Insurance: Mathematics and Economics 59 (2014) 78–86

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Optimal investment, consumption and proportional reinsurance foran insurer with option type payoff

Xingchun Peng a,∗, Linxiao Wei b, Yijun Hu b

a Department of Statistics, Wuhan University of Technology, Wuhan, 430070, PR Chinab School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, PR China

h i g h l i g h t s

• An investment, consumption and proportional reinsurance problem with option type payoff is considered.• Our problem is discussed in a general diffusion framework.• Very general constraints are imposed on the investment and reinsurance rate.• The problem is solved by using backward stochastic differential equations.• Some special cases are studied in detail by using Malliavin calculus.

a r t i c l e i n f o

Article history:Received November 2013Received in revised formJuly 2014Accepted 1 September 2014Available online 10 September 2014

MSC:97M3091G80s60H30

Keywords:InvestmentConsumptionReinsuranceBackward stochastic differential equationMalliavin calculus

a b s t r a c t

This paper is devoted to the study of optimization of investment, consumption and proportionalreinsurance for an insurer with option type payoff at the terminal time under the criterion of exponentialutility maximization. The surplus process of the insurer and the financial risky asset process are assumedto be diffusion processes driven by Brownian motions which are non-Markovian in general. Very generalconstraints are imposed on the investment and the proportional reinsurance processes. Based on themartingale optimization principle, we use BSDE and BMO martingale techniques to derive the optimalstrategy and the optimal value function. Some interesting particular cases are studied inwhich the explicitexpressions for the optimal strategy are given by using the Malliavin calculus.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

In order to control andmanage risk, reinsurance and investmentare two effectiveways for insurers. As a result, optimal reinsuranceand investment problems for insurers have attracted much atten-tion in the actuarial literature in recent years. To study the optimalinvestment–reinsurance policies for an insurer, maximization ofexponential utility, among many other criteria, is widely adoptedby researchers. See Bai andGuo (2008), Cao andWan (2009), Zhanget al. (2009), Liang et al. (2011), Liang and Bayraktar (2014) and thereferences therein.

This work was supported in part by the National Science Foundation of China(No. 11371284).∗ Corresponding author.

E-mail address: pxch@whu.edu.cn (X. Peng).

http://dx.doi.org/10.1016/j.insmatheco.2014.09.0020167-6687/© 2014 Elsevier B.V. All rights reserved.

In the papers mentioned above, the main approaches to studythe optimization problems are stochastic control theory and re-lated methodologies, especially the HJB equation method. That isa quite effective method for many stochastic optimization prob-lems in insurance. However, the Bellman programming principle,which leads to the HJB equations, can only be applied toMarkovianstate processes. This implies that the HJB equation method cannotbe used directly when the state process is assumed to be a non-Markovian process which is the case we are interested in.

In the present paper, we consider an exponential utility maxi-mization problem for an insurer with a consumption process, whoinvests in financialmarkets and takes the proportional reinsurancebusiness. Meanwhile, we also consider the case where the wealthprocess is of a terminal option type payoff, since option type payoffcan be used to reduce risk. We will allow the parameter processesto be general stochastic processes. Therefore, thewealth process of

X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86 79

the insurer with investment, consumption and proportional rein-surance is non-Markovian in general. Instead of adopting the HJBequation method, we use BSDE and BMO martingale techniquesto solve our problem. Under very general constraints on the pro-portional reinsurance rate and the investment, we can derive theoptimal strategy in a closed form. These constraints are weakerthan those employed by the papers mentioned above. In particu-lar, when there is no constraint on the investment process and thereinsurance rate is assumed to be nonnegative, or no reinsurance isconsidered, we can obtain the explicit expressions for the optimalstrategies by using the Malliavin calculus.

As a matter of fact, the BSDE and BMO martingale techniqueswere used to handle the utility maximization problem with op-tion type payoff in finance. Hu et al. (2005) first studied the utilitymaximization problem in an incomplete financial market by us-ing BSDE and BMOmartingale techniques. Later, Morlais (2009a,b)extended the risk asset process driven by the Brownian motion tothat driven by general continuous martingales and the Lévy pro-cess respectively. By using the BSDE with jumps, Lim and Quenez(2011) studied the exponential utility maximization problem in anincomplete financialmarketwith defaults. Cheridito andHu (2011)studied an optimal consumption and investment problem underutility maximization with general stochastic constraints by usingBSDE and BMO martingale techniques. However, so far, we havenot found any report in the literature on the study of exponen-tial utility maximization problem for an insurer by using BSDE andBMOmartingale techniques.

It should be mentioned that there exist some works about theoptimal investment and proportional reinsurance problem underthe criterion of utility maximization in the general non-Markovianframeworkwhere the parameter processes are supposed to be verygeneral stochastic processes. For example, by a modified ‘‘dualitymethod’’, Liu and Ma (2009) studied a utility optimization prob-lem for a general proportional reinsurance and investment model,where the reverse process of the insurer is non-Markovian in gen-eral. Peng and Hu (2013) investigated the optimal investment andproportional reinsurance under partial information with the crite-rion of utility maximization in a non-Markovian framework by us-ing theMalliavin calculus. However, themethods used in these twopapers are not applicable to the problem in the present paper, sincethe investment and proportional reinsurance rate in our model aresubject to very general constraints, not necessarily convex.

The rest of the paper is organized as follows. Section 2 gives themodel dynamics and states the problem. Section 3 is devoted tothe solution to the problem by using BSDE and BMO martingaletechniques. Section 4 studies the optimal strategies for someparticular cases by using the Malliavin calculus. Finally, Section 5concludes this paper.

2. The dynamics and the optimization problem

We consider a continuous time model where all uncertaintiescome from a complete probability space (Ω, F , P). Here P is a realworld probability. Fix T > 0 as the terminal time. Suppose that thesurplus process before investment of an insurer satisfiesdlt = p(t)dt + q(t)dW 1

t ,l0 = x.

Here, the constant x > 0 is the initial surplus, W 1 is a standardBrownian motion on (Ω, F , P). Denote F 1

t = σ(W 1s , 0 6 s 6

t)∨N , whereN is the P null sets. Assume that p and q are bounded(F 1

t )06t6T adapted cadlag processes.The insurer can invest in the financial market with no

transaction cost and tax where two investment possibilities areavailable:

• a risk free asset A(t) with price dynamics:dA(t) = rA(t)dt,A0 = 1;

• a risky asset S(t) with price dynamics:dS(t) = S(t)[µ(t)dt + σ(t)dW 2

t ],S(0) > 0,

where W 2 is a standard Brownian motion on (Ω, F , P) indepen-dent of W 1. Denote F 2

t = σ(W 2s , 0 6 s 6 t) ∨ N . Suppose that

the interest rate r > 0 is a constant, while the return rate µ(·)and the volatility σ(·) are bounded (F 2

t )06t6T adapted cadlag pro-cesses. Let bt denote the total amount of money invested in therisky asset at time t , and Rt − b(t) is invested in the risk free asset.We also consider the consumption of the insurer. Let ct be the con-sumption rate at time t . Besides investment and consumption, theinsurer is allowed to purchase proportional reinsurance or acquirenew business (for example, acting as a reinsurer of other insurers,see Bäuerle, 2005) at each moment in order to reduce insurancebusiness risk. The proportional reinsurance/new business level isassociated with the value of risk exposure a(t) ∈ [0, +∞) at anytime t ∈ [0, T ]. a(t) ∈ [0, 1] corresponds to a proportional rein-surance cover and implies that the insurer should pay a(t)Y for theclaim Y occurring at time t while the reinsurer should pay the rest(1 − a(t))Y . a(t) > 1 means that the insurer can take an extra in-surance business from other companies (i.e., act as a reinsurer forother cedents). For convenience, we call the process of risk expo-sure a(t), t ∈ [0, T ], as a reinsurance policy. Under the above as-sumptions, the wealth process with investment, consumption andreinsurance evolves over time as follows:

dRt =

p(t) −

1 − a(t)

λ(t) + r(Rt − b(t))

+ µ(t)b(t) − c(t)dt + a(t)q(t)dW 1

t + b(t)σ (t)dW 2t (2.1)

where λ(t) represents the safety loading premium of reinsurancesatisfying λ(t) > p(t) > 0 a.s., for all t ∈ [0, T ]. Assumethat σ(t), q(t) > 0 a.s., for all t ∈ [0, T ]. Let θ1(t) =

λ(t)q(t)

and θ2(t) =µ(t)−rσ(t) . θ1 and θ2 are assumed to be bounded

processes. Denote u1(t) = q(t)a(t), u2(t) = σ(t)b(t), π(t) =u1(t), u2(t), c(t)

, 0 6 t 6 T . In what follows, we consider π as

the optimal strategy. Then (2.1) can be rewritten as

dRπt = [p(t) − λ(t) − c(t)]dt + rRπ

t dt

+ u1(t)[dW 1t + θ1(t)dt] + u2(t)[dW 2

t + θ2(t)dt]. (2.2)

If T

0

|p(t) − λ(t) − c(t)| + u2

1(t) + u22(t)

dt < ∞, P-a.s.,

then the stochastic differential equation (2.2) has a uniquecontinuous solution Rπ

t .The insurer must provide at maturity T an option type payoff

represented by a boundedF 2T -measurable randomvariable ξ . Sup-

pose that its risk aversion is characterized by an exponential utility:

U(x) = − exp(−vx),

where v > 0 represents the risk aversion rate. This utility functionplays a crucial role in insurance context, since it is the only functionunder which the principle of zero utility gives a fair premium thatis independent of the level of reserve of an insurance company (seeGerber, 1979). Let K1, K2 be closed (not necessarily convex) subsetsof R+ and R respectively. The distance between the point a ∈ R tothe set Ki is defined by

dist(a, Ki) = minx∈Ki

|a − x| , i = 1, 2.

80 X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86

Let ΠKi(a) be the set of all points in Ki such that the minimum at-tains, that is,

ΠKi(a) =x ∈ Ki : |a − x| = dist(a, Ki)

, i = 1, 2.

Clearly, ΠKi(a) is nonempty andmay contain more than one point.For all t ∈ [0, T ], ω ∈ Ω , define the sets Ki(ω) ⊆ Rm, i = 1, 2,

by

K1(t, ω) = K1 · q(t, ω), K2(t, ω) = K2 · σ(t, ω).

Since q andσ are boundedprocesses,we can conclude thatK1(t, ω)and K2(t, ω) are all closed sets for all (t, ω) ∈ [0, T ] × Ω . Let A bethe set of all admissible strategies which are defined below.

Definition 2.1. An admissible strategy consists of a triple π =

(u1, u2, c) such that

(1) ui(t, ω) ∈ Ki(t, ω), i = 1, 2, c(t, ω) ∈ R, for all (t, ω) ∈

[0, T ] × Ω;(2) u1, u2 and c are all predictable processes satisfying

T0

u21(t) + u2

2(t) + |ct |dt < ∞, P-a.s.;

(3)exp(−vh(t)Rπ

t )06t6T is of class D;

(4) E T

0 e−vct dt

< ∞.

Here, h is a nonnegative bounded function on [0, T ] defined by

h(t) =

r

1 − (1 − r) exp(−r(T − t))(r > 0),

11 + T − t

(r = 0).

It is easy to verify that h satisfies the following ordinary differentialequation:h′(t) = h(t)(h(t) − r),h(T ) = 1. (2.3)

The objective of the insurer is to obtain the optimal investment,consumption and proportional reinsurance strategy with optiontype payoff ξ , that is, to solve the followingmaximization problem:

V (π) = supπ∈A

E T

0αe−βtU(ct)dt + e−βTU(Rπ

T + ξ)

, (2.4)

where α > 0 and β ∈ R are given constants.

3. Solution to the optimization problem

In order to solve the optimization problem (2.4), we need someresults about the BMOmartingale. For convenience,we give a shortreview about the definition and some properties about the BMOmartingale. We restrain our introduction on R1, which is enoughfor our study.

Definition 3.1. Given a filtered, complete probability space (Ω, F ,F, P), where the filtration F = (Ft)06t6T is assumed to be rightcontinuous. Let M be a continuous local martingale on (Ω, F ,F, P)with the zero initial value. DenoteΞ = τ |τ as an F stoppingtime. For a given p > 1, if

∥M∥BMOp , supτ∈Ξ

E|MT − Mτ |

p|Fτ

1p < ∞,

thenM is called a BMOp martingale.

By Proposition 2.1 in Kazamaki (1994),M is a BMOp martingaleif and only if it is a BMOq martingale for all q > 1. Therefore, wesimply call it a BMO martingale. In particular, M is a BMO martin-

gale if and only if

∥M∥BMO2 = supτ∈Ξ

E[⟨M⟩T − ⟨M⟩τ |Fτ ]12 < ∞. (3.1)

Denote P as the set of all F-predictable processes. Let X ∈ P ,and Mt =

t0 XsdWs, where W is a standard Brownian motion on

(Ω, F , F, P). It is clear from (3.1) thatM is a BMOmartingale if andonly if

∥M∥BMO2 = supτ∈Ξ

E T

τ

X2s ds

Fτ

12

< ∞. (3.2)

Denote the set of all predictable processes satisfying (3.2) byPBMO.We have the following important result about PBMO. For its proof,see Kazamaki (1994).

Proposition 3.2. If X ∈ PBMO, then E(X · W )t , 0 6 t 6 T , is amartingale. Furthermore, if both X and Y belong to PBMO, then Y ∈

PBMO under the Girsanov transformed measure Q = E(X · W )T · P.

Consider the backward stochastic differential equation (BSDE)

Yt = ξ +

T

tg(s, Ys, Zs)ds +

T

tZsdW 2

s , (3.3)

where ξ is a bounded F 2T -measurable random variable, and the

generator

g(s, y, z) = −v

2dist2

θ1(s)

v, h(t)K1(t)

−

v

2dist2

z +

θ2(t)v

, h(t)K2(t)

+ zθ2(t)

+θ21 (t)2v

+θ22 (t)2v

+ h(t)[p(t) − λ(t)]

+h(t)v

log

h(t)α

− 1

− h(t)y +β

v. (3.4)

Suppose that there exist bounded processes k1(t) and k2(t) that,respectively, satisfy k1(t) ∈ K1(t) and k2(t) ∈ K2(t) a.s., for allt ∈ [0, T ]. Then there exists a constant L > 0, such that|g(t, y, z)| 6 L(1 + |y| + |z|2)and|g(t, y1, z1) − g(t, y2, z2)|

6 L (|y1 − y2| + (1 + |z1| + |z2|) (|z1 − z2|)) .

By the results inMorlais (2009a), (3.3) has a unique solution (Y , Z).Furthermore, Y is bounded and Z ∈ PBMO.

We adopt the martingale optimization principle to solve (2.4).For solving the optimization problem by using this method, see forexample Hu et al. (2005), Ankirchner et al. (2008), Pham (2009),Delong (2013) and so on. The outline of this method is to constructa process Gπ , such that

(1) for all π ∈ A, GπT =

T0 αe−βtU(ct)dt + e−βtU(Rπ

T + ξ);(2) Gπ

0 is a constant independent of π ∈ A;(3) for allπ ∈ A, Gπ is a supermartingale. In addition, there exists

π ∈ A, such that Gπ is a martingale.

A strategy π satisfying the above three conditions is an optimalstrategy. In fact, for all α ∈ A, we have

E T

0αe−βtU(ct)dt + e−βtU(Rπ

T + ξ)

= E(Gπ

T ) 6 Gπ0

= E(GπT ) = E

T

0αe−βtU(ct)dt + e−βtU(Rπ

T + ξ)

,

which yields that π is an optimal strategy.We state the main result of this section.

X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86 81

Theorem 3.3. The optimal value function for the optimizationproblem (2.4) is

− exp−v(h(0)x + Y0)

. (3.5)

π = (u1, u2, c) is an optimal admissible strategy if and only if

u1(t) ∈ ΠK1(t)

θ1(t)vh(t)

, (3.6)

u2(t) ∈ ΠK2(t)

Zt + θ2(t)/v

h(t)

, a.s., ∀ t ∈ [0, T ], (3.7)

c = hRπ+ Y −

1vlog

hα

, dt × dP-a.e. (3.8)

Moreover, an optimal admissible strategy always exists, and it isunique if the sets K1 and K2 are convex.

Proof. For each fixed π ∈ A, define Gπ by

Gπt = −e−βte−v(h(t)Rπ

t +Yt ) −

t

0αe−βse−vcsds.

Then,

Gπ0 = −e−v(h(0)x+Y0), (3.9)

GπT = −e−βT e−v(Rπ

T +ξ)−

T

0αe−βse−vcsds. (3.10)

By the Itô formula,

dGπt = ve−βte−v(h(t)Rπ

t +Yt )h(t)u1(t)dW 1

t + (h(t)u2(t)

− Z(t))dW 2t + Aπ

t dt,

where

Aπt = −

v

2

h2(t)u2

1(t) + (h(t)u2(t) − Zt)2

− g(t, Yt , Zt) + Rπt h

′(t)

+ h(t)p(t) − λ(t) − c(t)

+ rh(t)Rπ

t + h(t)u1(t)θ1(t)

+ h(t)u2(t)θ2(t) +β

v−

α

vev(h(t)Rπ

t +Yt )e−vct .

Note that

−v

2

h2(t)u2

1(t) + (h(t)u2(t) − Zt)2

+ h(t)u1(t)θ1(t) + h(t)u2(t)θ2(t)

= −v

2

h(t)u1(t) −θ1(t)

v

2 −v

2

h(t)u2(t) − Zt −θ2(t)

v

2+ Ztθ2(t) +

θ21 (t)2v

+θ22 (t)2v

6 −v

2dist2

θ1(t)

v, h(t)K1(t)

−

v

2dist2

Zt +

θ2(t)v

, h(t)K2(t)

+ Ztθ2(t) +θ21 (t)2v

+θ22 (t)2v

,

and the inequality becomes equality if and only if

u1(t) ∈ ΠK1(t)

θ1(t)vh(t)

, u2(t) ∈ ΠK2(t)

Zt + θ2(t)/v

h(t)

,

t ∈ [0, T ], P-a.s.

Moreover, for fixed (t, ω) ∈ [0, T ] × Ω ,

c −→ −h(t)c −α

vev(h(t)Rπ

t +Yt)e−vc

is a strictly concave function that is equal to its maximum

h(t)v

logh(t)α

− h2(t)Rπt − h(t)Yt −

h(t)v

if and only if

c = h(t)Rπt + Yt −

1vlog

h(t)α

.

Therefore, recalling (2.3), we have

Rπt h

′(t) + h(t)[p(t) − λ(t) − c(t)] + rh(t)Rπt

+β

v−

α

vev(h(t)Rπ

t +Yt )e−vct

6 h(t)[p(t) − λ(t)] + Rπt h

′(t) + rh(t)Rπt

+β

v+

h(t)v

logh(t)α

− h2(t)Rπt − h(t)Yt −

h(t)v

= h(t)[p(t) − λ(t)] +h(t)v

logh(t)α

− h(t)Yt −h(t)v

+β

v,

where the equality is attained if and only if

c(t) = h(t)Rπt + Yt −

1vlog

h(t)α

.

It follows that for every π ∈ A, Gπ is a local supermartingale. Byitems (III) and (IV) in Definition 2.1 and the boundedness of Y , wecan conclude that Gπ is of class D. Therefore, Gπ is a supermartin-gale. By the martingale optimization principle, taking into accountthat (3.9) and (3.10) hold, a strategy π satisfying (3.6)–(3.8) is op-timal if and only if π ∈ A and Gπ is a martingale. In what follows,we verify that π ∈ A and Gπ is a martingale.

If π = (u1, u2, c) satisfies (3.6)–(3.8), then c is continuous. Inparticular, it is predictable and

T0

ct dt < ∞ a.s. Moreover, sinceθ1 and h are bounded and K1 contains a bounded process k1, by(3.6)we have that u1 is bounded. Since θ2 and h are bounded and K2contains a bounded process k1, by (3.7), we have

u2 6 L(1+ |Z |),

where L > 0 is a constant. Since Z ∈ PBMO, u2 ∈ PBMO. Thus, T0

u22 dt < ∞, P-a.s., that is, the item (II) in Definition 2.1 holds.

By the forgoing arguments, we have Aπ= 0 and −Gπ is a positive

local martingale. Hence it is a supermartingale. Consequently, weobtain

Ee−vRπ

T

+ E

T

0e−vct dt

6 L′E

−Gπ

T

6 L′E

−Gπ

0

< ∞,

where L′ is a positive constant. So the item (IV) in Definition 2.1holds.

Since θ1 and θ2 are boundedprocesses,D = E(−θ1·W 1−θ2·W 2)

is a martingale. Define a measure Q by:

Q = DT · P.

By the Girsanov theorem, W 1t , W 1

t + t0 θ1(s)ds and W 2

t ,

W 2t +

t0 θ2(s)ds are two independent Brownian motions under Q .

By the integration by parts formula,

d(h(t)Rπt ) = h(t)dRπ

t + Rπt h

′(t)dt

= h(t)[p(t) − λ(t) − c(t)] + rh(t)Rπt dt

+ h(t)u1(t)dW 1t + h(t)u2(t)dW 2

t + h′(t)Rπt dt

= h(t)p(t) − λ(t) − h(t)Rπ

t − Yt +1vlog

h(t)α

dt

+ h(t)rRπt dt + h′(t)Rπ

t dt

+ h(t)u1(t)dW 1t + h(t)u2(t)dW 2

t

82 X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86

= h(t)p(t) − λ(t) − Yt +

1vlog

h(t)α

dt

+ h(t)u1(t)dW 1t + h(t)u2(t)dW 2

t . (3.11)

Since u1 and h are bounded, and u2 ∈ PBMO, by Proposition 3.2, wehave ζt =

t0 h(s)u1(s)dW 1

s + h(s)u2(s)dW 2s which is a BMO mar-

tingale under Q . It can be easily seen from (3.11) that there existtwo positive constants L1 and L2 > 0, such that

L1e−vh(t)Rπt 6 e−vζt 6 L2e−vh(t)Rπ

t , t ∈ [0, T ].

Set Ft = F 1t ∨ F 2

t , t ∈ [0, T ]. For every stopping time τ 6 T , bythe conditional Jensen inequality, the Bayes formula and the con-ditional Hölder inequality, we obtain

e−vh(τ )Rπτ 6

1L1

e−vζτ 61L1

EQ

e−

v2 ζT |Fτ

2

=1L1

E

e−

v2 ζTDT |Fτ

2D−2

τ

61L1

E[e−vζT |Fτ ] · E[D2T |Fτ ] · D−2

τ

6L2L1

E[e−vRπT |Fτ ] · E[D2

T |Fτ ] · D−2τ .

Since θ1 and θ2 are bounded, there exists a constant L3 > 0, suchthat

E[D2T |Fτ ] · D−2

τ = E

E(−2θ1 · W 1− 2θ2 · W 2)T

E(−2θ1 · W 1 − 2θ2 · W 2)τ

· exp T

τ

(θ21 (s) + θ2

2 (s))ds Fτ

6 L3.

Therefore, for every stopping time τ 6 T ,

e−vh(τ )Rπτ 6

L2L3L1

Ee−vRπ

T |Fτ

.

This implies thatexp(−vh(t)Rπ

t )06t6T is of class D, that is, the

item (III) in Definition 2.1 holds. In addition, it is clear that the item(I) is satisfied. Hence, π = (u1, u2, c) ∈ A. Moreover, by the factAπ

= 0 and the arguments above, we can conclude that Rπ is amartingale.

By themeasurable selection theorem (see Lemma11 inHu et al.,2005), u1 and u2 satisfying (3.6) and (3.7) exist and they are pre-dictable processes. When K1 and K2 are convex sets, u1 and u2 areunique. As we have seen above, u1 is bounded and u2 ∈ PBMO. Sothere exists a unique solution Rπ satisfying

Rπt = x +

t

0rRπ

s ds +

t

0[p(s) − λ(s)]ds

+

t

0

−h(s)Rπ

s − Ys +1vlog

h(s)α

ds

+

t

0u1(s)

dW 1

s + θ1(s)ds

+

t

0u2(s)

dW 2

s + θ2(s)ds. (3.12)

Let

c(t) = h(t)Rπt + Yt −

1vlog

h(t)α

. (3.13)

By the uniqueness of solution to (3.12), we have Rπt = Rπ

t . Thus,(3.13) can be rewritten as

c(t) = h(t)Rπt + Yt −

1vlog

h(t)α

.

This implies that there exists a unique c satisfying (3.8).

Remark 3.4. (1) From expression (3.6), we can see that u1(t) isirrelevant to the parameters of the risky asset process. So is theoptimal proportional reinsurance rate a(t).

(2) It is easily seen from (3.3) and (3.4) that Zt may depend both onthe parameters of insurance market and the risk asset process.Thus, by expression (3.7) we can deduce that u2(t) dependsboth on the parameters of the insurance market and the riskasset process. So does the optimal investment b(t).

(3) From the expression (3.8) we can conclude that the optimalconsumption process c(t) depends both on the parameters ofinsurance market and the risk asset process.

Remark 3.5. Theorem3.3 gives the characterization of the optimalstrategy under general constraint conditions. However, we do notobtain very explicit expressions of the optimal strategy, since Zand Y are involved in the expressions for u2 and c respectively.Generally speaking, it is very difficult to derive the solution (Y , Z)of the BSDE (3.3) with the quadratic growth generator g(ω, t, y, z)given by (3.4) in an analytic form except for some very particularcases (see Delong, 2013). Wewill provide some examples in whichthe optimal strategies can be derived in a very implicit form.

4. Optimal strategies for some particular cases

In this section, we use the Malliavin calculus to study someparticular cases in order to obtain the concrete expressions of theoptimal investment, consumption, and proportional reinsurancestrategies. For convenience of reader, we first review briefly theconcept of Malliavin calculus on a Wiener space as related to ourmodeling framework. Some key results are stated without proofs.For details, interested readers may refer to Nualart (2006).

We assume that a Wiener process (Wt)06t6T is defined ona filtered, complete probability space (Ω, F W , FW , P), and thatFW

= (F Wt )06t6T is generated by W . Let C∞

p (Rn) be the set of allinfinitely continuously differentiable functions f : Rn

→ R suchthat f and all of its partial derivatives have polynomial growth.Denote S by the class of smooth random variables such that arandom variable F ∈ S has the form

F = f (W (h1), . . . ,W (hn)),

where f belongs to C∞p (Rn), h1, . . . , hn are in L2([0, T ]; R),

n > 1, and W (hi) = t0 hi(u)dW (u), so that F is F W

t -measurable.Then the Malliavin derivative of F ∈ S, denoted by D, is defined asthe following process

DuF =

ni=1

∂ f∂xi

(W (h1),W (h2), . . . ,W (hn))hi(u), u ∈ [0, t].

For each F ∈ S and p > 1, we define the norm:

∥F∥1,p =

E

|F |

p+

T

0|DuF |

2du p

2 1

p

.

It has been shown in Nualart (2006) that the Malliavin derivativeoperator D has a closed extension to the space D1,p, which is theclosure of S by the above norm ∥·∥1,p. Note that DuF = 0 foru ∈ (t, T ]. We give some important properties about the Malliavinderivative which will be used later. For proofs, see Nualart (2006)or Di Nunno et al. (2009).

X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86 83

Proposition 4.1 (Chain Rule). Let F ∈ D1,2. Suppose that ϕ isa continuously differentiable function and that E

T0

ϕ′(F)DtF

2 dt< ∞. Then ϕ(F) ∈ D1,2, and

Duϕ(F) = ϕ′(F)DuF .

Proposition 4.2 (Product Formula). Let F ∈ D1,p, G ∈ D1,q, 1 <p < ∞, and let r be such that 1

p +1q =

1r . Then FG ∈ D1,r , and

Du(FG) = FDuG + GDuF .

Proposition 4.3. If F ∈ D1,2, then E(F |Fu) ∈ D1,2, and

DsE(F |Fu) = E(DsF |Fu)I(s6u).

Next, we introduce a very useful result about the Malliavinderivatives of solutions of BSDEs. Before we do so, we intro-duce some notations. Let L1,p be the set of all real-valued, FW -progressively measurable processes (Z(t))06t6T such that(1) for a.e. t ∈ [0, T ], Zt ∈ D1,p;(2) (t, w) → DZt ∈ L2([0, T ], R) admits an FW -progressive

measurable version;

(3) ∥Z∥1,p = E T

0 Z2t dt

p2

+

T0

T0 (DuZt)2 dudt

p2 1

p

< ∞.

Note thatDuZt is defineduniquely up to sets of du×dt×dP measurezero. Let H4 be the space of all real-valued, FW -progressivelymeasurable, processes (h(t))06t6T such that

E T

0h4(t)dt

< ∞.

The following theorem was due to El Karoui et al. (1997). It statesthat under somemild conditions, the solution of a BSDE is differen-tiable inMalliavin’s sense and that itsMalliavin derivative is a solu-tion of a linear BSDE. We state this result without giving the proof.

Proposition 4.4. Suppose ξ ∈ D1,2 and g : Ω × [0, T ] × R2→ R

is continuously differentiable in (y, z) ∈ R2, with uniformly boundedand continuous derivatives. Assume, for each (y, z) ∈ R2, g(·, ·, y, z)is in L1,2 with the Malliavin derivative denoted by Dg(ω, t, y, z). Let(Y , Z) be the solution of the BSDEwith terminal condition ξ and driverfunction g. Suppose, further, that the following conditions hold:(1) g(ω, t, 0, 0) ∈ H4 and ξ is a real-valued random variable in

L4(Ω);(2)

T0 E

(Dtξ)2

dt < ∞ and

T0 E

(Dtg(·, t, y, z))2

dt < ∞;

(3) for any t ∈ [0, T ] and any (y1, z1), (y2, z2) ∈ R2,

|Dtg(ω, t, y1, z1) − Dtg(ω, t, y2, z2)|6 Ku(w, t)(|y1 − y2| + |z1 − z2|),

where for a.e. u, (Ku(t))06t6T is an R-valued, FW -adapted processsatisfying

T0 E

K 4u

du < ∞.

Then (Y , Z) ∈ L2([0, T ], R2), and a version of (DuY (t),DuZ(t)) |

u, t ∈ [0, T ] is given by

DuY (t) = Duξ +

T

t[∂xg(s, Y (s), Z(s))DuY (s)

+ ∂zg(s, Y (s), Z(s))DuZ(s) + Dug(s, Y (s), Z(s))]ds

−

T

tDuZ(s)dW (s), u 6 t 6 T ,

and DuY (t) = 0, DuZ(t) = 0, 0 6 t < u 6 T . Furthermore,DtY (t) | t ∈ [0, T ] is a version of Z(t) | t ∈ [0, T ].

In what follows, we study some particular cases of our problemby using the above results. First, when the investment has noconstraint and the reinsurance rate is nonnegative, we can obtainthe explicit expressions for the optimal strategy.

Proposition 4.5. Assume that the reinsurance rate a(t) ∈ [0, ∞),a.s., for all t ∈ [0, T ]. The amount of investment b and consumptionrate c have no constraint. Denote D2

t by the Malliavin derivative withrespect to W 2. Let

Mt = E(−θ2 · W 2)t

= exp

−

t

0θ2(s)dW 2

s −12

t

0θ22 (s)ds

, (4.1)

Jt =12v

θ21 (t) + θ2

2 (t)+ h(t)(p(t) − λ(t))

+h(t)v

log

h(t)α

− 1

+β

v. (4.2)

If the technical conditions of Proposition 4.4 are satisfied withg(ω, t, y, z) = −h(t)y + θ2(t)z + Jt , 0 6 t 6 T , then the optimalreinsurance rate and investment are

a(t) =λ(t)

vh(t)q2(t)(4.3)

and

b(t) =

−Ee−

Tt h(s)dsD2

t

MTMt

ξ

+ Tt e−

st h(u)duD2

t

MsMt

Jsds

F 2t

h(t)σ (t)

+µ(t) − r

vh(t)σ 2(t), (4.4)

respectively.The optimal consumption rate c(t) satisfies

c(t) = −erth(t) t

0c(s)ds + erth(t)Φ(t) + Yt −

1vlog

h(t)α

(4.5)

where

Yt = Ee−

Tt h(s)dsMT

Mt· ξ +

T

te−

st h(u)du Ms

MtJsds

F 2t

, (4.6)

Φ(t) = x +

t

0

p(s) − λ(s)

ds

+

t

0

λ(s)vh(s)q(s)

dW 1

s + θ1(s)ds

+

t

0b(s)σ (s)

dW 2

s + θ2(s)ds. (4.7)

Proof. Since λ(t)vh(t)q2(t)

> 0, a.s., for all t ∈ [0, T ], we get (4.3) from(3.6). By (3.7),

σ(t)b(t) = u2(t) =Zt + θ2(t)/v

h(t)=

Zth(t)

+µ(t) − rvh(t)σ (t)

. (4.8)

Next, we turn to the computation of Zt . Noting the expression (4.2)for Jt , the BSDE (3.3) can be rewritten as

Yt = ξ +

T

t

θ2(s)Zs − h(s)Ys + Js

ds +

T

tZsdW 2

s .

It is easy to verify that the unique solution Y to this linear backwardstochastic differential equation has the expression (4.6) with Mgiven by (4.1). Here we sketch the proof. For more details, seeProposition 2.2 in El Karoui et al. (1997). Applying the Itô formula tothe process ζ (t) = e−

t0 h(s)dsMtYt +

t0 e

s0 h(u)duMsJsds, 0 6 t 6 T ,

we can deduce that (ζ (t))06t6T is a local martingale. Moreover, wecan show that E

sup06t6T |ζ (t)|

< ∞. Hence, (ζ (t))06t6T is a

84 X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86

uniformly integrable martingale and ζ (t) = Eζ (T )|F 2

t

, which

implies (4.6). By Propositions 4.4 and 4.3, we have

Zt = −D2t Yt = −D2

t Ee−

Tt h(s)dsMT

Mt· ξ

+

T

te−

st h(u)du Ms

MtJsds

F 2t

= −E

e−

Tt h(s)dsD2

t

MT

Mtξ

+

T

te−

st h(u)duD2

t

Ms

MtJs

ds

F 2t

. (4.9)

A combination of (4.8) and (4.9) yields (4.4). Finally, (4.5) can beeasily deduced from (3.8) and (2.2) with Y (t) and Φ(t) given by(4.6) and (4.7) respectively.

Remark 4.6. Denote

∆1(t) = −

Ee−

Tt h(s)dsD2

t

MTMt

ξ F 2

t

h(t)σ (t)

, (4.10)

and

∆2(t) = −

E T

t e− st h(u)duD2

t

MsMt

Jsds

F 2t

h(t)σ (t)

. (4.11)

Then (4.4) can be rewritten as

b(t) = ∆1(t) + ∆2(t) +µ(t) − r

vh(t)σ 2(t). (4.12)

(1) If ξ = 0, then ∆1(t) = 0, 0 6 t 6 T . In our problem, ∆1(t)plays the role to hedge the option type payoff ξ .(2) If p(·), q(·), λ(·), µ(·), and σ(·) are all assumed to bedeterministic functions, then ∆2(t) = 0, 0 6 t 6 T . In fact, byProposition 4.1, we have

D2t

Ms

Mt

= D2

t exp

−

s

tθ2(u)dW 2

u −12

s

tθ22 (u)du

= exp

−

s

tθ2(u)dW 2

u −12

s

tθ22 (u)du

×D2

t

−

s

tθ2(u)dW 2

u

= 0,

for t < s 6 T . Thus, noting that J(·) is deterministic, by Proposi-tion 4.2 we have

D2t

Ms

MtJs

=

Ms

MtD2t Js + JsD2

t

Ms

Mt

= 0, (4.13)

for t < s 6 T . Substituting (4.13) into (4.11) yields∆2(t) = 0, 0 6t 6 T . Here ∆2(t) can be interpreted as a correction term for theoptimal investment b(t) due to the randomness of the parameterprocess p(·), q(·), λ(·), µ(·), and σ(·) in the insurance market andrisky asset process. Still in the case that all these parameter pro-cesses are deterministic, by Proposition 4.2, we have

D2t

MT

Mtξ

=

MT

MtD2t ξ + ξD2

t

MT

Mt

=

MT

MtD2t ξ . (4.14)

Substituting (4.14) into (4.10) yields

∆1(t) = −e−

Tt h(s)ds

h(t)σ (t)MtE

MTD2

t ξ |F 2t

.

Therefore, by (4.12), the optimal investment strategy has the fol-lowing explicit form

b(t) = −e−

Tt h(s)ds

h(t)σ (t)MtE

MTD2

t ξ |F 2t

+

µ(t) − rvh(t)σ 2(t)

. (4.15)

(3) If ξ = 0 and p(·), q(·), λ(·), µ(·), and σ(·) are all deterministic.That is to say, we only consider an optimal investment, consump-tion and proportional reinsurance problem under exponential util-ity maximization for an insurer where all the parameter processesare deterministic. From the deduction above, we have the optimalinvestment strategy

b(t) =µ(t) − r

vh(t)σ 2(t). (4.16)

The optimal reinsurance rate a(t) still has the form given by (4.3).We mention that (4.6) that is related to the optimal consumptionc simplifies to

Y (t) = E T

te−

st h(u)du Ms

MtJsds

F 2t

=

T

te−

st h(u)duJsds.

As a byproduct, Peng and Hu (2013) got the optimal investmentand proportional reinsurance rate for an insurer under exponentialutility maximization where all the parameter processes are deter-ministic. Specifically, the optimal reinsurance level is

a(t) = exp(−r(T − t))λ(t)

vq2(t), (4.17)

and the optimal investment strategy is

a(t) = exp(−r(T − t))µ(t) − rvσ 2(t)

. (4.18)

See Remarks 5.4 and 5.5 in Peng and Hu (2013) for details. For sim-ilar results, see also Cao and Wan (2009). We can easily find that(4.3) and (4.16) have similar forms as (4.17) and (4.18) respectively.However, from the expression for h(t), we can easily verify that

1h(t)

> exp(−r(T − t)), 0 6 t < T .

Therefore, a(t) > a(t), and b(t) > b(t), for 0 6 t 6 T . Itmeans thatbefore time T the insurer should invest more in the risky asset andtake a higher reinsurance level when considering consumption.

In Proposition 4.5, we consider the case a(t) ∈ [0, ∞), t ∈

[0, T ], where a(t) ∈ [0, 1] corresponds to proportional reinsuranceand a(t) ∈ (1, ∞) corresponds to acquiring new business. Next,we shall consider another case that the insurer neither purchasesreinsurance nor acquiring new business, that is to say, the valueof risk exposure a(t) ≡ 1 for all t ∈ [0, T ] and the set K1(t, ω)becomes q(t, ω) for all (t, ω) ∈ [0, T ] × Ω . In this case, thedynamics for the wealth process Rπ

t is given by

dRπt = [p(t) − c(t)]dt + rRπ

t dt + q(t)dW 1t

+ u2(t)[dW 2t + θ2(t)dt] (4.19)

with π(t) = (1, u2(t), c(t)) for t ∈ [0, T ]. In this setting,we have the following result about the optimal investment andconsumption strategies.

Proposition 4.7. Assume that the reinsurance rate a(t) ≡ 1 for allt ∈ [0, T ]. The amount of investment b and consumption rate c

X. Peng et al. / Insurance: Mathematics and Economics 59 (2014) 78–86 85

have no constraint. As in Proposition 4.5, denote D2t by the Malliavin

derivative with respect to W 2, and Mt is given by (4.1). Let

J∗t =12v

(θ21 (t) + θ2

t (t)) + h(t)p(t)

−v

2q2(t)h2(t) +

h(t)v

log

h(t)α

− 1

+β

v. (4.20)

If the technical conditions of Proposition 4.4 are satisfied withg(ω, t, y, z) = −h(t)y + θ2(t)z + J∗t , 0 6 t 6 T , then the optimalinvestment strategy is given by

b(t) =

−Ee−

Tt h(s)dsD2

t

MTMt

ξ

+ Tt e−

st h(u)duD2

t

MsMt

J∗sds

F 2t

h(t)σ (t)

+µ(t) − r

vh(t)σ 2(t). (4.21)

The optimal consumption rate c(t) satisfies

c(t) = −erth(t) t

0c(s)ds + erth(t)Φ∗(t)

+ Yt −1vlog

h(t)α

, (4.22)

where

Yt = Ee−

Tt h(s)dsMT

Mt· ξ

+

T

te−

st h(u)du Ms

MtJ∗s ds

F 2t

, (4.23)

Φ∗(t) = x +

t

0p(s)ds +

t

0q(s)dW 1

s

+

t

0b(s)σ (s)[dW 2

s + θ2(s)ds].

Proof. By (3.7), we have

σ(t)b(t) = u2(t) =Zt + θ2(t)/v

h(t)=

Zth(t)

+µ(t) − rvh(t)σ (t)

. (4.24)

Noting the expression (4.20) for J∗t , the BSDE (3.3) becomes

Yt = ξ +

T

t(θ2(s)Zs − h(s)Ys + J∗s )ds +

T

tZsdW 2

s .

As in Proposition 4.5, we can conclude that Yt is given by (4.23) and

Zt = −Ee−

Tt h(s)dsD2

t

MT

Mtξ

+

T

te−

st h(u)duD2

t

Ms

MtJ∗s

ds

F 2t

. (4.25)

Substituting (4.25) into (4.24) gives rise to (4.21). The optimalconsumption rate determined by (4.22) can be easily deduced from(3.8) and (4.19).

Remark 4.8. Denote

∆∗

2(t) = −

E T

t e− st h(u)duD2

t

MsMt

J∗sds

F 2t

h(t)σ (t)

and recall the expression for ∆1(t) in (4.10). We have

b(t) = ∆1(t) + ∆∗

2(t) +µ(t) − r

vh(t)σ 2(t).

Here ∆1(t) can be seen as the part to hedge the option type payoffξ , while ∆∗

2(t) is a correction term due to the randomness of

the parameter processes in the insurance market and risky assetprocess. If p(·), q(·), µ(·), and σ(·) are assumed to be deterministicfunctions, as Remark 4.8 shows, (4.21) can be rewritten as

b(t) = −e−

Tt h(s)ds

h(t)σ (t)MtE

MTD2

t ξ |F 2t

+

µ(t) − rvh(t)σ 2(t)

, (4.26)

which coincides with (4.10). But the optimal consumption processgiven by (4.22) may be different from that given by (4.5).Furthermore, if we assume ξ = 0, that is, no option type payoffis considered, then (4.26) simplifies to

b(t) =µ(t) − r

vh(t)σ 2(t).

Moreover, in this case, (4.23) that is linked to the optimalconsumption process simplifies to

Y (t) =

T

te−

st h(u)duJ∗s ds.

5. Conclusion

In this paper, we have investigated an insurer’s optimal invest-ment, consumption and proportional reinsurancewith option typepayoff. In our model, the wealth process with investment, con-sumption and proportional reinsurance is non-Markovian in gen-eral. The investment and proportional reinsurance rate are subjectto very general, not necessarily convex, constraints. We use BSDEand BMOmartingale techniques to derive the optimal strategy andthe optimal value function in the closed form. In some interestingparticular cases, we give the explicit expressions for the optimalstrategy by using the Malliavin calculus. Some economic explana-tions of our results are also given.

However, there are some aspects of the present paper whichdeserve to be investigated further in future study. First, we onlyconsider that the option type payoff ξ is bounded here, and do notconsider the unbounded case thatmay bemore realistic in practice.Second, the consumption process c has no constraints. Third, theprice process of the risky asset is continuous and without jumps.

Acknowledgments

The authors are very grateful to the Editor and two anonymousreferees for their helpful comments and suggestions on the originalversion of the manuscript, which led to an improvement of thepresentation of our work.

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