Insurance: Mathematics and Economics 46 (2010) 479–484

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Insurance: Mathematics and Economics

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

Optimal consumption, investment and insurance with insurable risk for aninvestor in a Lévy marketRyle S. Perera ∗Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

a r t i c l e i n f o

Article history:Received October 2009Received in revised formJanuary 2010Accepted 25 January 2010

MSC:60H3060G4691B28

Keywords:Investment–consumption-insurance modelIncomplete marketsLévy processesMartingale methodsUtility maximization

a b s t r a c t

Numerous researchers have applied the martingale approach for models driven by Lévy processes tostudy optimal investment problems. The aim of this paper is to apply the martingale approach to obtaina closed form solution for the optimal investment, consumption and insurance strategies of an individualin the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévyprocesses and the utility is a constant absolute risk aversion (CARA). The model developed in this papercan potentially be applied to absorb large insurable losses in the absence of insurance protection and toexamine the level of diminishing current utility and consumption.

Crown Copyright© 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction

Economic agents in the real world encounter random insurablelosses that are independent of risky asset returns. This paperconsiders an economic agent’s optimal consumption–investmentand insurance strategies over a fixed time horizon when theinsurable losses and security assets returns are driven by Lévyprocesses. We obtain a closed form solution via the martingaleapproachwhen the utility is CARA. The investor’s optimal responsewill be to translate this insurable risk through per-claim insurancein order to minimize a potential loss by simply paying a premiumin advance to the insurer to receive an additional revenue in theevent of a loss or an accident. Such insurance arrangements arecommon in private passenger automobile insurance, homeownersinsurance, building insurances and private health insurancepolicies. However, insurance premiums are costly and reduce theutility that could be derived from saving or consuming. On theother hand absorbing a large loss in the absence of insurance alsodiminishes the utility, the current consumption and reduces thefuture utility from a bequest function.Under fairly general conditions, the optimal insurance is de-

ductible (Arrow, 1963) and when the premium includes a positive

∗ Tel.: +61 2 9850 8578; fax: +61 2 9850 8497.E-mail address: rperera@efs.mq.edu.au.

0167-6687/$ – see front matter Crown Copyright© 2010 Published by Elsevier B.V. Aldoi:10.1016/j.insmatheco.2010.01.005

loading above the actuarially fair pricemargin then it is never opti-mal to purchase full insurance, Mossin (1968). Most of the existingmodels for deductible insurance are static. If the premium rate isproportional to the expected payout, then the optimal per-claiminsurance will be deductible insurance in a dynamic framework.Hence the insurance risk is compounded as a Poisson process andthis risk can be restricted into a form of indemnity or per-claim.Briys (1986) assumes that the loss is proportional to wealth andalludes to an optimal coinsurance. It eventuates that if the loss isproportional to the wealth and the utility function is isoelastic, theoptimal insurance will be a coinsurance. This implies that the de-ductible insurance ought to be a function ofwealth (and time). Gol-lier (1994) assumes that the loss function is not a function ofwealthbut only of time and seeks for an optimal coinsurance. Moore andYoung (2006) extended Briys (1986) analysis to show that the op-timal insurance is a deductible insurance for a risk-averse investorwithin a classical Merton framework and obtained a closed formsolutions for the optimal policies by setting the labor income tobe zero. By contrast to Moore and Young (2006) where a Markovchain approximation method is used in this study a martingaleapproach is used to derive the optimal portfolio-consumption-insurance strategies in a continuous time frame.The problem of utility maximization in incomplete markets

is drawn upon the developments of martingale methods stud-ied by Cox and Huang (1989), Pagés (1989), Karatzas et al.(1991) and the models driven by Lévy processes are studied by

l rights reserved.

480 R.S. Perera / Insurance: Mathematics and Economics 46 (2010) 479–484

Kramkov and Schachermayer (1999), Cont and Tankov (2003),Wang et al. (2007) and Zhou (2009).In this analysis we apply the martingale approach to obtain a

closed form solution for the optimal investment, consumption andinsurance strategy of an individual in the presence of insurablerisk when the insurable risk and risky asset returns are describedby Lévy processes and the utility is CARA. The key feature isthe transformation of an individual’s consumption–investment-insurance problem developed in this paper that can be potentiallyapplied when absorbing a large loss in the absence of insurancein order to examine the level of diminishing current utility andconsumption.The remainder of this paper is organized as follows. In Section 2,

we introduce the general economic framework describing thedynamics of the risky asset price and insurable risk that aregoverned by Lévy processes. In Section 3 the optimal strategies arederived for CARA utility and Section 4 concludes the study.

2. Formulation of the model

Our mathematical model for a frictionless market modelconsists of two tradable assets: one risk free asset (nominal bond)and a risky asset. We work with a filtered probability space(Ω, F, (Ft)t∈R+ , P

). The price of the risky asset S in the economy

is given by following stochastic differential equation (SDE):

dSt = (St)(α(t)dt + σ(t)dW (1)

t + σ(t, z) L(1)t

),

0 ≤ t ≤ T (2.1)

where µ > 0, σ > 0, W (1) is a one-dimensional standard Brow-nian motion and L(1) is a one-dimensional Lévy process defined ona filtered complete probability space

(Ω, F, (Ft)t∈R+ , P

)in a fixed

and finite time horizon T > 0.The price process of the nominal bond is given by

dBt = r Btdt. (2.2)

The decision maker’s risk process is modeled as a compoundPoisson process

dRt = adt + βdWt − d

(N(t)∑i=1

Yi

), (2.3)

where a and β are constants, W is another one-dimensional stan-dard Brownian motion, and

∑N(t)i=1 Yi is the compounded Poisson

insurable risk process on (Ω, F, P). N(t) is a homogeneous Pois-son process with intensity λ and represents the number of ran-dom losses occurring over a time interval [0, t] and Yi are iidr.v. with a common cumulative distribution function ϕ satisfyingϕ(0) = 0 and

∫∞

0 z2dϕ(z) < ∞. The diffusion term βWt repre-

sents the uncertainty associated with the surplus of the economicagent at time t .

(W (1), W

)is a 2-dimensional Brownian motion

such that the correlation coefficient of the components is ρ. Sub-sequently Wt = ρW (1)

t +√1− ρ2W (2)

t , where W (2) is a anotherone-dimensional standard Brownian motion that is independentofW (1). If the economic agent purchases a per-claim insurance ofIt then in the event of a random loss of H(Xt , t) = l (x, t) at time t ,the indemnity pays It(l(x, t)). In order to hold this insurance pol-icy we assume that the premium Pt is proportional to the expectedpayout such as P(t) = [1+ θ(t)]φ(t)E [It(H)], where the loadingfactor θ ≥ 0 and payable to the insurer continuously. Such indem-nity rearrangements are common in the insurance industry. ThenYi = (Hit(Xt , t)− It (Hit(Xt , t))) is the size of the ith claim subjectto the per-claim insurance (insurable risk). Let L(2)t denote the com-pensated compounded Poisson process (subject to insurable risk)as∑N(t)i=1 (Hit(Xt , t)− Iit(Hit(Xt , t)))−λmϕ t , wheremϕ is themean

of ϕ. Then the economic agent’s risk process (2.3) in the presenceof the per-claim insurance can be written as

dRt = κ(t)dt + βρdW(1)t +

(β√1− ρ2

)dW (2)

t

− (H (Xt , t)− It (H (Xt , t))) dL(2)t , (2.4)

where β is a constant,(W (1),W (2)

)is a two-dimensional Brown-

ian motion and L(2) is a one-dimensional Lévy process defined ona filtered complete probability space

(Ω, F, (Ft)t∈R+ , P

)in a fixed

and finite time horizon T > 0. (Ft) is the usual augmentation ofthe natural filtration of

(W (1),W (2), L(1), L(2)

)with F = FT and

W (1),W (2), L(1) and L(2) are mutually independent. For i = 1, 2 wedenote Ni as the jump measure of Li and ϑi as the dual predictableprojection of Ni that has a form of ϑi (dt, dz) = dt × mi(dz) withmi 0 = 0 and

∫R

(z2 ∧ 1

)mi(dz) < ∞. We also assume that the

Lévymeasuremi satisfies∫R z2midz <∞. As a result L(i) is square-

integrable and exhibits the following Lévy decomposition

L(i)t =∫ t

0

∫Rz(Ni(ds, dz)− ϑi(ds, dz)

), (2.5)

and the compensated jump measure of Li is(Ni(ds, dz)− ϑi(ds,

dz)). The risk process (2.3)m2(dz) = λ ϕ(dz).Note. It is assumed that W (1)

t , W(2)t , N and Yi, i ≥ 1 are mutually

independent.Let Xt be thewealth at time t andπt be the net amount of capital

allocated in the risky asset with a consumption rate of ct and apremium rate of Pt in order to hold a per-claim insurance policyof It in a self-financing setting. Then, the economic agent’s wealthprocess evolves as

dXt =Xt − πtBt

dBt − ctdt + κdt − [1+ θ(t)]φ(t)

× E (It (H (Xt , t))) dt +πt

StdSt + βρdW

(1)t

+β√1− ρ2dW (2)

t − (H (Xt , t)− It (H (Xt , t))) dL(2)t , (2.6)

with initial capital X0 = x > 0. Formally, πt , Ft0≤t≤T is a tradingportfolio process if it is progressively measurable and E

∫ T0 |α(t)|

|π(t)| S(t)+ σ 2(t)π2(t)S2(t)+ π2(t)S2(t)∫R σ

2(t, z)ϑ1(dz)dt

< ∞, where as ct , Ft0≤t≤T is a consumption process andIt , Ft0≤t≤T is the per-claim insurance that is nonnegative and pro-gressively measurable with

∫ T0 ctdt <∞ and

∫ T0 Itdt <∞.

Remark 2.1. The economic agent’s wealth process (2.6) withoutper-claim insurance can be written as

dXt =Xt − πtBt

dBt − ctdt + κdt +πt

StdSt + βρdW

(1)t

+β√1− ρ2dW (2)

t − (H (Xt , t)) dL(2)t .

2.1. Statement of the optimization problem

The associated wealth process, denoted by Xπ,c,I is the solutionto the integral equation

Xπ,c,It +

∫ t

0csds+

∫ t

0[1+ θ(s)]φsE (Is (H (Xs, s))) ds−

∫ t

0κsds

=

∫ t

0r Xπ,c.Is +

∫ t

0(αs − r) πsds+

∫ t

0(σπs + βρ) dW (1)

s

+

∫ t

0β√1− ρ2dW (2)

t +

∫ t

0σ(s, z)πsdL(1)s

−

∫ t

0(H (Xs, s)− Is (H (Xs, s))) dL(2)s . (2.7)

R.S. Perera / Insurance: Mathematics and Economics 46 (2010) 479–484 481

Trading strategy(Xπ,c,It , t ≥ 0

)is admissible if

(Xπ,c,It ≥ 0

)and a

set of such a strategy is denoted as A(x).That is

X x,π,c,It = exp(rt)(x+

∫ t

0(α − r) πs − cs + κs − [1+ θ(s)]φs

× E (Is (H (Xs, s)) ds) e−rsds+∫ t

0(σπs + βρ) e−rsdW (1)

s

+

∫ t

0β√1− ρ2e−rsdW (2)

s +

∫ t

0σ(s, z)πse−rsdL(1)s

−

∫ t

0(H (Xs, s)− Is (H (Xs, s))) e−rsdL(2)s

)= exp(rt)x−

∫ t

0cser(t−s)ds+

κ(ert − 1

)r

+ (α − r)∫ t

0πser(t−s)ds−

∫ t

0[1+ θ(s)]φs

× E (Is (H(Xs, s))) er(t−s)ds+∫ t

0(σπs + βρ) er(t−s)dW (1)

s

+

∫ t

0

(β√1− ρ2

)er(t−s)dW (2)

s +

∫ t

0σ(s, z)πser(t−s)dL(1)s

−

∫ t

0(H (Xs, s)− Is (H (Xs, s))) er(t−s)dL(2)s . (2.8)

The economic agent’s problem consists of

Maximizing EU1(Xπ,c,IT

)+

∫ T

0U2(ct)dt

,

over (π, c, I) ∈ A(x), (2.9)

as well as finding an optimal trading strategy (π∗, c∗, I∗). The util-ity functions U1 and U2 : R+ → R are differentiable, strictly in-creasing and concave. In addition to these fundamental properties,it is assumed that U ′i (∞) ∼= limb→∞ U

′

i (b) ∼= 0 and U′

i (0+) ∼=limb→0 U ′i (b) ∼= ∞, for i = 1, 2.Note. The utility function U2 measures the utility of consumption,while U1 measures the utility of the terminal wealth. We alsoassume that the utility for consumption and the utility for terminalwealth are additively separable.

Proposition 2.1. If there exists a strategy (π∗, c∗, I∗) ∈ A(x) thenE[(U ′1X

x,π∗,c∗,I∗T

)X x,π,c,IT

]is constant over (π, c, I) ∈ A(x). This

implies (π∗, c∗, I∗) ∈ A(x) is the optimal trading strategy.

Note. These conditions are widely applied in the martingaleapproach literature while analyzing the optimal investmentstrategies in complete and incomplete markets, Karatzas et al.(1991), Wang et al. (2007) and Zhou (2009).In order to apply Proposition 2.1 in to our analysis we introduce

some notations and a martingale representation theorem asfollows:

(i) Let P: be predictable σ -algebra on Ω × [0, T ], which isgenerated by all left-continuous and Ft-adapted processes.

(ii) P := P⊗ B(R)where B is the Borel σ -algebra onΩ × [0, T ].(iii) L(P) be the set of all Ft-predictable, R-valued processes θ1

such that∫ T0 |θ1(t)|

2 dt <∞ a.s.(iv) L(P)(L(P)) is the set of all P-measurable, R-valued func-

tions. We define θ3 on Ω × [0, T ] × R such that√∑0≤s≤t |θ3 (s,∆Ls)|

2I∆Ls 6=0 is locally integrable and in-

creasing ∀t ∈ [0, T ],∫R |θ3(t, z)|m1(dz) < ∞ a.s., where

I... is the indicator function.

(v)∫R z2mi(dz) < ∞, L

(i)t is square-integrable and measure ϑi

(dt, dz) = dt ×mi(dz) is deterministic.(vi) We describe two sets L,M of measures that are Ft-

predictable as(∑2

i=1 θi(t))and

∑4i=3 θt(t, z), t ≥ 0, z ∈ R.

(vii) Θ :=θi = (θ1, θ2, θ3, θ4) : (θ1, θ2, θ3, θ4) ∈ L(P) ×

L(P)× L(P)× L(P).

(viii) L2F as the set of all Ft-adapted processes (Xt)with cadlag pathssuch that E

[sup0≤t≤T |Xt |

2] <∞.Lemma 2.1 (Martingale Representation). For any local martingale(Zt) subject to θi = (θ1, θ2, θ3, θ4) ∈ Θ such that

Zt = Z0 +2∑i=1

θi(s)dW (i)s +

4∑i=3

∫ t

0

∫Rθi(s, z) (µi−2(ds, dz)

− ϑi−2(ds, dz)) , ∀t ∈ [0, T ] .

3. Optimal trading strategy

In this section we apply Proposition 2.1 in order to obtaina closed form optimal trading strategy (π∗, c∗, I∗) for theproblem (2.9) with CARA utility functions, U1(x) = U2(x) =−1γexp ((−γ x)) , γ > 0. In order to do this we need to show that

E[exp

(−γ X x,π

∗,c∗,I∗T

)X x,π,c,IT

]is constant over (π, c, I) ∈ A(x).

By (2.8) it is equivalent to

E

[exp

(−γ X x,π

∗,c∗,I∗T

) ∫ T

0(α − r) πse−rsds+ σπse−rsdW (1)

s

+ σ(s, z)πse−rsdL(1)s − cs exp(−rs)ds

− [1+ θ(s)]φ(s)E (Is (H(Xs, s))) e−rsds

](3.1)

which is constant over (π, c, I) ∈ A(x).In order to derive our results we first conjecture the form of

the optimal strategy (π∗, c∗, I∗) from the above condition and thenvary it.

Theorem 3.1. Let (π∗, c∗, I∗) are the optimal trading strategies forthe CARA utility function U(x) = − 1

γexp(−γ x), where U ′(x) =

exp(−γ x), γ > 0. Then the optimal policies are given by

π∗t = −θ1(t)e−r(T−t)

γ σ−βρ

σ, Q - a.s

I∗t (H (Xt , t)) = (H (Xt , t))−log(1+ θ4(t, z))e−r(T−t)

γ z,

Q - a.s.

c∗t = −1γln[ert − 1r

], Q - a.s.

Proof. Step 1. We first conjecture the form of (π∗, c∗, I∗) thatsatisfies the condition (3.1).Let Q be a measure which is absolutely continuous with

respect to P on (Ω, F) such that dQdP

∣∣FT= Z∗T , where Z

∗

T :=

exp(−γ Xx,π

∗,c∗,I∗T

)E[exp

(−γ Xx,π

∗,c∗,I∗T

)] . Z∗t := E [Z∗T |Ft], ∀ t ∈ [0, T ], implies Z∗τ :=E[Z∗T |Fτ

]a.s. for any stopping time τ ≤ T .

482 R.S. Perera / Insurance: Mathematics and Economics 46 (2010) 479–484

For any stopping time τ ≤ T a.s., define π τt = I[t≤τ ], cτt =

I[t≤τ ], Iτt = I[t≤τ ]. Substituting (πτ , cτ , Iτ ) into (3.1) we obtain

E

[Z∗T

∫ τ

0(α − r) e−rsds+ σe−rsdW (1)

s + σ(s, z)e−rsdL(1)s

− exp(−rs)ds− [1+ θ(s)]φ(s)e−rsds

]

= EQ

[∫ τ

0(α − r) e−rsds+ σe−rsdW (1)

s

+ σ(s, z)e−rsdL(1)s − exp(−rs)ds− [1+ θ(s)]φ(s)e−rsds

], (3.2)

which is constant over all stopping times τ ≤ T a.s. This implies∫ τ

0(α − r) e−rsds+ σe−rsdW (1)

s + σ(s, z)e−rsdL(1)s

− exp(−rs)ds− [1+ θ(s)]φ(s)e−rsds, (3.3)

is a Q -local martingale.Since

(Z∗t)is amartingale, Kt :=

∫ t01Z∗s−dZ∗s , t ∈ [0, T ] is implied

to be a local martingale and there exists some (θ1, θ2, θ3, θ4) ∈ Θsuch that Kθi(t) will be a solution of the stochastic differentialequation

dKt =2∑i=1

θi(t)dW(i)t

+ d

(4∑i=3

∫ t

0

∫Rθi(s, z) (µi−2(ds, dz)− ϑi−2(ds, dz))

), (3.4)

i.e.,

dZ∗t = Z∗

t−

[2∑i=1

θi(t)dW(i)t

+ d

(4∑i=3

∫ t

0

∫Rθi(s, z) (µi−2(ds, dz)− ϑi−2(ds, dz))

)].

Furthermore applying Doléans-Dade exponential formula, Jacodand Shiryaev (1987) we obtain

Z∗t = exp

∫ t

0

(θ1(s)dW

(s)1 + θ2(s)dW

(s)2

)−12

∫ t

0

(θ21 (s)+ θ

22 (s)

)ds

+

4∑i=3

∫ t

0

∫Rθi(s, z) (µi−2(ds, dz)− ϑi−2(ds, dz))

+

4∑i=3

∫ t

0

∫R(log(1+ θi(s, z))− θi(s, z)) µi−2(ds, dz)

.(3.5)

By Girsanov’s TheoremdW (1)t −θ1(t)dt , dW

(2)t −θ2(t)dt , are Brow-

nian motions, and∫R z (µ1(dt, dz)− (1− θ3(t, z)m1(dz)dt)), is a

martingale under Q together with (3.3) implies

θ1(t)−∫Rz θ3(t, z)m1(dz) = −

α − rσ

d∗t =1γexp

[1+ θ4(t, z)z exp (r(T − t))

]c∗t = −

1γln[ert − 1r

],

(3.6)

Karatzas and Shreve (1991) and Revuz and Yor (1991).Note. The optimal indemnity process I∗t , is either no insuranceor a per-claim deductible insurance and the deductibility mayvary with respect to time. Specifically at a given time the optimaldeductibility exists as d∗t =

(H (Xt , t)− I∗t (H (Xt , t))

). See Arrow

(1963).From Eq. (2.8), imply

exp(−γ X x,π

∗,c∗,I∗T

)= exp

−γ xerT + γ

∫ T

0[1+ θ(s)]φ(s)

× E(I∗s (H (Xs, s))

)e(T−s)ds+ γ

∫ T

0c∗s e

r(T−s)ds− γ (α − r)

×

∫ T

0π∗s e

r(T−s)ds− γ∫ T

0

(σπ∗s + βρ

)er(T−s)dW (1)

s

−γ κ

(erT − 1

)r

− γ β√1− ρ2

∫ T

0er(T−s)dW (2)

s

− γ

∫ T

0σ(s, z)π∗s e

r(T−s)dL(1)s + γ∫ T

0

(H (Xt , t)

− I∗s (H (Xt , t)))er(T−s)dL(2)s

. (3.7)

In order to satisfy Z∗T :=exp

(−γ Xx,π

∗,c∗,I∗T

)E[exp

(−γ Xx,π

∗,c∗,I∗T

)] condition, we comparedW (1)

t , dW (2)t , dµ1 and dµ2 terms in (3.5) with those in (3.7) and

taking (3.6) into account, we conjecture that

θ1(t) = −γ(σπ∗t + βρ

)er(T−t)

θ2(t) = −γ β√1− ρ2er(T−t)

log(1+ θ3(t, z)) = −γ σ(t, z)π∗t er(T−t)z

log(1+ θ4(t, z)) = γ(H (Xt , t)− I∗t (H (Xt , t))

)er(T−t)z

θ1(t)−∫Rzθ3(t, z)m1(dz) = −

α − rσ

d∗t =1γexp

[1+ θ4(t, z)z exp (r(T − t))

].

(3.8)

Finally we obtain

c∗t = −1γln[ert − 1r

], Q − a.s

π∗t = −θ1(t)e−r(T−t)

γ σ−βρ

σ, Q − a.s

I∗t (H (Xt , t)) = (H (Xt , t))−log(1+ θ4(t, z))e−r(T−t)

γ z,

Q − a.sθ3(t, z) = exp(−γ σ(t, z)π∗t ze

r(T−t))− 1θ4(t, z) = exp

(γ d∗t

)zer(T−t) − 1

θ1(t)−∫Rzθ3(t, z)m1(dz) = −

α − rσ

θ2(t) = −γ β√1− ρ2er(T−t).

(3.9)

Step 2. It is clear that (π∗, c∗, I∗) and (θ1, θ2, θ3, θ4) defined in (3.9)are all deterministic processes.In order to verify Z∗T defined in (3.5) satisfies Z

∗

T :=

exp(−γ Xx,π

∗,c∗,I∗T

)E[exp

(−γ Xx,π

∗,c∗,I∗T

)] , substituting (3.9) into (3.7) we have exp(−γ X x,π

∗,c∗,I∗T

)= JTHT , where

JT = exp

−γ xerT −

γ κ(erT − 1

)r

−

∫ T

0ln(1− ers

r

)ds

R.S. Perera / Insurance: Mathematics and Economics 46 (2010) 479–484 483

+ γ

∫ T

0[1+ θ(s)]φ(s)

(H (Xs, s)

−log(1+ θ4(t, z))e−r(T−t)

γ z

)e(T−s)ds

+ (α − r)∫ T

0θ1(s)ds+

∫ T

0

γ βρ (α − r)σ

er(T−s)ds

HT = exp∫ T

0θ1(s)dW (1)

s − γ β√1− ρ2

∫ T

0er(T−s)dW (2)

s

+

∫ T

0θ1(s)dL(1)s +

∫ T

0γ βρer(T−s)dL(1)s

+ γ

∫ T

0

(log(1+ θ4(t, z))e−r(T−t)

γ z

)er(T−s)dL(2)s

.

Hence θ1 and θ4 are defined and it is evident that JT is a constant andZ∗ is a martingale. Substituting (3.9) into (3.5) we have Z∗T = JTHT ,where

JT = exp

−12

∫ T

0θ21 (s)ds−

γ 2β2(1− ρ2

)2

∫ T

0e2r(T−s)ds

+

∫ T

0

∫R

(γ βρer(T−s)z + θ1(s)z − θ3(s, z)

)ϑ1(ds, dz)

+

∫ T

0

∫R

(γ er(T−s)z − θ4(s, z)

)ϑ2(ds, dz)

+

∫ T

0

∫R(log(1+ θ3(s, z))− θ3(s, z)) µ1(ds, dz)

+

∫ T

0

∫R(log(1+ θ4(s, z))− θ4(s, z)) µ2(ds, dz)

,

is a constant and E[Z∗T]= 1, hence Z∗ is a martingale. This

implies that E [HT ] = J−1T . Finally we obtain our final resultsexp

(−γ Xx,π

∗,c∗,I∗T

)E[exp

(−γ Xx,π

∗,c∗,I∗T

)] = JTHTJT E[HT ]

= Z∗T .

Step 3. In order to examine the conditions on (3.1) subject to theoptimal controlled variables (π∗, c∗, I∗) given in (3.9) we provethat

Mπ,c,It :=

∫ t

0

((α − r) πse−rsds+ σπse−rsdW (1)

s

+ σ(s, z)πse−rsdL(1)s − cs exp(−rs)ds− [1+ θ(s)]

× φ(s)E (Is (H (Xs, s))) e−rsds), t ∈ [0, T ] , (3.10)

is a local martingale under Q by Girsanov’s Theorem E[(Z∗T)2]

<

∞ for any (π, c, I) ∈ A(x) and by Burkholder–Davis–Gundyinequality, Karatzas and Shreve (1991) and Revuz and Yor (1991)we obtain

EQ

[sup0≤t≤T

∣∣∣Mπ,c,It

∣∣∣] = E [Z∗T sup0≤t≤T

∣∣∣Mπ,c,It

∣∣∣]

≤

√E(Z∗T)2√E [ sup

0≤t≤T

∣∣∣Mπ,c,It

∣∣∣2] <∞.This implies, for a family of stopping times

Mπ,c,Iτ : τ is a stopping

time and τ ≤ Tis uniformly integrable under Q and Mπ,c,I is a

Q -martingale.Hence EQ

[Mπ,c,IT

]= 0, for any (π, c, I) ∈ A(x).

Next, we consider an example in the absence of per-claiminsurance.

Example 3.1 (CARA Utility). . Consider U1(x) = U2(x) = − 1γ exp(−γ x), γ > 0 and assume that the economic agent doesnot purchase per-claim insurances but all other assumptions ofSection 2 exist. Then applying Theorem 3.1 and Remark 2.1 weobtain the economic agent’s optimal trading strategies (π∗, c∗) as

c∗t = −1γln[ert − 1r

], Q − a.s

π∗t = −θ1(t)e−r(T−t)

γ σ−βρ

σ, Q − a.s

θ2(t) = −γ β√1− ρ2er(T−t)

θ3(t, z) = exp(−γ σ(t, z)π∗t zer(T−t))− 1

θ4(t, z) = exp(γ z (H (Xt , t)) er(T−t)

)− 1

θ1(t)−∫Rz θ3(t, z)m1(dz) = −

α − rσ

.

From these results it is evident that in the absence of insuranceprotection the level of diminishing current and consumption issolely due to Remark 2.1. On the other hand, in the presenceof per-claim insurance, as shown in Theorem 3.1, Eq. (2.6) wasutilized to derive the optimal policies, Eq. (3.9). Note that as the riskaversion of the economic agent (which ismeasured by γ ) increasesthe allocation to the risky asset (π ) decreases. Moreover thedeductable (d) decreases, i.e. the demand for insurance increases.Also note that, as the load θ(t) increases, the demand for insurancedecreases because it becomes more expensive. More specificallythe optimal deductibility at anytime for per-claim insuranceshould satisfy, d∗t =

1γexp

[1+θ4(t,z)z exp(r(T−t))

].

4. Conclusion

This paper is focused on a subset of a portfolio choice problem,where the emphasis placed on the demand for insurance withrandom insurable risks that are independent of a risky asset’sreturn over a finite time interval. The primary research question inthis paper was to investigate the interaction between the demandfor insurance and a risky asset at an optimal level of consumptionwhen the insurable risks and the risky asset price process aregoverned by Lévy processes.From a technical point of view by applying the martingale

approach we obtain a closed form solution to the maximizationproblem when the utility is CARA. The model developed in thispaper can potentially be applied to absorb large insurable losses inthe absence of insurance and examine the level of the diminishingcurrent utility and the consumption in this context.

Acknowledgements

The author would like to thank the referees for their carefulreading and helpful suggestions to improve an earlier version ofthis manuscript.

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