , City University London , London , UKd Faculty of Business and Economics, Department of AppliedFinance and Actuarial Studies , Macquarie University , Sydney ,AustraliaPublished online: 14 May 2013.

To cite this article: Jingzhen Liu , Ka-Fai Cedric Yiu & Tak Kuen Siu (2014) Optimal investmentof an insurer with regime-switching and risk constraint, Scandinavian Actuarial Journal, 2014:7,583-601, DOI: 10.1080/03461238.2012.750621

To link to this article: http://dx.doi.org/10.1080/03461238.2012.750621

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Scandinavian Actuarial Journal, 2014Vol. 2014, No. 7, 583–601, http://dx.doi.org/10.1080/03461238.2012.750621

Original Article

Optimal investment of an insurer with regime-switching andrisk constraint

JINGZHEN LIU†‡, KA-FAI CEDRIC YIU∗‡ and TAK KUEN SIU§¶

†School of Insurance, Central University Of Finance and Economics, Beijing 100081, P.R. China

‡Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon,

Hong Kong, P.R. China

§Cass Business School, City University London, London, UK

¶Faculty of Business and Economics, Department of Applied Finance and Actuarial Studies, Macquarie

University, Sydney, Australia

(Accepted November 2012)

We investigate an optimal investment problem of an insurance company in the presence of risk constraintand regime-switching using a game theoretic approach. A dynamic risk constraint is considered where weconstrain the uncertainty aversion to the ‘true’model for financial risk at a given level. We describe the surplusof an insurance company using a general jump process, namely, a Markov-modulated random measure. Theinsurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach,where a family of probability measures is cosidered and the insurance company maximizes the expectedutility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is thenformulated as a constrained two-player, zero-sum, stochastic differential game between the insurance companyand the market. Different from the other works in the literature, our technique is to transform the probleminto a deterministic differential game first, in order to obtain the optimal strategy of the game problemexplicitly.

Keywords: optimal investment; entropy risk; risk constraint; regime-switching; model uncertainty; stochasticdifferential game

1. Introduction

Portfolio allocation is one of the key problems in the interplay between financial mathematicsand insurance mathematics. Due to the rapid convergence of insurance and financial markets,many insurance companies are actively involving in investment activities in capital markets.An important issue in optimal investment problem for an insurer is model uncertainty whichis attributed to uncertainty in the ‘true’ data generating processes for financial prices andinsurance liabilities. Indeed, model uncertainty is an important issue in all modeling exercisesin finance and insurance, (see, e.g. Derman 1996 and Cont 2006). Mataramvura & Oksendal(2008) investigated a risk-minimizing portfolio selection problem, where an investor is to select

∗Corresponding author. E-mail: macyiu@polyu.edu.hk

© 2013 Taylor & Francis

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584 J. Liu et al.

an optimal portfolio strategy so as to minimize a convex risk measure of terminal wealth ina jump-diffusion market. Note that model uncertainty is incorporated in the penalty functionof the convex risk measure. It seems that Zhang & Siu (2009) were the first who attempted toincorporate model uncertainty in an optimal investment–reinsurance model for an insurer, wherethe insurer select an optimal investment-reinsurance strategy so as to maximize the expectedutility in the ‘worst-case’ scenario. Elliott & Siu (2010) studied the optimal investment of aninsurer in the presence of both the regime-switching effect and model uncertainty. To incorporatemodel risk, Elliott and Siu considered a robust approach where a family of probability measures,or scenarios, was used in the formulation of the optimal investment problem of an insurer. Thegoal of the insurer was then to maximize the expected utility in the ‘worst-case’ scenario. Foran overview on robust representations of preferences, please refer Follmer & Schied (2002).

In the past few decades or so, the insurance sector has grown significantly making it the second-largest in the European financial services industry. Consequently, any negative disturbances inthe insurance industry can have adverse impacts on the entire financial system and the wholeeconomy. In response to this, regulators in the European Union region introduced the SolvencyII, where some requirements are imposed for insurance companies to minimize the risk ofinsolvency. In quite a substantial amount of the existing literature, much emphasis has been puton investigating the ultimate, or long-run, risk of insurance companies. However, as pointedout in Bingham (2000), a significant loss from the risky investment within a short-term horizoncould be more serious than what will happen in the long term. Inspired by this fact, we willconsider the situation in the presence of risk constraint. In order to monitor the financial strengthof an insurer, assume that the regulator will regularly evaluate the potential current risk of thecompany. For risk evaluation, the regulator will look at the risk based on the current portfolioholdings of the insurer and potential future fluctuation in the risky asset.

Traditionally, volatility is used as a measure of risk in finance while ruin probability is used asa risk measure in insurance. Value at Risk (VaR) has emerged as a popular risk measure in boththe finance and insurance industries. It has become a benchmark for risk measurement in bothindustries. However, as pointed out by Artzner et al. (1999), VaR does not, in general, satisfy thesubadditivity property. In other words, the merge of two risky positions can increase risk, whichis counter-intuitive. Artzner et al. (1999) introduced an axiomatic approach for constructingrisk measures and the concept of coherent risk measures. A risk measure satisfying a set offour desirable properties, namely, translation invariance, positive homogeneity, monotonicityand subadditivity, is said to be coherent. Though theoretically sound, the notion of coherentrisk measures cannot incorporate the impact of liquidity risk of large financial positions inrisk measurement. To articulate the problem, Follmer & Schied (2002) and Frittelli & Gianin(2002) introduced the class of convex risk measures which can incorporate the nonlinearityattributed to the liquidity risk of large financial positions. They relaxed the subadditive andpositive homogeneous properties of coherent risk measures and replaced them with the convexityproperty. They extended the notion of coherent risk measures by the class of convex riskmeasures and obtained a representation for convex risk measures. Here, we adopt a particularform of a convex risk measure in a short time horizon as a risk constraint. In particular,the convex risk measure with the penalty function being a relative entropy is adopted herein the short time horizon. Consequently, the convex risk measure used here may be related to the

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Optimal investment of an insurer 585

entropic risk measure which is an important example of convex risk measures and correspondsto an exponential utility function (see, e.g. Frittelli & Gianin 2002, Barrieu & El-Karoui 2005;Barrieu & El-Karou in press).

Dynamic VaR risk constraint, first proposed in Yiu (2004), has been considered in the liter-ature for financial portfolio optimization under a Markovian regime-switching environment inYiu et al. (2010) or with Poisson jumps in Liu et al. (2011). When taking risk constraint intoconsideration for an insurer, Liu et al. (2012) considered the problem of minimizing the ruinprobability with the classic diffusion model. In this paper, we study the optimal investmentproblem for an insurer, whose objective is to maximize the expected utility of the terminalwealth in the worst-case scenario with the risk constraint. To incorporate model uncertainty,here, we use a set of probability models surrounding an approximating model, where the set ofprobability models is characterized by a family of probability measures, which are absolutelycontinuous with respect to a real-world probability measure, introduced by a Girsanov-typetransformation for jump-diffusion processes. Moreover, we consider a general model for theaggregate insurance claims, namely, a Markov regime-switching random measure. This modelis flexible enough to incorporate the regime-switching effect in modeling both the mean rateof claims arrivals and the distribution of claim sizes. To model the price process of a riskyfinancial asset, we adopt a Markov, regime-switching, geometric Brownian motion, wherekey model parameters such as the appreciation rate and the volatility of the risky asset aremodulated by a continuous-time, finite-state, observable Markov chain. Some works using thisprice process model include Zariphopoulou (1992), Zhou & Yin (2003), Sass & Haussmann(2004), Yin & Zhou (2004), Zhang & Yin (2004), Elliott & Siu (2010) and Elliott et al. (2010).Here, we interpret the states of the observable Markov chain as proxies of different levels ofsome observable economic indictors, such as Gross Domestic Product and Retail Price Index.Indeed, the states of the observable Markov chain may also be interpreted as the credit ratingsof a country, (or a corporation).

In our modeling framework, we incorporate four important sources of risk in our model,namely, financial risk due to price fluctuations, insurance risk due to uncertainty about insuranceclaims, economic risk (or regime-switching risk) that is attributed to structural changes ineconomic conditions, and model risk due to uncertainty about the ‘true’model. We formulate theoptimal investment problem of the insurer with risk constraint as a constrained Markov, regime-switching, zero-sum, stochastic differential game with two players, namely, the insurer and thenature (i.e. the market). In such a game, the insurer chooses an optimal investment strategyso as to maximize the expected utility of the terminal wealth in the worst-case probabilityscenario subject to a risk constraint. The market can be viewed as a ‘fictitious’ player in thesense that it responds antagonistically to the choice of the insurer by choosing the worst-caseprobability scenario which minimizes the expected utility. This is a standard assumption inthe robust approach to model uncertainty used in economics. One popular approach to discussstochastic differential games is based on solving the Hamilton–Jacobi–Bellman–Issacs (HJBI)equations, (see, e.g. Evans & Souganidis 1984, Mataramvura & Oksendal 2008, Zhang & Siu2009, Elliott & Siu 2010). One of the key assumptions of the HJBI dynamic programmingapproach is that the value function of the game problem should be C2 function and it is attained ata saddle point(Nash equilibrium) so as to obtain a classical solution to the problem. However,

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586 J. Liu et al.

it appears that in the situation with constraint, such a C2 solution to the HJBI equation orthe Nash equilibrium cannot be found and it could be difficult to verify the underlyingassumptions even though it holds. For the Markov, regime-switching, stochastic differentialgame, the problem becomes more involved due to the system of nonlinear differential equations.In this work, we consider the class of exponential utility functions as which plays an importantrole for pricing insurance contracts. Here, we transform the original game control problem withuncertainties in financial prices, insurance risk, and market uncertainty, (i.e. the Markov chain),into one with only market uncertainty being included and the problem is simplified greatly. Thenwe derive a closed-form optimal strategy of the transformed control problem.

The paper is organized as follows. The next section first describes the model dynamics inthe financial and insurance markets. Then we describe the worst-case scenario risk and use itas a dynamic risk constraint in the optimal investment problem, where the expected utility ofthe terminal wealth in the worst-case probability scenario is maximized under the dynamic riskconstraint. We then formulate the problem as a zero-sum stochastic differential game between theinsurer and the market. We consider an important case based on an exponential utility and derivea closed-form solution to the game problem in Section 3. The final section gives concludingremarks.

2. Model dynamics, optimal investment and stochastic differential game

We start with a complete probability space (�, F, P), where P represents a reference probabilitymeasure from which a family of probability measures absolutely continuous with respect to Pare generated. We suppose that the probability space (�, F, P) is rich enough to incorporateall sources of uncertainties, such as financial and insurance risks, in our models. We call such aprobability space a reference model, or an approximating model. Suppose T is a time parameterset [0, T ], where T < ∞.

To describe economic risk attributed to transitions in economic, or environmental, conditions,we consider a continuous-time, N -state, observable Markov chain Z := {Z(t)|t ∈ T } withstate space Z := {z1, z2, · · · , zN } ⊂ �N . These states are interpreted as proxies of differentlevels of observable, (macro)-economic, conditions. For mathematical convenience, we followthe convention in Elliott et al. (1994) and identify the state space of the chain Z with a finiteset of standard unit vectors E := {e1, e2, . . . , eN } ∈ �N , where the j th-component of ei is theKronecker delta δi j for each i, j = 1, 2, · · · , N . This is called the canonical representation ofthe state space of the chain Z. Let Q be the constant rate matrix, or the generator, [qi j ]i, j=1,2,··· ,N

of the chain under P so that the probability law of the chain Z under P is characterized by Q.With the canonical representation of the state space of the chain Z, Elliott et al. (1994) derivedthe following semimartingale dynamics for the chain Z:

Z(t) = Z(0) +∫ t

0QZ(u)du + M(t) , t ∈ T . (2.1)

Here {M(t)|t ∈ T } is an �N -valued, (FZ, P)-martingale; FZ := {FZ(t)|t ∈ T } is the right-

continuous, P-completed natural filtration generated by the Markov chain Z.

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Optimal investment of an insurer 587

In what follows, we shall specify how the state of the economy described by the chain Zinfluences the price process of a risky financial asset and the insurance surplus process.

2.1. The model

2.1.1. The surplus of an insurer

We model the risk process of the insurer by a Markov, regime-switching, random measure whichis flexible enough to allow the Markov chain influencing the distribution of claim sizes and theclaims arrival rate. Let C := {C(t)|t ∈ T } be the aggregate claim process, where C(t) is theaggregate claim up to time t . We suppose that C is a real-valued, Markov, regime-switchingpure jump process on (�, F, P). Then for each t ∈ T ,

C(t) =∑

0<u≤t

�C(u) , C(0) = 0 , P-a.s.

Here the claim size �C(u) ∈ (0,∞).Suppose C is the state space of the claim size (0,∞). Let γ (·, ·) be a random measure defined

on the product space M := T ×C, which selects random claims arrival times and random claimsizes. Then

γ (dc, ds) =∑k≥1

δ(�C(Tk ),Tk )(dc, ds)I {�C(Tk )=0,Tk<∞} , (2.2)

where

(1) Tk is the arrival time of the kth claim;(2) �C(Tk) is the size of the kth claim;(3) δ(�C(Tk ),Tk )(·, ·) is the random delta functions at the point (�C(Tk), Tk) ∈ C × T ;(4) I E is the indicator function of an event E .

Then the aggregate insurance claims process C can be written as:

C(t) =∫ t

0

∫ ∞

0cγ (dc, ds) , t ∈ T . (2.3)

For each t ∈ T ,

N (t) :=∫ t

0

∫ ∞

0γ (dc, ds) , (2.4)

which counts the number of claim arrivals up to time t .Suppose, under P , N := {N (t)|t ∈ T } is a conditional Poisson process on (�, F, P) with

intensity process λ := {λ(t)|t ∈ T } modulated by the chain Z as:

λ(t) := 〈λ, Z(t)〉 , t ∈ T , (2.5)

where λ := (λ1, λ2, · · · , λN )′ ∈ �N with λi > 0; λi is the jump intensity of N when theeconomy is in the i th state.

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588 J. Liu et al.

When Z(s−) = ei , Fi (c) is the probability distribution of the claim size c := C(s) − C(s−).Then the compensator of the Markov, regime-switching, random measure γ (·, ·) under P isgiven by:

νZ(s−)(dc, ds) :=N∑

i=1

〈Z(s−), ei 〉 λi Fi (dc)ds. (2.6)

Consequently, the compensated version of the Markov, regime-switching, random measure,denoted by γ̃ (·, ·), is given by:

γ̃ (dc, ds) := γ (dc, ds) − νZ(s−)(dc, ds). (2.7)

Let FC := {FC (t)|t ∈ T } be the right-continuous, P-completed filtration generated by the

aggregate claim process C .We suppose that the premium rate at time t , denoted by p(t), is modulated by the chain Z as:

p(t) := 〈p, Z(t)〉 , t ∈ T . (2.8)

Here p := (p1, p2, · · · , pN )′ ∈ �N with pi > 0.Let R := {R(t)|t ∈ T } denote the surplus process of the insurer. Then

R(t) := x0 +∫ t

0p(s)ds − C(t)

= x0 +N∑

i=1

pi

∫ t

0〈Z(s), ei 〉 ds −

∫ t

0

∫ ∞

0cγ (dc, ds), t ∈ T . (2.9)

2.1.2. The investment process

We assume that the insurer is allowed to invest in a risky financial asset S, say a stock. Supposethat the price process {S(t)|t ∈ T } of the risky financial asset S is governed by the followingMarkov, regime-switching, geometric Brownian motion:

d S(t) = S(t)

(μ(t)dt + σ(t)dW (t)

), S(0) = s. (2.10)

Here W := {W (t)|t ∈ T } is a standard Brownian motion on (�, F, P) with respect to FW :=

{FW (t)|t ∈ T }, the P-augmentation of the natural filtration generated by the standard Brownianmotion W , μ(t) and σ(t) are the appreciation rate and the volatility of the risky financial assetS at time t , respectively. We assume that μ(t) and σ(t) are modulated by the chain Z as:

μ(t) := 〈μ, Z(t)〉 ,

σ (t) := 〈σ , Z(t)〉 , (2.11)

where μ := (μ1, μ2, . . . , μN )′ ∈ �N and σ := (σ1, σ2, . . . , σN )′ ∈ �N with μi > 0 andσi > 0, for each i = 1, 2, . . . , N .

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Optimal investment of an insurer 589

For each t ∈ T , let π(t) be the amount of money allocated to the risky asset S at time t .Suppose that {X (t)|t ∈ T } is the surplus process of the insurer investing in the financial marketwith initial wealth X (0) = x0 > 0. Then, the surplus process of the insurer after taking intoinvestment is governed by the following Markov, regime-switching, jump-diffusion process:⎧⎨⎩ d X (t) = (p(t) + π(t)μ(t))dt + π(t)σ (t)dW (t) −

∫ ∞

0cγ (dc, dt),

X (0) = x0.

(2.12)

Let G(t) := FZ(t) ∨ FW (t) ∨ FC (t), the minimal σ -field containing FZ(t), FW (t) andFC (t). Write, then, G := {G(t)|t ∈ T }.

A portfolio process π := {π(t)|t ∈ T } is said to be admissible if it satisfies the followingthree conditions:

(1) π(·) is a G-adapted, measurable, process;(2) for each t ∈ T ,∫ t

0

(|p(u) + π(u)μ(u)| + (π(u)σ (u))2 +

∫ ∞

0c2νZ(u−)(dc, du)

)du < ∞ ,

P − a.s.; (2.13)

(3) the stochastic differential equation governing the surplus process X has a unique,strong solution.

Write for the space of all admissible portfolio processes without any constraint.

2.1.3. Model risk by change of measures

In this section, we incorporate model uncertainty, or model risk, by a family of probabilitymeasures introduced via Girsanov’s transformation of jump-diffusion processes.

Let θ := {θ(t)|t ∈ T } be a G-progressively measurable process characterizing a probabilitymeasure in the family. Suppose θ(·) satisfies the following conditions:

(1) for each t ∈ T , θ(t) ≤ 1, P-a.s.;(2)

∫ T0 θ2(t)dt < ∞, P-a.s.

Write � for the space of all such processes θ(·). Then, the family of probability measures forcapturing model risk is parameterized, or indexed, by �. Note that � is the space of admissiblestrategies adopted by the market without constraint. For each θ(·) ∈ �, we define a G-adaptedprocess θ := { θ(t)|t ∈ T } by putting:

θ(t) = exp

(−∫ t

0θ(u)dW (u) − 1

2

∫ t

0θ(u)2du +

∫ t

0

∫ ∞

0ln(1 − θ(u))γ̃ (dc, du)

+∫ t

0

∫ ∞

0(ln(1 − θ(u)) + θ(u))νZ(u−)(dc, du)

). (2.14)

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590 J. Liu et al.

Or equivalently, {d θ(t) = θ(t−)(−θ(t)dW (t) − ∫∞

0 θ(t)γ̃ (dc, dt)), θ (0) = 1, P − a.s.

(2.15)

Then, for each θ(·) ∈ �, θ is a (G, P)-(local)-martingale. For each θ(·) ∈ �, a real worldprobability measure Pθ is defined as:

dPθ

dP

∣∣∣∣G(T )

:= θ(T ). (2.16)

Suppose, the insurer is averse to model uncertainty. He/she may consider a family of probabilitymeasures which are introduced by perturbing the approximating model described by the referenceprobability P . Here we suppose that the family of probability measures is given by {Pθ |θ(·) ∈�}. The following lemma is a well-known result which can be obtained from applying theGirsanov theorem. We state the result without giving the proof.

Lemma 2.1 For each θ ∈ �, under Pθ ,

W θ (t) := W (t) −∫ t

0θ(u)du , t ∈ T , (2.17)

is a standard Brownian motion and the random measure γ θ has the following compensator:

νθZ(u−)(dc, du) :=

N∑i=1

(1 − θ(u−)) 〈Z(u−), ei 〉 λi Fi (dc)du. (2.18)

Then under Pθ ,{d X (t) = [p(t) + π(t)(μ(t) − σ(t)θ(t))]dt + π(t)σ (t)dW θ (t) − ∫∞

0 cγ θ (dc, dt),X (0) = x0.

(2.19)

2.2. The model uncertainty constraint and risk constraint

In this work, the risk is evaluated according to the current information and the model parametersat time t . As the insurer is averse to model uncertainty, he/she will take into account the maximalrisk in the family of probability measures {Qθ |θ(·) ∈ �†} to be defined later in this section andamong all regimes of the Markov chain.

2.2.1. The model uncertainty constraint

For each t ∈ T , let H := FZ(t) ∨ FW (t). Write H := {H(t)|t ∈ T }. Suppose �† is the spaceof all processes θ := {θ(t)|t ∈ T } such that

(1) θ(·) is H-progressively measurable;(2)

∫ T0 |θ(t)|2dt < ∞;

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Optimal investment of an insurer 591

(3) θ(·) satisfies the Novikov condition, (i.e., E[exp( 12

∫ T0 |θ(t)|2dt)] < ∞).

Consider, for each θ(·) ∈ �†, the following H-adapted process θ1 := { θ

1(t)|t ∈ T } on(�, F, P):

θ1(t) := exp

(−∫ t

0θ(u)dW (u) − 1

2

∫ t

0θ2(u)du

).

Then

d θ1(t) = −θ(t) θ

1(t)dW (t).

Consequently, θ1 is a (H, P)-(local)-martingale. Since θ satisfies the Novikov condition, θ

1 isa (H, P)-martingale.

For each θ(·) ∈ �†, we define the new probability measure Qθ which is absolutely continuouswith respect to P on H(T ) by putting:

dQθ

dP

∣∣∣∣H(T )

:= θ1(T ).

We now define the relative entropy between Qθ and P in the short duration [t, t +�t) as follows:

Kt,t+�t (Qθ , P) := E

[ θ

1(t + �t)

θ1(t)

ln

( θ

1(t + �t)

θ1(t)

)|H(t)

].

Here E[·] is expectation under P .By Bayes’ rule,

Kt,t+�t (Qθ , P) =E[ θ

1(t + �t) ln( θ

1(t+�t)

θ1(t)

)|H(t)]E[ θ

1(t)|H(t)]= Eθ

[ln

( θ

1(t + �t)

θ1(t)

)|H(t)

]= Eθ

[ ∫ t+�t

t

(1

2θ2(u)du

)|H(t)

].

For the evaluation in the short duration [t, t + �t), we use the current regime and model (i.e.Z(u) = Z(t) and θ(u) = θ(t), for all u ∈ [t, t + �t)). Consequently, Kt,t+�t (Qθ , P) can beapproximated as follows:

Kt,t+�t (Qθ , P) ≈ �tN∑

i=1

〈Z(t−), ei 〉(

1

2θ2

i

).

For each i = 1, 2, · · · , N , write

K̃t,t+�t (Qθ , P, ei ) := 1

2θ2

i �t.

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592 J. Liu et al.

Constraint 2.1 We impose the following constraint for the aversion to the model uncertainty:

maxi

K̃t,t+�t (Qθ , P, ei ) ≤ R1,t .

For each i = 1, 2, · · · , N , let θ(t,i,−) and θ(t,i,+) be the negative and positive roots, respectively,of the following nonlinear equation in θ :

K̃t,t+�t (Qθ , P, ei ) = R1,t .

Let

Kt := [θ(t,−), θ(t,+)] =⋂

i=1,2,··· ,N

[θ(t,i,−), min(1, θ(t,i,+))].

Define the subspace �†1 of �† as follows:

�†1 := {θ(·) ∈ �†|θ(t) ∈ Kt ,∀t ∈ T }.

Then the space �̃ of constrained admissible strategies adopted by the market is given by:

�̃ := �†1 ∩ �.

2.2.2. The risk constraint on investment

Suppose that the insurer evaluate the risk based on the current portfolio holdings and potentialfuture fluctuations in the risky asset. In order to evaluate the potential investment risk at time t ,we evaluate the change in the surplus of the insurer in the horizon [t, t + �t) under P , whichcan be approximated by

�X (t) ≈ μ(t)π(t)�t + σ(t)π(t)(W (t + �t) − W (t)).

Consequently, under Qθ ,

�X (t) ≈ (μ(t) − θ(t)σ (t))π(t)�t + σ(t)π(t)(W θ (t + �t) − W θ (t)).

For each i = 1, 2, · · · , N and each θ ∈ �̃, let

Lθt (�X (t), ei ) := Eθ [−�X (t)|H(t), X(t) = ei ]

and Eθ is expectation under Qθ . We define a convex risk measure as

ρ(�X (t)|G(t)) = maxi

maxθ∈�̃

[Lθ

t (�X (t), ei ) − K̃t,t+�t (Qθ , P, ei )

].

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Under the time horizon, the risk measure can be approximated by

ρ(�X (t)|G(t)) ≈ maxi

maxθ(t)∈�̃(t)

�t

[−(μi − θ(t)σi )π(t) − 1

2θ2(t)

]= max

ihi (π(t)), (2.20)

where

hi (π(t)) :=

⎧⎪⎪⎨⎪⎪⎩�t[−(μi − θ(t,+)σi )π(t) − 1

2θ2(t,+)], if π(t) ≥ θ(t,+)

σi,

�t[ (σi π(t))2

2 − μiπ(t)], θ(t,−)

σi≤ π(t) ≤ θ(t,+)

σi,

�t[−(μi − θ(t,−)σi )π(t) − 12θ2

(t,−)], if π(t) ≤ θ(t,−)

σi.

Constraint 2.2 Supposes that, for each t ∈ T , the risk level described by the convex risk measureis constrained below R2,t , i.e.,

maxi

hi (π) < R2,t . (2.21)

For each i = 1, 2, · · · , N , let r(t,i,±) := μi ±√

μ2i +2R2,t

σ 2i

. Obviously, hi (π(t)) decreases with π(t)

when π ≤ θ(t,−)

σi. Here, we assume that μi − θ(t,+)σi < 0, then hi (π(t)) increases with π(t)

when π(t) ≥ θ(t,+)

σi. Write, for each t ∈ T and i = 1, 2, · · · , N ,

π(t,i,−) :=⎧⎨⎩ r(t,i,−), if�t[ θ2

(t,−)

2 − μi θ(t,−)

σi] ≥ R2,t ,

−R2,t�t + 1

2 θ2t

μi −θσi, otherwise,

(2.22)

and

π(t,i,+) :=⎧⎨⎩ r(t,i,+), if�t[ θ2

(t,+)

2 − μi θ(t,+)

σi] ≥ R2,t ,

−R2,t�t + 1

2 θ2t

μi −θσi, otherwise,

(2.23)

then the constraint (2.21) leads to: π(t) ∈ [π(t,−), π(t,+)] := ⋂i=1,··· ,N [π(t,i,−), π(t,i,+)].

2.3. Optimal investment problem as a stochastic differential game

Problem 2.1 Let ̃ := {π ∈ |πt ∈ [π(t,−), π(t,+)]}. Denote the expected utility on theterminal wealth

J (t, x, π(·), θ(·)) = Eθ [U (Xπ (T ))|Xπ (t) = x]. (2.24)

Then the optimal investment problem of the insurer is to select an investment process in ̃ so asto maximize the following minimal, (or ‘worst-case’), expected utility on the terminal wealth:

supπ∈̃

infθ∈�̃

J (t, x, π(·), θ(·)). (2.25)

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This is a two-player, zero-sum, stochastic differential game between the insurer and the marketin the sense that the insurer tries to maximize the value of J (t, x, π(·), θ(·)) and the market triesto minimize this value. We call

V+(t, x) = supπ∈̃

infθ∈�̃

J (t, x, π(·), θ(·))

the upper value of the game and

V−(t, x) = infθ∈�̃

supπ∈̃

J (t, x, π(·), θ(·))

the lower value of the game. If

V+(t, x) = V−(t, x),

it is called the value of the game. If the value of the game exists, a popular approach, in theliterature, is applying the dynamic programming principle, then the problem is reduced to finda C2 solution to the resulted HJBI equations and the C2 solution is attained at a Markov controlpoint. However, when there is a constraint on the strategy, it is often difficult to find the C2

solution due to the constraint on the strategy.In the insurance literature, the class of exponential utility functions play an important role for

pricing insurance contracts. In this work by an technique in stochastic analysis, we derive anoptimal solution of the investment problem of the insurer when the insurer has an exponentialutility of the following form:

U (x) = − exp(−αx) , (2.26)

where α is a positive constant.

3. The optimal strategy

For each t ∈ T and (π(·), θ(·)) ∈ ̃ × �̃, we define the following quantities:

Xπ,θ1,t (T ) := − exp

{− αX (t) +

∫ T

t

(− α(p(s) + π(s)(μ(s) − σ(s)θ(s)))

+1

2α2π2(s)σ 2(s)

)ds +

∫ T

t(1 − θ(s))λ(s)

∫ ∞

0(eαc − 1)F(dc, ds)

},

Xπ,θ2,t (T ) := exp

(− 1

2

∫ T

tα2π2(s)σ 2(s)ds +

∫ T

tαπ(s)σ (s)dW θ (s)

)×

exp

(−∫ T

t

∫ ∞

0αcγ θ (dc, ds) − (1 − θ(s))λ(s)

∫ ∞

0(eαc − 1)F(dc, ds)

).

(3.1)

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Optimal investment of an insurer 595

Let Eθt be the conditional expectation under Qθ given G(t). Then

Eθt [U (Xπ (T )]

= −Eθt

[exp

(− αX (t) − α

∫ T

t[p(s) + π(s)(μ(s) − σ(s)θ(s))]ds

−α

∫ T

tπ(s)σ (s)dW θ (s) − α

∫ T

t

∫ ∞

0cγ θ (dc, ds)

)]= −Eθ

t

[exp

(− αX (t) − α

∫ T

t[p(s) + π(s)(μ(s) − σ(s)θ(s))]ds

+1

2α2∫ T

tπ(s)2σ 2(s)ds +

∫ T

t(1 − θ(s))λ(s)

∫ ∞

0(eαc − 1)F(dc, ds)

)× exp

(− 1

2α2∫ T

tπ2(s)σ 2(s)ds − α

∫ T

tπ(s)σ (s)dW θ (s) −

∫ T

t

∫ ∞

0αcγ θ (dc, ds)

−∫ T

t(1 − θ(s))λ(s)

∫ ∞

0(eαc − 1)F(dc, ds)

)]= Eθ

t [X1,t (T )X2,t (T )]. (3.2)

For each t ∈ T , let G†(t) := FZ(T ) ∨ FW (t) ∨ FC (t). Write G† := {G†(t)|t ∈ T }. Then

for each fixed t ∈ T , it is not difficult to see that {X2,t (u)|u ≥ t} is a (G†, Qθ )-martingale.Consequently,

Eθt [X2,t (T )|G†(t)] = 1 , P-a.s.

Now for any (π(·), θ(·)) ∈ (̃, �̃),

Eθt [Xπ,θ

1,t (T )Xπ,θ2,t (T )]

= Eθt {Eθ [Xπ,θ

1,t (T )Xπ,θ2,t (T )|G†(t)]}

= Eθ {Xπ,θ1,t (T )Eθ [Xπ,θ

2,t (T )|G†(t)]}= Eθ

t [Xπ,θ1,t (T )]. (3.3)

Consequently, Problem 2.1 can be simplified into the following problem:

Problem 3.1 Find

V (θ(·), π(·), t, x) := supπ∈̃

infθ∈�̃

Eθt [Xπ,θ

1,t (T )]. (3.4)

It is important to note that the only source of uncertainty involved in Problem 3.1 is fromMarkov chain Z.

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596 J. Liu et al.

Write �(Z) for the path space generated by the sample paths of the Markov chain Z. For eachωz ∈ �(Z), t ∈ T , (π, θ) ∈ ̃ × �̃, we define:

H(t, ωz, π(t, ωz), θ(t, ωz))

:= α[p(t, ωz) + π(t, ωz)(μ(t, ωz) − σ(t, ωz)θ(t, ωz))] − 1

2π(t, ωz)

2σ 2(t, ωz)α2

−(1 − θ(t, ωz))λ(t, ωz)

∫ ∞

0(eαc − 1)F(dc, dt).

For each t ∈ T , let ̃(t) and �̃(t) be the coordinate spaces of ̃ and �̃ at time t , respectively.Now if, for each ωz ∈ �(Z) and t ∈ T , we can find (π∗(t, ωz), θ

∗(t, ωz)) ∈ ̃(t) × �̃(t) suchthat

H(t, ωz, π∗(t, ωz), θ

∗(t, ωz)) = supπ(t,ωz)∈̃(t)

infθ(t,ωz)∈�̃(t)

H(t, ωz, π(t, ωz), θ(t, ωz)), (3.5)

then

Eθt [Xπ∗,θ∗

1,t (T )] = supπ(·)∈̃

infθ(·)∈�̃

Eθt [Xπ,θ

1,t (T )].

In other words, to solve the stochastic differential game, we can solve the correspondingpathwise minimax problem with respect to the path space of the Markov chain �(Z). Thepathwise minimax problem is presented as follows:

Problem 3.2 If, for each t ∈ T and ωz ∈ �(Z), we can find (π∗(t, ωz), θ∗(t, ωz)) such that

H(t, ωz, π∗(t, ωz), θ

∗(t, ωz)) = supπ(t,ωz)∈̃(t)

infθ(t,ωz)∈�̃(t)

H(t, ωz, π(t, ωz), θ(t, ωz)), (3.6)

and (π∗(·), θ∗(·)) ∈ × �, then (π∗(·), θ∗(·)) solves the original Problem 2.1.

3.1. The Nash equilibrium

For (π(·), θ(·)) ∈ × �, let

�(θ(t), π(t)) = α

(p(t) + π(t)(μ(t) − σ(t)θ(t)) − α

1

2π2(t)σ 2(t)

)−(1 − θ(t))λ(t)

∫ ∞

0(eαc − 1)F(dc, dt). (3.7)

and�i (θ(t), π(t)) = �(θ(t), π(t)|Z(t) = i)

= α

(p(t) + π(t)(μi − σiθ(t)) − α

1

2π2(t)σ 2

i

)−(1 − θ(t))λ(t)

∫ ∞

0(eαc − 1)F(dc, dt). (3.8)

for each i = 1, 2, . . . , N .

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Then Problem 3.2 is equivalent to the following set of N minimax problems:

supπ(t)∈̃(t)

infθ(t)∈�̃(t)

�i (θ(t), π(t)) , i = 1, 2, · · · , N . (3.9)

for any time t ∈ T .For each i = 1, 2, · · · , N , a pair (π∗

i (t), θ∗i (t)) is said to achieve an Nash equilibrium of

i th problem in (3.9), or equivalently, a saddle point of i th problem in (3.9) if it satisfies thefollowing Issac’s condition:

�i (θ∗i (t), π(t)) ≤ �i (θ

∗i (t), π∗

i (t)) ≤ �i (θ(t), π∗i (t)),∀(θ(t), π(t)) ∈ ̃(t) × �̃(t).

(3.10)

We have the following lemma which gives an Nash equilibrium of i th problem in (3.9).From the saddle-point theorem, the problem is equivalent to the following minmax problem:

infθ(t)∈�̃(t)

supπ(t)∈̃(t)

�i (θ(t), π(t)), t ∈ T . (3.11)

In the following, we solve (3.9) by considering (3.11).Firstly, we present the results without constraint, namely, when (π(·), θ(·)) ∈ (,�).

Lemma 3.1 For each i = 1, 2, · · · , N and t ∈ T , let (π(0,∗)i (t), θ(0,∗)

i (t)) denote the Nashequilibrium of the game when there are no constraints on π(·) and θ(·) and Z(t) = ei . Then

(π(0,∗)i (t), θ(0,∗)

i (t)) =(

λi∫∞

0 (eαz − 1)F(dz)

σiα,μi − π

(0,∗)i (t)ασ 2

i

σi

).

Proof According to (3.8), (π(0,∗)i (t), θ(0,∗)

i (t)) can be obtained from the following first-orderconditions:

{−μi + σ 2

i π(t)α + θ(t)σi = 0,

σiπ(t)α + λi∫∞

0 (1 − eαz)F(dz) = 0.(3.12)

�

The following lemma will be needed in the proof of Theorem 3.1.

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Lemma 3.2 Denote , for each i = 1, 2, · · · , N , π̄i (θ) := − θσi −μi

ασ 2i

. Then

infθ(t)∈�(t)

supπ(t)∈(t)

�i (θ(t), π(t))

= infθ(t)∈�(t)

α[p(t) + π̄i (θ(t))(μ(t) − σ(t)θ(t)) − 1

2απ̄2

i (θ(t))σ 2(t)]

+(1 − θ(t))λ∫ ∞

0(eαz − 1)F(dz, dt)

= infθ(t)∈�(t)

g0i (θ(t))

= g0i (θ

(0,∗)i (t)), (3.13)

where

g0i (θ) := α[p(t) − θσi − μi

ασ 2i

(μi − σiθ) − 1

2α

(−θσi − μi

ασ 2i

)2

σ 2i ]

+(1 − θi )λ

∫ ∞

0(eαc − 1)F(dc, dt).

Note that g0i (θ) is increasing with θ when θ(t) > θ

(0,∗)i and decreasing with θ when θ(t) <

θ(0,∗)i . This property will be used later in Theorem 3.1.

We now state the main result of this paper:

Theorem 3.1 Denote θ̂i (π(t)) := μi −π(t)ασ 2i

σand

π̌i (θ(t)) =

⎧⎪⎨⎪⎩θ(t)σi −μi

αi σ2i

, if θ̂i (π(t,+)) ≤ θ(t) ≤ θ̂i (π(t,−)),

π(t,+), if θ(t) ≤ θ̂i (π(t,+)),

π(t,−), if θ(t) ≥ θ̂i (π(t,−)).

(3.14)

If θ(0,∗)i (t) ∈ [θ̂i (π(t,+)), θ̂i (π(t,−))] ∩ [θ(t,−), θ(t,+)], i.e., the constraint is inactive, then the

optimal strategy

(π∗i (t), θ∗

i (t)) = (π(0,∗)i (t), θ(0,∗)

i (t)).

When the constraint is active, we have that

(1) if π(0,∗)i (t) > π(t,+), then

(θ∗i (t), π∗

i (t)) = (θ(t,−), π̌(θ(t,−))), (3.15)

(2) if π(0,∗)i (t) < π(t,−), then

(θ∗i (t), π∗

i (t)) = (θ(t,+), π̌(θ(t,+)), (3.16)

(3) if π(0,∗)i (t) ∈ [π(t,−), π(t,+)] and

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Optimal investment of an insurer 599

(a) if θ∗0 (t) < θ(t,−), then (θ∗(t), π∗(t)) = (θ(t,−), π̌(θ(t,−))),

(b) if θ∗0 (t) > θ(t,+), then (θ∗(t), π∗(t)) = (θ(t,+), π̌(θ(t,+))).

Proof Since

infθ(t)∈�̃(t)

supπ(t)∈(t)

�i (θ(t), π(t))

= infθ(t)∈�̃(t)

�i (θ(t), π̌i (θ(t)))

= infθ(t)∈�̃(t)

[p(t) + π̌i (θ(t))(μi − σiθ(t))

+1

2π̌2

i (θ(t))σ 2i + (1 − θ(t))λ(t)

∫ ∞

0(eαz − 1)F(dz, dt)], (3.17)

when the constraint is inactive, the result is obvious. If the constraint is active, we just prove 1and 3, while the proof of 2 is similar to 1. The definition of θ̂i (·) shows that θ̂i (π) is a decreasingfunction of π . Thus, if π(t,+) < π∗

0 (t), we have

θ̂i (π) > θ̂i (π(0,∗)i (t)) = θ

(0,∗)i (t) (3.18)

for any π ∈ [π(t,−), π(t,+)]. Denote

gi (θ(t)) := �i (θ(t), π̌(θ(t))),

then we have gi (θ) = g0i (θ) in the interval [θ̂i (π(t,+)), θ̂i (π(t,−))]∩[θ(t,−), θ(t,+)] and it follows

from Lemma 3.3, together with (3.18), that gi (θ) increases with θ in this interval.On the other hand, if θ(t) < θ̂i (π(t,+)), then π̌i (θ(t)) = π(t,+) and if θ(t) > θ̂i (π(t,−)), then

π̌i (θ(t)) = π(t,−). We clare that in these two cases gi (θ) still increases with θ . In fact, whenπ < π

(0,∗)i (t), it follows from the definition of �i (·, ·) that ∂�i (θ,π)

∂θ> 0, i,e, �i (θ, π) increases

with θ . Thus π(t,+) < π(0,∗)i (t) implies that both �i (θ, π(t,+)) and �i (θ, π(t,−)) increase with

θ .The arguments above shows that gi (θ) increases with θ in [θ(t,−), θ(t,+)]. As a result, gi (θ)

attains its minimum at θ∗i (t) = θ(t,−).

For 3, if θ(0,∗)i (t) < θ(t,−), then π̌(θ) < π

(0,∗)i (t) for all θ ∈ [θ(t,−), θ(t,+)], thus we have

θ∗(t) = θ(t,−), π∗(t) = π̌(θ(t,−)). The proof of the case θ(0,∗)i (t) > θ(t,+) is similar. �

As (π∗i (t), θ∗

i (t)) denotes the Nash equilibrium at time t and Z(t) = ei , i = 1, · · · , N ,we let (π∗(t), θ∗(t)) = (π∗

i (t), θ∗i (t)) if Z(t) = ei , i = 1, · · · , N , it is easily seen that

(π∗(·), θ∗(·)) ∈ (̃, �̃), which solves the original Problem 3.1. The results of Theorem 3.1show that

(1) when the risk with π(0,∗)i (·) is beyond the risk constraint, i.e., π

(0,∗)i (t) > π+ or

π(0,∗)i (t) < π−, t ∈ T , the statements of 1 and 2 show that risk is stabilized by

π(t,−) < π̌(t) < πt,+;

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600 J. Liu et al.

(2) the utility is improved if the risk constraint is inactive, but the constraint on θ isactive (from (3)).

Remark 3.1 When there is no constraint and p(t), λ(t), μ(t), σ (t) are constants, our resultcoincides with Zhang & Siu (2009). We transform the stochastic differential game of (2.25) intoa deterministic game of Problem (3.6), which is much simpler than the original one. Furthermore,although the regime-switching model in this work is Markovian, there is no need to assumethat the parameters p(t), λ(t), μ(t), σ (t) are also Markovian, which can be generalized by G-progressively measurable process and our approach still works.

4. Conclusion

We considered an optimal investment problem of an insurer in the presence of risk constraintand regime-switching. By adopting a robust approach, we incorporated model uncertainty. Weimposed a risk constraint on the investment taken by the insurer using the notion of convex riskmeasures as well as a constraint on the uncertainty aversion of the insurer on the ‘true’investmentmodel. The goal of the insurer was to maximize the expected utility on terminal wealth in theworst-case scenario subject to the risk constraint. The optimal investment problem of the insurerwas then formulated as a two-person, zero-sum, stochastic differential game between the insurerand the market. Under the assumption of an exponential utility, we transformed the original gameproblem into one where the only source of uncertainty is due to the modulating Markov chain. Theproblem was then equivalent to a pathwise minimax problem. By solving the pathwise minimaxproblem, we obtained a closed-form expression of the Nash equilibrium of the game. Unlike theHJBI dynamic programming approach, our approach does not require smoothness assumptionson the value function. Indeed, in many practical situations, the smoothness assumptions on thevalue function may hardly be satisfied. In these situations, one may consider our approach todiscuss the optimal investment problem. Our approach works well whether the risk constraint ispresent or not. Moreover, our technique can be applied to a general model when the parameters ofthe model are not Markov. This is impossible to achieve using the HJBI dynamic programmingapproach.

Acknowledgements

The authors would like to thank the referee for helpful comments and suggestions. The first andsecond authors were supported by the Research Grants Council of HKSAR (PolyU 5001/11P)and the research committee of the Hong Kong Polytechnic University.

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