Accepted Manuscript

Optimal investment, consumption and proportional reinsurance undermodel uncertainty

Xingchun Peng, Fenge Chen, Yijun Hu

PII: S0167-6687(14)00128-0DOI: http://dx.doi.org/10.1016/j.insmatheco.2014.09.013Reference: INSUMA 1993

To appear in: Insurance: Mathematics and Economics

Received date: September 2013Revised date: July 2014Accepted date: 30 September 2014

Please cite this article as: Peng, X., Chen, F., Hu, Y., Optimal investment, consumption andproportional reinsurance under model uncertainty. Insurance: Mathematics and Economics(2014), http://dx.doi.org/10.1016/j.insmatheco.2014.09.013

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OPTIMAL INVESTMENT, CONSUMPTION AND PROPORTIONALREINSURANCE UNDER MODEL UNCERTAINTY ∗

XINGCHUN PENG1,†, FENGE CHEN1, YIJUN HU2

Abstract. This paper considers the optimal investment, consumptionand proportional reinsur-ance strategies for an insurer under model uncertainty. Thesurplus process of the insurer beforeinvestment and consumption is assumed to be a general jump diffusion process. The financialmarket consists of one risk–free asset and one risky asset whose price process is also a generaljump diffusion process. We transform the problem equivalently into atwo–person zero–sumforward–backward stochastic differential game driven by two–dimensional Levy noises. Themaximum principles for a general form of this game are established to solve our problem. Somespecial interesting cases are studied by using Malliavin calculus so as to give explicit expressionsof the optimal strategies.

Key words: Investment, Consumption, Reinsurance, Model uncertainty, Stochastic maximumprinciple, Malliavin calculus

2010 Mathematics Subject classification: 91G80; 97M30; 93E20; 60H30

1. Introduction

Recently, optimal investment and reinsurance problems forinsurers have attracted great at-tention due to the facts that investment is an interesting important element in the insurancebusiness and reinsurance is an effective way to spread risk. For example, Schmidli (2002), Baiand Guo (2008), Luo et al. (2008), and Chen et al. (2010) studied the optimal investmentand reinsurance strategies for insurers to minimize the ruin probability under different marketassumptions; Bai and Zhang (2008), Zeng (2011), Li et al. (2012), and Bi and Guo (2013)investigated the optimal investment and reinsurance strategies for insurers with mean–variancecriteria in different situations. In addition, some scholars have recentlystudied the optimal in-vestment and reinsurance strategies to maximize insurers’expected utility from the terminalwealth, see Cao and Wan (2009), Zhang et al. (2009), Liang et al. (2011), and so on.

On the other hand, it can be drawn from the facts of the global financial crisis and the Eu-ropean debit crisis that there exist many uncertainties in financial markets and insurance in-dustries. This motivates many researchers to consider problems with model uncertainties infinancial markets or insurance industries. See for example Maenhout (2006), Roma (2006), Xuet al. (2010) and Cairns (2000). One popular approach to describe model uncertainty is a robustapproach, where a family of probability models are introduced via perturbing an approximatingmodel and an optimal decision under uncertainty is made in the “worst–case” scenario overthe perturbed probability models. The rationale of this approach is to incorporate ambiguity

∗Supported in part by the National Natural Science Foundation of China.1Department of Statistics, Wuhan University of technology,Wuhan, 430070, P.R.China.2School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R.China.†Corresponding author. E-mail address: pxch@whu.edu.cn (Xingchun Peng).

1

ManuscriptClick here to view linked References

aversion. For details, see Leippold et al. (2008). In the case of a continuous–time optimiza-tion problem, the family of probability models, or probability measures, is usually introducedvia Girsanov–type transformations and the maximization ofexpected utility in the worst–casescenario is formulated as a two–person zero–sum stochasticdifferential game.

There has been a considerable works on studying optimal investment problems of insurerswith or without reinsurance under model uncertainties via stochastic differential games. Forexample, Zhang and Siu (2009) investigated an optimal investment and reinsurance problemwith model uncertainty, and formulated the problem as a zero–sum stochastic differential game.They solve the stochastic differential game by the HJBI equation method. Lin et al. (2012)discussed an optimal portfolio selection problem for an insurer facing model uncertainty in ajump–diffusion risk process and in its diffusion approximation by using some techniques instochastic linear–quadratic control. Yin et al. (2013) considered a robust optimal investmentand reinsurance problem for an Ambiguity–Averse Insurer (AAI) under Heston’s stochasticvolatility model by adopting the stochastic dynamic programming approach.

In this paper, we investigated an optimal investment, consumption and proportional rein-surance problem for an insurer facing model uncertainty in ajump diffusion surplus process.Unlike many existing works, the exogenous parameter process in our model are not assumedto be constants or Markov diffusion processes (such as the Heston’s stochastic volatility case).Thus the HJBI equation method cannot be used directly in our model because the dynamic pro-gramming principle no longer works in the non–Markov setting. One of the effective methods tohandle this kind of problem is to use backward stochastic differential equations (BSDEs), wherethe comparison theorem for BSDEs plays a key role. In the non–Markov setting, Øksendal andSulem (2011) studied an optimal investment problem under model uncertainty by using BS-DEs. Also by means of BSDEs, Lim and Quenez (2011) consideredan optimal investmentproblem under model uncertainty in markets with defaults. But the approaches of these two pa-pers are strongly linked to the exponential utility case. For general utility functions, Øksendaland Sulem (2012) studied an optimal investment and consumption problem under model uncer-tainty. They solved their problem in two steps. First, wrotethe performance functional as thevalue at time zero of the solution of an associated backward stochastic differential game of asystem of forward–backward stochastic differential equations. Then, they studied this stochasticdifferential game by the maximum principle approach. Inspired by Øksendal and Sulem (2012),we study the similar optimization problem for an insurer with general utility functions, wherethe model uncertainties not only come from the financial market, but also the insurance industry.Due to the importance of reinsurance for insurers to reduce risk, we consider the proportionalreinsurance besides investment and consumption. By a modified approach of Øksendal andSulem (2012), we use the stochastic maximum principles (including the necessary maximumprinciple and the sufficient maximum principle) to solve our problem. In some special cases, weobtain the explicit expressions of the optimal strategies by using Malliavin calculus, especiallythe generalized Clark–Ocone formula.

The remainder of this paper is organized as follows. Section2 gives the model we are inter-ested in. The optimization problem we aim to solve is transformed into a forward–backwardstochastic differential game driven by two dimensional Levy noises. Section 3 provides thestochastic maximum principles for the general two dimensional forward–backward stochasticdifferential game with jumps. The proofs are given in Appendix B.Section 4 is devotes tothe solution of the problem by using stochastic maximum principles established, where someinteresting special cases are studied by techniques of Malliavin calculus. Finally, Section 5concludes the study.

2

2. Model formulation

In our model, we assume that all the uncertainties come from afixed complete probabilityspace (Ω,F ,P) defined by

Ω = Ω1 × Ω2 × Ω3 × Ω4, F = F 1 × F 2 × F 3 × F 4, P = P1 × P2 × P3 × P4,

where (Ωi ,F i,Pi)i=1,2,3,4 are the probability spaces generated byW1, W2, N1 and N2, respec-tively. Here, W1 and W2 are Brownian motions, andN1 and N2 are compensated Poissonstochastic measures which have, respectively, the expressionsNi(dt, dz) = Ni(dt, dz)− νi(dz)dt,i = 1, 2, whereNi(dt, dz), i = 1, 2, are Poisson stochastic measures, andνi(dz), i = 1, 2, are theirLevy measures. We assume thatνi(dz) satisfy

∫R0

z2νi(dz) < ∞, whereR0 = R \ 0, i = 1, 2.

W1, W2, N1 andN2 are assumed to be mutually independent. LetFi = (F it )t>0, i = 1, 2, 3, 4, be

the filtration generated byW1, W2, N1 andN1, respectively, and augmented by the sets of nullprobability underPi, i = 1, 2, 3, 4, correspondingly. We setFt = F 1

t × F 2t × F 3

t × F 4t , t > 0.

Fix a constantT > 0 as the terminal time. Suppose that the surplus process of the insurerbefore investment is the following jump diffusion process:

dR(t) = p(t)dt+ q(t)dW1

t +

∫

R0

γ1(t, z)N1(dt, dz),

R(0) = x.

Here, the constantx > 0 is the initial surplus, ¯p(t), q(t) andγ1(t, z) are bounded (F 1t × F 3

t )t>0-adapted and caglad exogenous parameter processes.

Suppose that the insurer can invest in financial market dynamically with no transaction costand tax. Two investment possibilities are available:

(I) a risk free asset with price dynamics

dA(t) = r(t)A(t)dt,

A0 = 1,

(II) a risky asset with price dynamics

dS(t) = S(t−)

[µ(t)dt+ σ(t)dW2

t +

∫

R0

γ2(t, z)N2(dt, dz)

],

S(0) > 0.

Here, ¯r(t) is a deterministic function andµ(t),σ(t) andγ2(t, z) are all bounded (F 2t ⊗F 4

t )–adaptedand caglad exogenous parameter processes. Letb(t) denote the total amount of money investedin the risky asset at timet, andR(t) − b(t) is invested in the risk free asset. We assume that theinsurer can short sell the risky asset (b(t) < 0), and can also borrow money for investment inrisky asset (b(t) > R(t)). Besides investment, we also consider the consumption ofthe insurer.The consumption rate at timet is assumed to bec(t). In order to reduce its risk, the insurer willpurchase reinsurance or acquire new business. The proportional reinsurance/new business levelis associated with the value of risk exposurea(t) ∈ (−∞,+∞) at timet ∈ [0,T]. a(t) ∈ [0, 1]corresponds to a proportional reinsurance cover and implies that the insurer should paya(t)Yfor the claimY occurring at timet while the reinsurer should pay the rest (1− a(t))Y. a(t) > 1means that the insurer can take an extra insurance business from other companies (i.e., act as areinsurer for other cedents) anda(t) < 0 implies that the insurer acquires other new business.For convenience, we call the process of risk exposurea(t), t ∈ [0,T], as a reinsurance policy.

3

In this paper, we consider a partial information optimization problem in that the insurer hasonly partial information at its disposal. In other words, the investment, consumption and rein-surance proportion processesb(t), c(t) anda(t) are assumed to adapted to some filtration (G1

t )t>0

which is smaller than (Ft)t>0. (that isG1t ⊂ Ft, for all t ∈ [0,T]). HereG1

t represents the infor-mation obtained by the insurer at timet. All the decisions of the insurer are based on the flow ofinformationG1

t . The optimal portfolio problem under partial information in financial marketshas been studied by Di Nunno and Øksendal (2009). Due to the existence of relevant acronymsIBNR (Incurred But Not Reported) or even IBNyR (Not yet Reported) or IBNeR (Not enoughReported) in the insurance industry, we think the study of partial information problem in insur-ance is also valuable and interesting. For the optimizationproblem of proportional reinsuranceand investment under partial information, see Peng and Hu (2013).

Under the above assumptions, the surplus process with investment, consumption and reinsur-ance evolves over time as follows:

dRν(t) =[p(t) − (

1− a(t))λ(t) + r(t)(Rν(t) − b(t)) + µ(t)b(t) − c(t)

]dt+ a(t)q(t)dW1

t

+b(t)σ(t)dW2t +

∫R0

a(t)γ1(t, z)N1(dt, dz) +∫R0

b(t)γ2(t, z)N2(dt, dz),

Rν(0) = x,

(2.1)

whereλ(t) represents the safety loading premium of reinsurance satisfying λ(t) > p(t) > 0 a.s.,for all t ∈ [0,T], andν(·) = (a(·), b(·), c(·)) denotes the strategy process for the insurer.

Consider a model uncertainty setup, where a family of probability measures equivalent tothe original probability measureP is assumed to exist in reality. We may regard eachQκ asa scenario measure controlled by the financial market and insurance industry or the economicenvironment. DenoteGκ(t) as the Radon–Nikodym derivative ofQκ with respect toP on Ft,that is d(Qκ |Ft)

d(P|Ft)= Gκ(t). Assume thatGκ(t), t ∈ [0,T] is a martingale that satisfies the following

stochastic differential equation:

dGκ(t) = Gκ(t−)

[θ1(t)dW1

t + θ2(t)dW2t +

∫

R0

ζ1(t, z)N1(dt, dz) +

∫

R0

ζ2(t, z)N2(dt, dz)

],

Gκ(0) = 1.

Hereκ = (θ1, θ2, ζ1, ζ2) can be seen as the control process of the market. With respect to κ (orsay to the market), we also consider a partial information problem as follows. Given a filtration(flow of information)G2

t ⊂ Ft, t ∈ [0,T], suppose thatκ is a (G2t )06t6T–adapted process. Denote

A1 andA2 as the sets of all admissibleν andκ, respectively. The admissibleν andκ are assumedto satisfy the following conditions:(1) ν andκ are caglad and adapted to the filtrationG = (G1

t )06t6T andG = (G2t )06t6T respectively,

(2) a(t), b(t) andc(t) belong toR, for all t ∈ [0,T],(3)

E

[ ∫ T

0

(∣∣∣λ(t)a(t) + (µ(t) − r(t))b(t) − c(t)∣∣∣ + a2(t)q2(t) + b2(t)σ2(t)

+

∫

R0

a2(t)γ21(t, z)ν1(dz) +

∫

R0

b2(t)γ22(t, z)ν2(dz)

)dt

]< ∞,

(4)

E

∫ T

0

θ21(t) + θ

22(t) +

2∑

i=1

∫

R0

ζ2i (t, z)νi(dz)

dt

< ∞,4

(5) ζ1, ζ2 > −1+ ǫ (ǫ > 0).We aim to solve the following stochastic differential game problem between the insurer and

the market. Find (ˆν, κ) ∈ A1 × A2, such that

supν∈A1

infκ∈A2

EQκ [H(ν, κ)] = EQκ [H(ν, κ)] = infκ∈A2

supν∈A1

EQκ [H(ν, κ)] , (2.2)

whereH(ν, κ) = U2(Rν(T)) +∫ T

0U1(s, c(s))ds+

∫ T

0ρ(κ(t))dt, U1 : [0,T] × A1 −→ R andU2 :

R −→ R are two given functions, concave and increasing with a strictly decreasing derivative,

andρ is a convex function called penalty function. The termΛ(κ) = EQκ

[∫ T

0ρ(κ(t))dt

]can be

considered as a penalty term adopted to penalize the difference betweenQκ and the originalmeasureP.

Let F(t, u) = U1(t, c) + ρ(κ), u = (ν, κ) = (a, b, c, θ1, θ2, ζ1, ζ2). Then

EQκ(H(ν, κ)) = E

[Gκ(T)U2(R

ν(T)) +∫ T

0Gθ(s)F(s,Rν(s), u(s))ds

].

DefineX(t) = X(ν,κ)(t) by

X(t) = E

[Gκ(T)Gκ(t)

U2(Rν(T)) +

∫ T

t

Gκ(s)Gκ(t)

F(s,Rν(s), u(s))ds

∣∣∣∣∣∣Ft

], t ∈ [0,T]. (2.3)

Lemma 2.1. X(t) defined by (2.3) is the unique solution of the following linerbackward sto-chastic differential equation (BSDE).

dX(t) = −F(t, u(t)) +

2∑

i=1

θi(t)Yi(t) +2∑

i=1

∫

R0

ζi(t, z)Ki(t, z)νi(dz)

dt

+∑2

i=1 Yi(t)dWit +

∑2i=1

∫R0

Ki(t, z)Ni(dt, dz),

X(T) = U2(Rν(T)).

(2.4)

Proof. The existence and uniqueness of the solution to BSDE (2.4) follows immediately fromLemma 2.4 in Tang and Li (1994). We need only to verify thatX(t) defined by (2.3) is really asolution to this BSDE. By the Ito formula,

d(Gκ(t)X(t)) = Gκ(t−)dX(t) + X(t−)dGκ(t) + d[Gκ,X](t)

= −Gκ(t)

F(t, u(t)) +2∑

i=1

θi(t)Yi(t) +2∑

i=1

∫

R0

ζi(t, z)Ki(t, z)νi(dz)

dt

+Gκ(t−)

2∑

i=1

Yi(t)dWit +

2∑

i=1

∫

R0

Ki(t, z)Ni(dt, dz)

+ X(t−)Gκ(t−)

2∑

i=1

θi(t)dWit +

2∑

i=1

∫

R0

ζi(t, z)Ni(dt, dz)

+Gκ(t)2∑

i=1

θi(t)Yi(t)dt+Gκ(t−)2∑

i=1

∫

R0

ζi(t, z)Ki(t, z)Ni(dt, dz)

= −Gκ(t)F(t, u(t))dt+Gκ(t−)

2∑

i=1

Yi(t)dWit +

2∑

i=1

∫

R0

Ki(t, z)Ni(dt, dz)

5

+ X(t−)Gκ(t−)

2∑

i=1

θi(t)dWit +

2∑

i=1

∫

R0

ζi(t, z)Ni(dt, dz)

+Gκ(t−)2∑

i=1

∫

R0

ζi(t, z)Ki(t, z)Ni(dt, dz)

which impliesGκ(t)X(t) +∫ t

0Gκ(s)F(s, u(s))ds is a martingale. Therefore,

Gκ(t)X(t) +∫ t

0Gκ(s)F(s, u(s))ds

= E

[U2(R

ν(T))Gκ(T) +∫ T

0Gκ(s)F(s, u(s))ds

∣∣∣∣∣Ft

].

Equivalently,X(t) = E[

Gκ(T)Gκ(t) U2(Rν(T)) +

∫ T

tGκ(s)Gκ(t) F(s, u(s))ds

∣∣∣∣Ft

], t ∈ [0,T].

Note thatX(0) = X(ν,κ)(0) = EQκ [H(ν, κ)]. Thus, (2.2) is equivalent to

supν∈A1

infκ∈A2

X(ν,κ)(0) = X(ν,κ)(0) = infκ∈A2

supν∈A1

X(ν,κ)(0), (2.5)

whereX(ν,κ)(t) is determined by the forward–backward stochastic differential equation (FBSDE)(2.1) and (2.4). This is a two–person zero–sum forward–backward stochastic differential game.Since the coefficients of this FBSDE are path dependent stochastic processes,X(ν,κ) is not neces-sary a Markov process. In this setting, it is not effective to derive the corresponding HJBI equa-tion by the dynamic programming principle. Inspired by Øksendal and Sulem (2012), we usethe stochastic maximum principles to solve our problem. TheFBSDEs considered in Øksendaland Sulem (2012) were driven by one dimensional Brownian motion and compensated Poissonstochastic measure. However, the FBSDE (2.1) and (2.6) is driven by two dimensional Brown-ian motions and compensated Poisson stochastic measures. Thus, it is necessary to generalizethe related theoretic results in Øksendal and Sulem (2012) so that they are applicable to ourinterested problem (2.5).

In the next sections, we first give the general stochastic maximum principles of forward–backward stochastic differential game driven by two dimensional Levy noises (including Brow-nian motions and compensated Poisson stochastic measures). Then we use them to solve ourproblem (2.5).

3. Stochastic maximum principles

The results in this section are derived in a more general setting than the problem we areinterested in. For convenience, we will adopt some definitions and notations introduced in theprevious section without any confusion. For example, the definitions of (Ω,F ,P), (Ft)06t6T , W1,W2, N1 andN2 are the same to those in the previous section. DenoteW =

(W1

W2

), N =

(N1

N2

).

First, we give the following forward stochastic differential equation (SDE):

dR(t) = dRu(t) = A(t,R(t), u(t)

)dt+ B

(t,R(t), u(t)

)dW(t)

+

∫

R0

C(t,R(t), u(t), z

)N(dt, dz),

R(0) = x.

(3.1)

6

Here,B = (B1, B2), C = (C1,C2), u = (u1, u2), ui(t) is the control process (or strategy process)of playeri, i = 1, 2. As in the previous section, letGi

t ⊆ Ft, t ∈ [0,T], represent the informationobtained by the playeri at timet. LetAi be the set of admissible strategies of playeri and itconsists of all theGi

t–predictable processes valued inAi ⊂ Rd, d > 1, i = 1, 2. SetU = A1 × A2.Next, we give the BSDE thatX(t), Y(t) = (Y1(t),Y2(t)) andK(t, ·) = (K1(t, ·),K2(t, ·)) satisfy:

dX(t) = −g (t,R(t),X(t),Y(t),K(t, ·), u(t))dt+ Y(t)dW(t)

+

∫

R0

K(t, z)N(dt, dz),

X(T) = h(R(T)).

(3.2)

Here,g(t, r, x, y, k, u) : [0,T] × R × R2 × R2 × U −→ R andh(r) : R −→ R are given functionssuch that (3.2) has a unique solution.

Let f (t, r, u) : [0,T] × R × U −→ R, ϕ(r) : R −→ R andψ(x) : R −→ R are given profitfunction, legacy function and risk valuation, respectively. Define

J(u) = E

[∫ T

0f(t,R(u)(t), u(t)

)dt+ ϕ

(R(u)(T)

)+ ψ(X(0))

].

A general form of two–person zero–sum forward–backward stochastic differential game underpartial information is to findu∗ = (u∗1, u

∗2) ∈ A1 ×A2 such that

J(u∗1, u∗2) = sup

u1∈A1

infu2∈A2

J(u1, u2) = infu2∈A2

supu1∈A1

J(u1, u2). (3.3)

If there exists au∗ = (u∗1, u∗2) such that (3.3) holds, then we call it an optimal strategy.

The intuitive meaning of this problem is that there are two players I and II. Player I controlsu1 and player II controlsu2. The actions of the two players are antagonistic, which means thatbetween I and II there is a payoff J(u1, u2) that is a reward for I and a cost for II.

The aim of this section is to establish the stochastic maximum principles for (3.3). Define theHamiltonians

H(t, r, x, y, k, u, λ, p, q, ξ) : [0,T] × R × R × R2 × R2 × A1 × A2 × R × R × R2 × R2 −→ Rby

H(t, r, x, y, k, u, λ, p, q, ξ(·)) = f (t, r, u) + λg(t, r, x, y, k, u)

+ A(t, r, u)p+ B(t, r, u)qT +

2∑

i=1

∫

R0

Ci(t, r, u, z)ξTi (z)νi(dz),

(3.4)

whereq = (q1, q2), ξ = (ξ1, ξ2), R2 represents the set of all functionsξ(·) from R0 to R2 suchthat the integral in (3.4) converges. For technical requirement, we assume thatH is Frechetdifferentiable with respect tor, x, y andu. Let us recall some elements about Frechet derivativethat will be used later.

Let V be an open subset of a Banach spaceX with norm‖·‖ and letF : V −→ R.(i) We say thatF has a directional derivative atx ∈ X in the directiony ∈ X if

DyF(x) = limǫ→0

1ǫ

(F(x+ ǫy) − F(x))

exists.(ii) We say thatF is Frechet differentiable atx ∈ V if there exists a linear mapL : X −→ R

such that

limh→0,h∈X

1‖h‖

∣∣∣F(x+ h) − F(x) − L(h)∣∣∣ = 0

7

In this caseL is called the gradient ofF at x and we writeL = ∇xF.(iii) If F is Frechet differentiable, thenF has a directional derivative in all directionsy ∈ X

and

DyF(x) = ∇xF(y).

For the sake of convenience, we simply write

∂H∂x=∂H∂x

(t,R(t),X(t),Y(t),K(t, ·), u(t), λ(t), p(t), q(t), ξ(t, ·)) ,and analogously for the other partial derivatives ofH.

By using the HamiltonianH, we can give a system of FBSDE in the adjoint processesλ(t),p(t), q(t) andξ(t, z) as follows.

(i) Forward SDE inλ(t):

dλ(t) =∂H∂x

(t)dt+ ∇yH(t)dW(t) +∫

R0

∇kH(t, z)N(dt, dz),

λ(0) = ψ′(X(0)).(3.5)

(ii) BSDE in p(t), q(t), ξ(t, ·):

dp(t) = −∂H∂r

(t)dt+ q(t)dW(t) +∫

R0

ξ(t, z)N(dt, dz),

p(T) = ϕ′ (R(T)) + h′ (R(T)) λ(T).(3.6)

Theorem 3.1. (Sufficient maximum principle) Letu = (u1, u2) ∈ A1 × A2 with correspondingsolutionsR(t), X(t), Y(t), K(t, z), λ(t), p(t), q(t), and ξ(t, z) of equations (3.1), (3.2), (3.5) and(3.6). Assume that the following conditions hold.

(1) The functions x7→ h(x), x 7→ ϕ(x) and x 7→ ψ(x) are linear.(2) (The conditional maximum principle)

supu1∈A1

E[H(t, R(t), X(t), Y(t), K(t, ·), u1, u2(t), λ(t), p(t), q(t), ξ(t, ·))

∣∣∣G1t

]

= E[H(t, R(t), X(t), Y(t), K(t, ·), u1(t), u2(t), λ(t), p(t), q(t), ξ(t, ·))

∣∣∣G1t

],

(3.7)

and similarly,

infu2∈A2

E[H(t, R(t), X(t), Y(t), K(t, ·), u1(t), u2, λ(t), p(t), q(t), ξ(t, ·))

∣∣∣G2t

]

= E[H(t, R(t), X(t), Y(t), K(t, ·), u1(t), u2(t), λ(t), p(t), q(t), ξ(t, ·))

∣∣∣G2t

].

(3.8)

(3) For each fixed t, the function(r, x, y, k, u1) 7→ H(t, r, x, y, k, u1, u2(t), λ(t), p(t), q(t), ξ(t, ·)

)

is concave a.s., and the function(r, x, y, k, u2) 7→ H(t, r, x, y, k, u1(t), u2, λ(t), p(t), q(t), ξ(t, ·)

)is

convex a.s..(4) Moreover, the following growth conditions are assumed to hold:

E

[ ∫ T

0

p2(t)

∣∣∣B(t) − B(t)

∣∣∣2 +2∑

i=1

∫

R0

∣∣∣ξi(t, z) − ξi(t, z)∣∣∣2 νi(dz)

+(R(t) − R(t)

)2 ·|q(t)|2 +

2∑

i=1

∫

R0

∣∣∣ξi(t, z)∣∣∣2 νi(dz)

8

+(X(t) − X(t)

)2∣∣∣∇yH(t)

∣∣∣2 +2∑

i=1

∫

R0

∣∣∣∇ki H(t, z)∣∣∣2 νi(dz)

+ λ2(t)

∣∣∣Y(t) − Y(t)

∣∣∣2 +2∑

i=1

∫

R0

∣∣∣Ki(t, z) − Ki(t, z)∣∣∣2 νi(dz)

dt

]< ∞. (3.9)

Thenu(t) = (u1(t), u2(t)) is a saddle point for J(u), that is,u(t) is an optimal control.

Proof. See Appendix B.

Remark 3.2. Condition (3) in Theorem 3.1 may be restrict in general. However, there doesexist some cases in practice that fulfill this condition. Forexample, suppose that

H(t, r, x, y, k, u1, u2, λ(t), p(t), q(t), ξ(t, ·))= H1(t, r, x, y, k, λ(t), p(t), q(t), ξ(t, ·)) + H2(t, u1, λ(t), p(t), q(t), ξ(t, ·))+ H3(t, u2, λ(t), p(t), q(t), ξ(t, ·)).

Here(r, x, y, k) 7→ H1(t, r, x, y, k, λ(t), p(t), q(t), ξ(t, ·)) is linear, u1 7→ H2(t, u1, λ(t), p(t), q(t), ξ(t, ·))is concave, and u2 7→ H3(t, u2, λ(t), p(t), q(t), ξ(t, ·)) is convex, for each fixed t. Then it is easy toverify that condition (3) holds.

In what follows, we give a necessary maximum principle whichdoes not require the concav-ity condition (3) in the previous theorem. However, it requires the following assumptions.

(1) For allt0 ∈ [0,T] and all boundedGit–measurable random variablesαi, the control process

βi(t) = X(t0,T)αi belongs toAi, i = 1, 2.(2) For allui andβi belong toAi with βi bounded, there existsδi > 0 such that the control

process ˆui(t) = ui(t) + sβi(t) belongs to∈ Ai, for all s ∈ (−δi , δi), i = 1, 2.(3) The following derivative processes exist and belong toL2([0,T] ×Ω):

r1(t) =dds

R(u1+sβ1,u2)(t)∣∣∣∣∣s=0, x1(t) =

dds

X(u1+sβ1,u2)(t)∣∣∣∣∣s=0,

y1(t) = (y11(t), y

12(t)) =

dds

Y(u1+sβ1,u2)(t)∣∣∣∣∣s=0, k1(t, z) = (k1

1(t, z), k12(t, z)) =

dds

K(u1+sβ1,u2)(t, z)∣∣∣∣∣s=0,

and similarlyr2(t) = ddsR

(u1,u2+sβ2)(t)∣∣∣∣s=0

etc. SinceR(u1,u2)(0) = x for all (u1, u2), we haver i(0) =

0, for i = 1, 2. For convenience, we use the shorthand notation∂A∂r (t) for ∂A

∂r (t,R(t), u(t)) etc. Itis easy to verify that

dr1(t) =

[∂A∂r

(t)r1(t) +∂A∂u1

(t)β1(t)

]dt+

[∂B∂r

(t)r1(t) +∂B∂u1

(t)β1(t)

]dW(t)

+

∫

R0

[∂C∂r

(t, z)r1(t) +∂C∂u1

(t, z)β1(t)

]N(dt, dz), (3.10)

dx1(t) = −[∂g∂r

(t)r1(t) +∂g∂x

(t)x1(t) + ∇yg(t)(y1(t))T +

2∑

i=1

∫

R0

∇ki g(t, z)k1i (t, z)νi(dz)

+∂g∂u1

(t)β1(t)]dt+ y1(t)dW(t) +

∫

R0

k1(t, z)N(dt, dz),

x1(T) = h′(R(T))r1(T),

(3.11)

and similarly fordr2(t), dx2(t).9

Theorem 3.3. (Necessary maximum principle) Let u= (u1, u2) ∈ A1 × A2, R(t), X(t), Y(t),K(t, z), λ(t), p(t), q(t), ξ(t, z) be the solution processes to (3.1), (3.2), (3.5), (3.6). Besides theassumptions given above, we further assume that

E

[ ∫ T

0

p2(t)

∣∣∣∣∣∂B∂r

(t)r i(t) +∂B∂ui

(t)βi(t)∣∣∣∣∣2

+

2∑

j=1

∫

R0

∣∣∣∣∣∣∂C j

∂r(t, z)r i(t) +

∂C j

∂ui(t, z)βi(t)

∣∣∣∣∣∣2

ν j(dz)

+ (r i(t))2

|q(t)|2 +2∑

j=1

∫

R0

∣∣∣ξ j(t, z)∣∣∣2 ν j(dz)

+ λ2(t)

∣∣∣yi(t)

∣∣∣2 +2∑

j=1

∫

R0

∣∣∣kij(t, z)

∣∣∣2 ν j(dz)

+ (xi(t))2

|∇H(t)|2 +2∑

j=1

∫

R0

∣∣∣∇kj H(t, z)∣∣∣2 ν j(dz)

dt

]< ∞, i = 1, 2. (3.12)

Then the following are equivalent:

(1) ddsJ(u1 + sβ1, u2)

∣∣∣∣s=0= d

dsJ(u1, u2 + sβ2)∣∣∣∣s=0= 0, for all boundedβ1 ∈ A1, β2 ∈ A2.

(2)

E

[∂

∂u1H (t,R(t),X(t),Y(t),K(t, ·), u1, u2(t), λ(t), p(t), q(t), ξ(t, ·))

∣∣∣∣∣G1t

] ∣∣∣∣∣∣u1=u1(t)

= E

[∂

∂u2H (t,R(t),X(t),Y(t),K(t, ·), u1(t), u2, λ(t), p(t), q(t), ξ(t, ·))

∣∣∣∣∣G2t

] ∣∣∣∣∣∣u2=u2(t)

= 0.

Proof. See Appendix B.

4. Solution to the problem

We now apply the stochastic maximum principles developed inthe previous section to solveproblem (2.5). First, by the necessary maximum principle (Theorem 3.3) we can give the con-ditions thatu = (ν, κ) must satisfy, which are described by some equations. Then,in someparticular cases, we try to obtainu explicitly.

From (3.4), The Hamiltonian to (2.5) is

H(t, r, x, y, k, u, λ, p, q, ξ) = λ

U1(t, c) + ρ(κ) +2∑

i=1

θiyi +

2∑

i=1

∫

R0

ξi(z)ki(z)νi(dz)

+ p[p(t) − (1− a)λ(t) + r(t)(r − b) + µ(t)b− c

]+ q(t)aq1 + σ(t)bq2

+

∫

R0

γ1(t, z)ξ1(z)aν1(dz) +∫

R0

γ2(t, z)ξ2(z)bν2(dz).

(4.1)

From (3.5), the SDE forλ is

dλ(t) = λ(t)

[θ1(t)dW1

t + θ2(t)dW2t +

∫

R0

θ1(t, z)N1(dt, dz) +

∫

R0

θ2(t, z)N2(dt, dz)

],

λ(0) = 1.(4.2)

10

From (3.6), the BSDE forp, q, ξ is

dp(t) = q1(t)dW1t + q2(t)dW2

t +

∫

R0

ξ1(t, z)N1(dt, dz) +

∫

R0

ξ2(t, z)N2(dt, dz),

p(T) = λ(T)U′2(R(T)).(4.3)

It is clear thatu = (ν, κ) satisfy (1) in Theorem 3.3. If all the assumptions in Theorem 3.3 aresatisfied, then (2) in Theorem 3.3 holds true by the theorem. Consequently, by (4.1), we obtainthe following equations forν = (a, b, c).

E[λ(t)

∣∣∣G1t

] ∂U1

∂c(t, c(t)) = E

[p(t)

∣∣∣G1t

], (4.4)

E

[λ(t)p(t) + q(t)q1(t) +

∫

R0

γ1(t, z)ξ1(t, z)ν1(dz)∣∣∣∣∣G1

t

]= 0, (4.5)

E

[(µ(t) − r(t)

)p(t) + σ(t)q2(t) +

∫

R0

γ2(t, z)ξ2(t, z)ν2(dz)∣∣∣∣∣G1

t

]= 0, (4.6)

and the following equations forκ = (θ1, θ2, ζ1, ζ2):

∂ρ(κ(t))∂θ1

+ E[Y1(t)

∣∣∣G2t

]= 0, (4.7)

∂ρ(κ(t))∂θ2

+ E[Y2(t)

∣∣∣G2t

]= 0, (4.8)

∇ζ1ρ(κ(t)) + E

[∫

R0

K1(t, z)ν1(dz)∣∣∣∣∣G2

t

]= 0, (4.9)

∇ζ2ρ(κ(t)) + E

[∫

R0

K2(t, z)ν2(dz)∣∣∣∣∣G2

t

]= 0. (4.10)

In summary, we have the following theorem.

Theorem 4.1. If u = (ν, κ) is a solution to the forward–backward stochastic differential game(2.5), and all the assumptions in Theorem 3.3 are satisfied, then u must satisfy (4.4), (4.5), (4.6),(4.7), (4.8), (4.9) and (4.10).

Although we get the equations that characterizeu , these equations are very difficult to solvein general due to the conditional expectation with respectstoGi

t, i = 1, 2, the nonlocal term, thatis, the integral with respect to the Levy measureνi, i = 1, 2, and the fact that all the parameterprocesses are stochastic. In what follows, we try to solve these equations in some particularcases in order to get the concrete expressions of the optimalstategies.

First, we consider the case of complete information withoutjumps, that is,N1 = N2 = ν1 =

ν2 = K1 = K2 = ζ1 = ζ2 = 0,G1t = G2

t = Ft, 0 6 t 6 T. Then, the equations from (4.2) to (4.10)simplify to

λ(t) = exp

(∫ t

0θ1(s)dW1

s +

∫ t

0θ2(s)dW2

s −12

∫ t

0θ2

1(s)ds− 12

∫ t

0θ2

2(s)ds

), (4.11)

dp(t) = q1(t)dW1

t + q2(t)dW2t ,

p(T) = λ(T)U′2(R(T)),(4.12)

λ(t)∂U1

∂c(t, c(t)) = p(t), (4.13)

11

λ(t)p(t) + q(t)q1(t) = 0, (4.14)

(µ(t) − r(t))p(t) + σ(t)q2(t) = 0, (4.15)

∂ρ(κ(t))∂θ1

+ Y1(t) = 0, (4.16)

∂ρ(κ(t))∂θ2

+ Y2(t) = 0. (4.17)

The forward–backward stochastic differential equations (2.1) and (2.4) simplify to

dR(t) =[p(t) − (1− a(t))λ(t) + r(t)(R(t) − b(t))

+ µ(t)b(t) − c(t)]dt+ a(t)q(t)dW1

t + b(t)σ(t)dW2t ,

R(0) = x

(4.18)

and

dX(t) = −U1(t, c(t)) + ρ(κ(t)) +

2∑

i=1

θi(t)Yi(t)

dt+2∑

i=1

Yi(t)dWit ,

X(T) = U2(R(T))

(4.19)

respectively.By Proposition A.4, we have

q1(t) = E[D1

t

(λ(T)U′2(R(T))

) ∣∣∣Ft

], (4.20)

q2(t) = E[D2

t

(λ(T)U′2(R(T))

) ∣∣∣Ft

], (4.21)

where the Malliavin derivativeD = (D1,D2) is defined in the white noise setting (see AppendixA). Let

G = p(T) = λ(T)U′2(R(T)). (4.22)

By (4.12), we getp(t) = E(G|Ft). (4.23)

Substituting this into (4.14) and (4.15) gives

λ(t)E(G|Ft) + q(t)E(D1

t G∣∣∣Ft

)= 0, (4.24)

and

(µ(t) − r(t))E(G|Ft) + σ(t)E(D2

t G∣∣∣Ft

)= 0, (4.25)

respectively.According to Theorem A.1 in Øksendal and Sulem (2009), the general solution to (4.24) and

(4.25) is

p(t) = E(G|Ft) = E(G) exp

−

∫ t

0

λ(s)q(s)

dW1s −

12

∫ t

0

λ2(s)q2(s)

ds

−∫ t

0

µ(s) − r(s)σ(s)

dW2s −

12

∫ t

0

(µ(s) − r(s))2

σ2(s)ds

. (4.26)

Write λ∗ = E(G) as an unknown constant. Define the functionI i(x) by

I i(x) :=

(U′i )

−1(x), 0 6 x 6 xi ,

0, x > xi ,12

wherexi = limt→0+ U′i (t), i = 1, 2. Denote

Mt = exp

−

∫ t

0

λ(s)q(s)

dW1s −

12

∫ t

0

λ2(s)q2(s)

ds−∫ t

0

µ(s) − r(s)σ(s)

dW2s −

12

∫ t

0

(µ(s) − r(s))2

σ2(s)ds

(4.27)Noting that the Novikov condition is satisfied byMt, 0 6 t 6 T due to the boundedness of theparameter processes ¯r(·), µ(·), σ(·), p(·), q(·), andλ(·). Thus,Mt, 0 6 t 6 T is a martingale. By(4.22), (4.26) and (4.27), the optimal terminal wealth is

R(T) = I2

(λ∗MT

λ(T)

). (4.28)

The optimal consumption rate, by (4.13), is

c(t) = I1

(t,λ∗Mt

λ(t)

), 0 6 t 6 T. (4.29)

We can conclude from (4.16) and (4.17) that

θ1(t) =

(∂ρ(κ(t))∂θ1

)−1

(−Y1(t)) , (4.30)

θ2(t) =

(∂ρ(κ(t))∂θ2

)−1

(−Y2(t)) , (4.31)

where (X(t),Y(t)) is the solution to BSDE (4.19).Let us further assume thatU1 = c = 0, that is, no consumption is considered, and that

ρ(θ) = 12 |θ|2. Substituting (4.30), (4.31) and (4.28) into (4.19) gives

dX(t) =12

2∑

i=1

θ2i (t)dt−

2∑

i=1

θi(t)dWit ,

X(T) = U2

(I2

(λ∗MT

λ(T)

)).

(4.32)

It can be seen from (4.32) that2∑

i=1

∫ T

0θi(t)dWi

t −12

∫ T

0

2∑

i=1

θ2i (t)dt = X(0)− U2

(I2

(λ∗MT

λ(T)

)). (4.33)

Combining this with (4.11) leads to

λ(T) =exp

(X(0)

)

exp(U2

(I2

(λ∗MTλ(T)

))) .

Therefore,λ(t) satisfies the following BSDE:

dλ(t) = λ(t)2∑

i=1

θi(t)dWit ,

λ(T) = Σ,

(4.34)

whereΣ is determined by the following equation:

exp

(U2

(I2

(λ∗MT

Σ

)))Σ = exp

(X(0)

). (4.35)

13

By Proposition A.4 and BSDE (4.34), we have

λ(t)θ1(t) = E(D1t Σ|Ft), 0 6 t 6 T, (4.36)

λ(t)θ2(t) = E(D2t Σ|Ft), 0 6 t 6 T. (4.37)

Combining (4.2), (4.36) and (4.37) we obtain

dλ(t) = E(D1t Σ|Ft)dW1

t + E(D2t Σ|Ft)dW2

t ,

λ(0) = 1,(4.38)

θ1(t) =E(D1

t Σ|Ft)λ(t)

, 0 6 t 6 T, (4.39)

and

θ2(t) =E(D2

t Σ|Ft)

λ(t), 0 6 t 6 T. (4.40)

DenoteZ(t) = (Z1(t),Z2(t)) = (a(t)q(t), b(t)σ(t)). Thena(t) = Z1(t)q(t) , b(t) = Z2(t)

σ(t) and (4.18)becomes, by (4.28),

dR(t) =

[p(t) − λ(t) + r(t)R(t) +

λ(t)q(t)

Z1(t) +µ(t) − r(t)σ(t)

Z2(t)

]dt+ Z(t)dWt,

R(T) = I2

(λ∗MT

λ(T)

)= I2

(λ∗MT

Σ

).

(4.41)

By similar arguments to Lemma 2.1, we can verify that the unique solution to this continuousBSDE is

R(t) = exp

(−

∫ T

tr(s)ds

)· E

[I2

(λ∗MT

Σ

)MT

Mt

∣∣∣∣∣Ft

]

+ exp

(−

∫ T

tr(s)ds

)· E

[∫ T

t

Ms

Mt

(λ(s) − p(s)

)ds

∣∣∣∣∣Ft

]. (4.42)

In particular, settingt = 0, we get

exp

(∫ T

tr(s)ds

)x = E

[I2

(λ∗MT

Σ

)MT

]+ E

[∫ T

0Ms

(λ(s) − p(s)

)ds

]. (4.43)

Taking conditional expectations on both sides of (4.33) gives

X(0) = E

U2

(I2

(λ∗MT

Σ

))− 1

2

∫ T

0

2∑

i=1

θ2i (t)dt

. (4.44)

Up to now, we can see thatΣ, λ∗ andX(0) are determined by (4.35), (4.43) and (4.44). It iswell known that, under some mild conditions on the parameterprocesses in (4.41), the solution(R(t),Z(t)) to the BSDE (4.41) satisfies

Z(t) = DtR(t), a.s., t ∈ [0,T]. (4.45)

The definition of the Malliavin derivativeD = (D1,D2) is given in Appendix A. For a detaileddiscussion of this result, see El Karoui et al. (1997, Proposition 5.3).

We summarize what we have derived above as the following theorem.14

Theorem 4.2.Consider the problem to find(ν, κ) such that

supν∈A1

infκ∈A2

X(ν,κ)(0) = X(ν,κ)(0) = infκ∈A2

supν∈A1

X(ν,κ)(0),

where X(ν,κ) satisfy the following FBSDE:

dR(t) =[p(t) − (

1− a(t))λ(t) + r(t)R(t)

+ (µ(t) − r(t))b(t)]dt+ a(t)q(t)dW1

t + b(t)σ(t)dW2t ,

R(0) = x.

dX(t) = −12

2∑

i=1

θ2i (t) +

2∑

i=1

θi(t)Yi(t)

+2∑

i=1

Yi(t)dWit ,

X(T) = U2(R(T)).

Suppose that (4.45) holds. If all the assumptions in Theorem3.2 are satisfied, and assumption(4) in Theorem 3.1 holds, then the optimal strategyu = (a, b, θ1, θ2) is given by

a(t) =D1

t R(t)

q(t), (4.46)

b(t) =D2

t R(t)

σ(t), (4.47)

θ1(t) =E

(D1

t Σ∣∣∣Ft

)

1+∑2

i=1

∫ T

0E

(Di

tΣ∣∣∣Ft

)dWi

t

, (4.48)

θ2(t) =E

(D2

t Σ∣∣∣Ft

)

1+∑2

i=1

∫ T

0E

(Di

tΣ∣∣∣Ft

)dWi

t

, (4.49)

whereR(t) is the optimal surplus process with respect tou(t) that is given by (4.42),Σ, λ∗ andX(0) are determined by (4.35), (4.43) and (4.44).

Proof. Suppose that the optimal strategy ˆu exists. By Theorem 3.3 and the arguments above,we can conclude that ˆu is given by (4.46), (4.47), (4.48) and (4.49).

Conversely, suppose that ˆu is given by (4.46), (4.47), (4.48) and (4.49). Assumption (1) inTheorem 3.1 is trivially satisfied. By the explicit expression of the HamiltonianH, that is (4.1),and Remark 3.2, it is easy to verify that the assumptions (2) and (3) of Theorem 3.1 are satisfied.Condition (4) also holds by our assumption. Consequently, ˆu is the optimal strategy accordingto Theorem 3.1.

Corollary 4.3. Let U2 be an exponential function, that is, U2(x) = mv e−vx, with m and v are pos-

itive constants. If the parameter processesr(·), µ(·), σ(·), p(·), q(·), andλ(·) in Theorem 4.2 areall bounded and deterministic such thatΣ determined by (4.35) belongs toD1,2. Then we havethe following more explicit expressions for the optimal strategies of proportional reinsuranceand investment:

a(t) = exp

(−

∫ T

tr(s)ds

)λ(t)

vq2(t)+

exp(−

∫ T

tr(s)ds

)

q(t)E

(D1

t Σ

vΣ· MT

Mt

∣∣∣∣∣Ft

),

15

b(t) = exp

(−

∫ T

tr(s)ds

)µ(t) − r(t)

vσ2(t)+

exp(−

∫ T

tr(s)ds

)

σ(t)E

(D2

t Σ

vΣ· MT

Mt

∣∣∣∣∣Ft

).

Proof. WhenU2(x) = mv e−vx, then I2

(λ∗MTΣ

)= −1

v (ln λ∗ + ln MT − lnΣ) ∈ D1,2. By the chainrules of Malliavin derivatives, we have

D1t

(I2

(λ∗MT

Σ

))=λ(t)vq(t)

+D1

t Σ

vΣ,

D2t

(I2

(λ∗MT

Σ

))=µ(t) − r(t)

vσ(t)+

D2t Σ

vΣ.

It is easy to verify thatD1t

(MT

Mt

)= D2

t

(MT

Mt

)= 0. Therefore, from (4.42), (4.46) and (4.47) we

have

a(t) =D1

t R(t)q(t)

=

exp(−

∫ T

tr(s)ds

)

q(t)

D1

t E

[I2

(λ∗MT

Σ

)MT

Mt

∣∣∣∣∣Ft

]+ D1

t

[∫ T

t

(λ(s) − p(s)

)ds

]

=

exp(−

∫ T

tr(s)ds

)

q(t)E

[D1

t

(I2

(λ∗MT

Σ

)MT

Mt

) ∣∣∣∣∣Ft

]

= exp

(−

∫ T

tr(s)ds

)λ(t)

vq2(t)+

exp(−

∫ T

tr(s)ds

)

q(t)E

(D1

t Σ

vΣ· MT

Mt

∣∣∣∣∣Ft

),

b(t) =D2

t R(t)σ(t)

=

exp(−

∫ T

tr(s)ds

)

σ(t)

D2

t E

[I2

(λ∗MT

Σ

)MT

Mt

∣∣∣∣∣Ft

]+ D2

t

[∫ T

t

(λ(s) − p(s)

)ds

]

=

exp(−

∫ T

tr(s)ds

)

σ(t)E

[D2

t

(I2

(λ∗MT

Σ

)MT

Mt

) ∣∣∣∣∣Ft

]

= exp

(−

∫ T

tr(s)ds

)µ(t) − r(t)

vσ2(t)+

exp(−

∫ T

tr(s)ds

)

σ(t)E

(D2

t Σ

vΣ· MT

Mt

∣∣∣∣∣Ft

).

At the end of this section, we consider the very particular case with no model uncertainty. Inthis case, we assume thatθ1 = θ2 = 0 andλ ≡ 1. The problem we are interested in reduces to

supν

Jν(0) = supν

E

[∫ T

0U1(t, c(t))dt+ U2(R(T))

], (4.50)

which is an optimal investment, consumption and proportional reinsurance problem for an in-surer. This is a problem of its own interest and value since the parameter processes we considerhere are assumed to be stochastic. The usually adopted HJB equation method is not effectivehere since our problem is not necessarily in the Markovian setting. Here we solve this problemaccording to the above arguments.

By (4.28) and (4.29), the optimal terminal wealth and consumption rate for the insurer are

R(T) = I2(λ∗MT), (4.51)

16

and

c(t) = I1(t, λ∗Mt), 0 6 t 6 T, (4.52)

respectively. Note thatλ∗ is an unknown constant. In order to findλ∗, we consider (4.18) thatR(t) satisfies. As above, letZ(t) =

(a(t)q(t), b(t)σ(t)

). Combining (4.18) and (4.51), we have

dR(t) =[r(t)R(t) + p(t) − λ(t) +

(λ(t)q(t)

,µ(t) − r(t)σ(t)

)Z(t)T

− I1(t, λ∗Mt)

]dt+ Z(t)dWt,

R(T) = I2(λ∗Mt).

(4.53)

Solving this continuous BSDE, we obtain

R(t) = exp

(−

∫ T

tr(s)ds

)· E

[I2(λ∗MT

) · MT

Mt

∣∣∣∣∣Ft

]

+ exp

(−

∫ T

tr(s)ds

)· E

[∫ T

t

Ms

Mt

(λ(s) − p(s) + I1(s, λ

∗Ms))ds

∣∣∣∣∣Ft

]. (4.54)

Settingt = 0, we get

x = E[I2(λ∗MT

) · MT]+ E

[∫ T

0Ms

(λ(s) − p(s) + I1(s, λ

∗Ms))ds

], (4.55)

which determinesλ∗. As the equality (4.45) shows, under some mild conditions onthe param-eter processes in (4.53) (see El Karoui et al. (1997, Proposition 5.3)), the optimal proportionalreinsurance and investment strategies are given by

a(t) =D1

t R(t)

q(t)(4.56)

and

b(t) =D2

t R(t)

σ(t)(4.57)

respectively, whereR(t) is given by (4.54).In particular, ifU1(x) = U2(x) = m

v e−vx, m,v > 0, and the parameter processes ¯r(·), µ(·), σ(·),p(·), q(·), andλ(·) are all bounded and deterministic. Then we can get more explicit expressionsfor the optimal strategies of investment, consumption and proportional reinsurance as Corollary4.3 shows. In this case, it is easy to verify that

D1t I i(λ

∗Ms) =λ(t)vq(t)

, t < s6 T, i = 1, 2;

D2t I i(λ

∗Ms) =µ(t) − r(t)

vσ(t), t < s6 T, i = 1, 2;

D1t

(Ms

Mt

)= D2

t

(Ms

Mt

)= 0, t < s6 T.

17

Therefore, from (4.54), (4.56), (4.57), and (4.52), we obtain the following concrete expressionsof the optimal strategies for problem (4.50).

a(t) =D1

t R(t)

q(t)=

1q(t)

exp

(−

∫ T

tr(s)ds

)D1

t E

[I2(λ

∗MT) · MT

Mt

∣∣∣∣∣Ft

]

+1

q(t)exp

(−

∫ T

tr(s)ds

)D1

t

E

[∫ T

t

Ms

MtI1(λ

∗Ms)ds∣∣∣∣∣Ft

]+

∫ T

t

(λ(s) − p(s)

)ds

=1

q(t)exp

(−

∫ T

tr(s)ds

) E

[D1

t

(I2(λ

∗MT) · MT

Mt

) ∣∣∣∣∣Ft

]+ E

[D1

t

∫ T

t

Ms

MtI1(λ

∗Ms)ds∣∣∣∣∣Ft

]

= exp

(−

∫ T

tr(s)ds

)λ(t)(T − t + 1)

vq2(t),

b(t) =D2

t R(t)

σ(t)=

1σ(t)

exp

(−

∫ T

tr(s)ds

)D2

t E

[I2(λ

∗MT) · MT

Mt

∣∣∣∣∣Ft

]

+1σ(t)

exp

(−

∫ T

tr(s)ds

)D2

t

E

[∫ T

t

Ms

MtI1(λ

∗Ms)ds∣∣∣∣∣Ft

]+

∫ T

t

(λ(s) − p(s)

)ds

=1σ(t)

exp

(−

∫ T

tr(s)ds

) E

[D2

t

(I2(λ

∗MT)MT

Mt

) ∣∣∣∣∣Ft

]+ E

[D2

t

∫ T

t

Ms

MtI1(λ

∗Ms)ds∣∣∣∣∣Ft

]

= exp

(−

∫ T

tr(s)ds

)(µ(t) − r(t))(T − t + 1)

vσ2(t),

c(t) =1v

(− ln λ∗ − ln MT + ln m) ,

whereλ∗ andMT are given by (4.55) and (4.27) respectively.

5. Conclusion

In this paper, we consider the optimal investment, consumption and proportional reinsur-ance strategies for an insurer under model uncertainty. Thewealth process with investment,consumption and proportional reinsurance is supposed to bea general jump diffusion processdriven by two–dimensional Levy noises. The model uncertainty is described by a family ofprobability measures equivalent to the original probability measure. We transform the probleminto a two–person zero–sum forward–backward stochastic differential game driven by two–dimensional Levy noises. Then, we establish the maximum principles for a general form ofthis game and use them to solve the problem. Last, we use Malliavin calculus to derive explicitexpressions of the optimal strategies in some interesting particular cases.

The issue of robustness on the optimal strategies has been investigated by several papers ina similar framework by numerical simulation. However, thisissue is not studied in the presentpaper, since we have not found an effective way to numerically solve the Malliavin derivativeincluded in the optimal strategies in Theorem 4.2 and Corollary 4.3. Hence, it would be inter-esting to see this issue to be worked out.

Acknowledgements:18

The authors are very grateful to the Editor and the referee for their helpful comments andsuggestions on the original version of the manuscript, which led to this improved version of themanuscript.

Appendix A. Some elements about theMalliavin calculus

We only give a brief review of the Malliavin calculus here. For more information, see Nualart(2006) and Ni Nunno et al. (2009). Denote the probability space on whichW(t), t > 0, is aWiener process by (Ω0,FW

T ,PW). Let (Ω1,F 1T ,P1) and (Ω2,F 2

T ,P2) be two independent copiesof (Ω0,FW

T ,PW). Set

Ω = Ω1 × Ω2, FT = F 1T × F 2

T , P = P1 × P2.

We consider the product of the form

Iα(ω) = Iα(1)(f1,α(1)

)(ω1) · Iα(2)

(f2,α(2)

)(ω2)

for anyα ∈ I2, which is the set of indices of the formα = (α(1), α(2)), with α(1) = 0, 1, ..., fork = 1, 2. HereIα(k)

(fk,α(k)

)(ωk) is theα(k)–fold iterated Ito integral with respect to the Wiener

processWk, for k = 1, 2. The elementsIα, α ∈ I2, constitute an orthogonal basis inL2(P). AnyrealFT–measurable random variableF ∈ L2(P) can be written as

F =∑

α∈J2

Iα

for an appropriate choice of deterministic symmetric integrands in the iterated Ito integrals.

Definition A.1. (1) We say that F∈ D1,2 if

‖F‖2D1,2=

2∑

k=1

∑

α∈J2

α(k)α(k)!∥∥∥ fk,α(k)

∥∥∥2

L2([0,T]α(k)

) < ∞.

(2) If F ∈ D1,2, we define the Malliavin derivative DF of F as the gradient

DF = (D1t F,D2

t F),

whereDk

t F =∑

α∈J2

α(k)Iα−ǫ(k)(t), t ∈ [0,T], k = 1, 2.

Hereǫ(k) = (0, ..., 0, 1, 0, ..., 0) with 1 in the kth position.

For the properties of the above defined Malliavin derivatives, see Nualart (2006) or Di Nunnoet al. (2009). Besides the approach to define Malliavin derivatives above, we also introduce theway to define Malliavin derivatives in the white noise setting that are usually called Hida–Malliavin derivatives. In the following, we sketch it.

Let W be a Wiener process on the white noise space (Ω0,FWT ,PW). Here,Ω0 = S′(R) is the

Schwartz space of tempered distributions, andPW is the Gaussian white noise measure. LetJ denote the set of all finite multi-indicesα = (α1, α2, ..., αm), m = 1, 2, ..., of non-negativeintegers. Ifα = (α1, α2, ..., αm) ∈ J , α , 0, we put

Hα(ω0) =m∏

j=1

hα j (θ j (ω0)), ω0 ∈ Ω0,

19

whereθ j(ω0) = 〈ω0, ej〉 =∫R

ej(x)dW(x, ω0),ej

j>1

constitutes an orthonormal basis forL2(R)

andej ∈ S(R) (S(R) represents the Schwartz space of rapidly decreasing smooth functionson R) for all j, andhα j is the Hermite polynomial of orderα j for all j. Let (Ω1,F 1

T , µ1) and(Ω2,F 2

T , µ2) be two copies of (Ω0,FWT ,PW). We set

Ω = Ω1 × Ω2, FT = F 1T × F 2

T , P = µ1 × µ2.

We denoteJ2 by the set of allβ = (β(1), β(2)) with β( j) ∈ J , j = 1, 2. Further, we put

Hβ(ω) = Hβ(1)(ω1)Hβ(2)(ω2).

Then the familyHβ, β ∈ J2, constitutes an orthogonal basis forL2(P). More precisely, for anyFT measurableX ∈ L2(P), there exist uniquely determined numberscβ ∈ R, such that

X =∑

β∈J2

cβHβ.

Based onHβ, β ∈ J2, we can define the corresponding stochastic distribution spaceG∗ by theinductive limit topology (see Holden et al. (2009)).

Definition A.2. We define the Hida–Malliavin derivativesDit as mappingsG∗ → G∗ by

DitX =

∑

β∈J2

∑

j>1

cββ(i)j Hβ−ǫ(i, j)ej(t), t ∈ [0,T], i = 1, 2,

whereǫ(1, j) = (ǫ( j), 0) andǫ(2, j) = (0, ǫ( j)).

The following proposition gives a very important relationship between the Hida–Malliavinderivative and the Malliavin derivative. For its proof, seeDi Nunno et al. (2009).

Proposition A.3. The Hida–Malliavin derivativeD coincides with the Malliavin derivative Don the spaceD1,2 given in Definition A.1.

Define the two independent Wiener processes on (Ω,FT ,P) as follows

Wj(t, ω) =W(t, ω j), ω = (ω1, ω2), j = 1, 2.

We give the following important generalized Clark–Ocone formula in the white noise setting.For its proof, see Aase et al. (2000) or Di Nunno et al. (2009).

Proposition A.4. Let ξ ∈ L2(P) be anFT–measurable random variable. Then

ξ = E[ξ] +∫ T

0E

[D1

t ξ∣∣∣Ft

]dW1(t) +

∫ T

0E

[D2

t ξ∣∣∣Ft

]dW2(t).

Appendix B. Proofs of the maximum principles

Proof of Theorem 3.1: First, we prove that for allu1 ∈ A1,

J(u1, u2) 6 J(u1, u2).

For each fixedu1 ∈ A1, denote∆ = J(u1, u2) − J(u1, u2) = I1 + I2 + I3, where

I1 = E

[∫ T

0

(f (t,R(t), u(t)) − f

(t, R(t), u(t)

))dt

],

I2 = E[ϕ (R(T)) − ϕ

(R(T)

)],

I3 = E[ψ (X(0)) − ψ

(X(0)

)].

20

By (3.4), we have

I1 = E

[ ∫ T

0

(H(t) − H(t) − λ(t) (g(t) − g(t)) − p(t)

(A(t) − A(t)

)

−(B(t) − B(t)

)qT(t) −

2∑

i=1

∫

R0

(Ci(t, z) − Ci(t, z)

)ξi(t, z)νi(dz)

)dt

]. (B.1)

By the linearity ofϕ, (3.6), (3.9) and the Ito formula, we have

I2 = E[ϕ′

(R(T)

) (R(T) − R(T)

)]= E

[(p(T) − h′

(R(T)

)λ(T)

) (R(T) − R(T)

)]

= E

[ ∫ T

0p(t−)d

(R(t) − R(t)

)+

∫ T

0

(R(t−) − R(t−)

)dp(t) +

∫ T

0

(B(t) − B(t)

)qT(t)dt

+

2∑

i=1

∫ T

0

∫

R0

(Ci(t, z) − Ci(t, z)

)ξi(t, z)νi(dz)dt

]− E

[λ(T)h′

(R(T)

) (R(T) − R(T)

)]

= E

[ ∫ T

0p(t)

(A(t) − A(t)

)dt+

∫ T

0

(R(t) − R(t)

) (−∂H(t)

∂r

)dt+

∫ T

0

(B(t) − B(t)

)qT(t)dt

+

2∑

i=1

∫ T

0

∫

R0

(Ci(t, z) − Ci(t, z)

)ξi(t, z)νi(dz)dt

]− E

[λ(T)h′

(R(T)

) (R(T) − R(T)

)]. (B.2)

By the linearity ofψ andh, (3.5), (3.9), and the Ito formula, we have

I3 = E[ψ (X(0)) − ψ

(X(0)

)]

= E[ψ′ (X(0))

(X(0)− X(0)

)]

= E[λ(0)

(X(0)− X(0)

)]

= E[λ(T)

(X(T) − X(T)

)]− E

[∫ T

0

(X(t−) − X(t−)

)dλ(t) +

∫ T

0λ(t−)d

(X(t) − X(t)

)]

− E

∫ T

0∇yH(t)

(Y(t) − Y(t)

)Tdt+

2∑

i=1

∫ T

0

∫

R0

∇ki H(t, z)(Ki(t, z) − Ki(t, z)

)νi(dz)dt

= E[λ(T)h′

(R(T)

) (R(T) − R(T)

)]

− E

[∫ T

0

(∂H(t)∂x

) (X(t) − X(t)

)dt+

∫ T

0λ(t) (g(t) − g(t)) dt

]

− E

∫ T

0∇yH(t)

(Y(t) − Y(t)

)Tdt+

2∑

i=1

∫ T

0

∫

R0

∇ki H(t, z)(Ki(t, z) − Ki(t, z)

)νi(dz)dt

(B.3)

By (B.1), (B.2) and (B.3), we get

∆ = I1 + I2 + I3

6 E

[ ∫ T

0

H(t) − H(t) − ∂H(t)

∂r(R(t) − R(t)

) − ∂H(t)∂x

(X(t) − X(t)

)

− ∇yH(t)(Y(t) − Y(t)

)T −2∑

i=1

∫

R0

∇ki H(t, z)(Ki(t, z) − Ki(t, z)

)νi(dz)

dt

]

21

= E

[ ∫ T

0E[

H(t) − H(t) − ∂H(t)∂r

(R(t) − R(t)

) − ∂H(t)∂x

(X(t) − X(t)

)

− ∇yH(t)(Y(t) − Y(t)

)T −2∑

i=1

∫

R0

∇ki H(t, z)(Ki(t, z) − Ki(t, z)

)νi(dz)

∣∣∣∣∣G1t

]dt

]. (B.4)

Since (r, x, y, k, u1) −→ H(t, r, x, y, k, u1, u2(t), λ(t), p(t), q(t), ξ(t, ·)) is a concave function, we

can conclude that

H(t) − H(t) 6∂H(t)∂r

(R(t) − R(t)

)+∂H(t)∂x

(X(t) − X(t)

)+ ∇yH(t)

(Y(t) − Y(t)

)T

+∂H(t)∂u1

(u1(t) − u1(t)

)+

2∑

i=1

∫

R0

∇ki H(t, z)(Ki(t, z) − Ki(t, z)

)νi(dz). (B.5)

According to (3.7) we have

∂

∂u1E

[H(t, R(t), X(t), Y(t), k(t, ·), u1, u2(t), λ(t), p(t), q(t), ξ(t, ·))

∣∣∣∣G1t

] ∣∣∣∣∣u=u1(t)

· (u1(t) − u1(t))6 0,

Furthermore, we have

E

[∂H(t)∂u1

(u1(t) − u1(t)

)∣∣∣∣∣G1t

]6 0, (B.6)

Combining (B.4), (B.5) and (B.6) we get that for allu1 ∈ A1,

∆ = J(u1, u2) − J(u1, u2) 6 0.

By similar arguments we can conclude that for allu2 ∈ A2,

J(u1, u2) > J(u1, u2).

Thus, for all (u1, u2) ∈ A1 ×A2, we have

J(u1, u2) 6 J(u1, u2) 6 J(u1, u2).

Therefore,J(u1, u2) 6 inf

u2∈A2

J(u1, u2) 6 supu1∈A1

infu2∈A2

J(u1, u2)

andJ(u1, u2) > sup

u1∈A1

J(u1, u2) > infu2∈A2

supu1∈A1

J(u1, u2).

On the other hand, we always have

supu1∈A1

infu2∈A2

J(u1, u2) 6 infu2∈A2

supu1∈A1

J(u1, u2).

In conclusion, we get

J(u1, u2) = supu1∈A1

infu2∈A2

J(u1, u2) = infu2∈A2

supu1∈A1

J(u1, u2).

Proof of Theorem 3.3: Clearly,

dds

J(u1 + sβ1, u2)∣∣∣∣∣s=0

22

= E

[∫ T

0

(∂ f (t)∂r

r1(t) +∂ f (t)∂u1

β1(t)

)dt+ ϕ′(R(T))r1(T) + ψ′(X(0))x1(0)

](B.7)

By (3.6), (3.10), (3.12) and the Ito formula, we have

E[ϕ′

(R(T)

)r1(T)

]= E

[p(T)r1(T)

]− E

[h′

(R(T)

)r1(T)λ(T)

]

= E

[ ∫ T

0

p(t−)dr1(t) + r1(t−)dp(t) +

[∂B∂r

(t)r1(t) +∂B∂u1

(t)β1(t)

]qT(t)dt

+

2∑

i=1

∫

R0

[∂Ci

∂r(t, z)r1(t) +

∂Ci

∂u1(t, z)β1(t)

]ξi(t, z)νi(dz)

dt

]− E

[h′

(R(T)

)r1(T)λ(T)

]

= E

[ ∫ T

0

p(t)

[∂A∂r

(t)r1(t) +∂A∂u1

(t)β1(t)

]− r1(t)

∂H∂r

(t) +

[∂B∂r

(t)r1(t) +∂B∂u1

(t)β1(t)

]qT(t)

+

2∑

i=1

∫

R0

[∂Ci

∂r(t, z)r1(t) +

∂Ci

∂u1(t, z)β1(t)

]ξi(t, z)νi(dz)

dt

]− E

[h′

(R(T)

)r1(T)λ(T)

]. (B.8)

By (3.5), (3.11), (3.12) and the Ito formula, we have

E[ψ′(X(0))x1(0)

]= E

[λ(0)x1(0)

]

= E[λ(T)x1(T)

]− E

[ ∫ T

0λ(t−)dx1(t) +

∫ T

0x1(t−)dλ(t) +

∫ T

0∇yH(t)(y1(t))Tdt

+

2∑

i=1

∫ T

0

∫

R0

k1i (t, z)∇ki H(t, z)νi(dz)dt

]

= E[λ(T)h′(R(T))r1(T)

]− E

[ ∫ T

0

− λ(t)

[∂g(t)∂r

r1(t) +∂g(t)∂x

x1(t) + ∇yg(t)(y1(t))T

+

2∑

i=1

∫

R0

k1i (t, z)∇ki g(t, z)νi(dz) +

∂g(t)∂u1

β1(t)]

+∂H(t)∂x

x1(t) + ∇yH(t)(y1(t))T +

2∑

i=1

∫

R0

k1i (t, z)∇ki H(t, z)νi(dz)

dt

]. (B.9)

Substituting (B.8) and (B.9) into (B.7) gives

dds

J(u1 + sβ1, u2)∣∣∣∣s=0

= E

[ ∫ T

0

[∂ f∂r

(t) + p(t)∂A∂r

(t) +∂B∂r

(t)qT(t) +2∑

i=1

∫

R0

∂Ci

∂r(t, z)ξi(t, z)νi(dz)

− ∂H∂r

(t) + λ(t)∂g∂r

(t)]r1(t) +

[− ∂H∂x

(t) + λ(t)∂g∂x

(t)]x1(t)

+

[− ∇yH(t) + λ(t)∇yg(t)

] (y1(t)

)T+

2∑

i=1

∫

R0

[− ∇ki H(t, z) + λ(t)∇ki g(t, z)

]k1

i (t, z)νi(dz)

+

[∂ f∂u1

(t) + p(t)∂A∂u1

(t) +∂B∂u1

(t)qT(t) +2∑

i=1

∫

R0

∂Ci

∂u1(t, z)ξi(t, z)νi(dz) + λ(t)

∂g∂u1

(t)]β1(t)

dt

]

23

= E[ ∫ T

0

∂H∂u1

(t)β1(t)dt]

= E[ ∫ T

0E[∂H∂u1

(t)β1(t)∣∣∣∣∣G1

t

]dt

]. (B.10)

Suppose thatddsJ(u1 + sβ1, u2)∣∣∣s=0= 0 holds for any boundedβ1 ∈ A1. Let β1 ∈ A1 be of the

following form

β1(t) = I(t0,t0+h](t)α1, (B.11)

whereα1 is a boundedG1t0–measurable random variable,t0 > 0, h > 0 such thatt0 + h 6 T.

Substituting (B.11) into (B.10) gives

E

[∫ t0+h

t0

E

[∂H(t)∂u1

∣∣∣∣∣G1t0

]α1dt

]= 0, (B.12)

Dividing by h (h > 0) on both sides and lettingh tend to 0, by the dominated convergencetheorem and the Lebesgue differential theorem, we obtain

E

[E

[∂H(t0)∂u1

∣∣∣∣∣G1t0

]α1

]= 0, a.e. t0 ∈ [0,T].

Since this equation holds for all boundedG1t0–measurable random variables, we can conclude

that

E

[∂H(t)∂u1

∣∣∣∣∣G1t

]= 0, a.e. t ∈ [0,T].

On the other hand, suppose thatddsJ(u1, u2 + sβ2)

∣∣∣s=0= 0 holds for all boundedβ2 ∈ A2. By

similar arguments as above we can deduce that

E

[∂H(t)∂u2

∣∣∣∣∣G2t

]= 0, a.e. t ∈ [0,T].

So the proof of (1)=⇒ (2) is complete.Conversely, if

E

[∂H(t)∂u1

∣∣∣∣∣G1t

]= 0,

then (B.12) holds. That is to say, whenβ1(t) = I(t0,t0+h](t)α1 we have

dds

J(u1 + sβ1, u2)∣∣∣∣s=0= E

[∫ T

0E

[∂H(t)∂u1

β1(t)∣∣∣∣∣G1

t

]dt

]= 0. (B.13)

It is easy to verify that (B.13) still holds for any step processβ1 ∈ A1 that is a linear combinationof some processes of the form (B.11). For any boundedβ1 ∈ A1, there exists a sequence ofuniformly bounded step processes

β(n)

1

contained inA1 such thatβ(n)

1 converges toβ1 asn tendto +∞, dP× dt a.e.. By the dominated convergence theorem, (B.13) holds for any boundedβ1 ∈ A1. On the other hand, suppose thatE

[∂H(t)∂u2

∣∣∣G2t

]= 0. By similar arguments as above,

ddsJ(u1, u2 + sβ2)

∣∣∣s=0= 0 holds for all boundedβ2 ∈ A2. So the proof of (2)=⇒ (1) is complete.

24

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26

HIGHLIGHTS

• An investment consumption, and proportional reinsurance problem under model uncer-tainty with general utility function is studied.

• Our problem is discussed in a general jump diffusion framework.

• Stochastic maximum principles are used to solve the problem.

• Some special cases are considered in detail by using Malliavin calculus.

1

Highlights (for review)