A Lévy insurance risk process with tax

www.ricam.oeaw.ac.at

A Lévy insurance risk process

with tax

H. Albrecher, J.-F. Renaud, X Zhou

HANSJÖRGALBRECHER,JEAN-FRANÇOISRENAUD,ANDXIAOWENZHOU Abstra t. Using u tuation theory, we identify the ruinprobability ofageneralspe trallynegativeLévyriskpro esswithtaxpaymentsof loss arryforwardtype. Westudyarbitrarymomentsofthedis ounted totalamountoftaxpaymentsand determinethesurpluslevelto start taxation whi h maximizes the expe ted dis ounted aggregate in ome forthetaxauthorityinthismodel. Theresults onsiderablygeneralize thosefortheCramér-Lundbergriskmodelwithtax.

1. Introdu tion

The lassi al riskmodeldes ribes the surplus pro ess of an insuran e om-panybya sto hasti pro ess

U

0

= (U

0

(t))

t≥0

with

U

0

(t) = u + ct

− S(t),

where

S(t)

isa ompound Poisson pro ess withjump intensity

θ

and jump distribution

F

(representing the aggregate laim payments up to time

t

),

u > 0

denotes the initial surplusand

c > 0

is a onstant premium intensity. Usually itis assumedthatthenet prot ondition

c > θµ

holds,where

µ

denotestheexpe tedvalueofthesingle laimsizedistribution

F

. This ondition ensuresthat ruinwillnot o uralmostsurely. AsaLévy pro ess,

U

0

hasa hara teristi exponent given by

Ψ(λ) =

_{− ln E}

h

e

iλU

0

(1)

i

₌

_{−icλ −}

Z

0

−∞

(e

iλz

_{− 1) θF (dz)}

for

λ

∈ R

.

One way to generalize the lassi al risk pro ess is to onsider an arbitrary spe trally negative Lévy pro ess, i.e. a pro ess

X = (X(t))

t≥0

with inde-pendent and stationary in rements and with hara teristi exponent given by

Ψ(λ) =

_{−icλ +}

1

2

σ

2

_λ

2

₋

Z

0

−∞

(e

iλz

_{− 1 − izI}

_{z>−1}

) Π(dz),

Date:November29,2007.

2000Mathemati s Subje tClassi ation. 60G51,91B30.

Key words and phrases. Lévypro ess, u tuation theory, s ale fun tions, insuran e risktheory,ruinprobability,taxpayments.

for

λ

∈ R

,

σ

≥ 0

andwhere

Π

isa measureon

(

−∞, 0)

su hthat

Z

0

−∞

(1

_{∧ z}

2

) Π(dz) <

_∞.

Here,

c > 0

again represents the onstant premium density. The netprot onditionfor this Lévyinsuran erisk pro essnowreads

E[X(1)] > 0,

whi h isequivalent to

lim

t→∞

X(t) =

∞

almost surely.

An interpretation of su h Lévy risk pro esses for the surplus modelling of large insuran e ompanies is for instan e given in Klüppelberg and Kypri-anou [12 ℄ and Kyprianou and Palmowski [15℄. This model has re ently at-tra ted a lot of resear h interest, see e.g. also Furrer [8℄, Yang and Zhang [18℄,Huzaketal. [11℄,Klüppelbergetal. [13 ℄,ChiuandYin[6℄andGarrido and Morales[9℄.

In a re ent paper, Albre her and Hipp [1℄ investigated how tax payments (a ordingtoaloss arryforward system)ae t thebehaviourofa Cramér-Lundbergsurpluspro ess. Intheir model, taxesare paidat a xed propor-tional rate

γ

whenever the ompany is in a protable situation, dened as being at a running maximum of thesurplus pro ess. It turned out that in this model there is a strikingly simple relationship between ruin probabili-ties with and without tax and one an also get an expli it formula for the expe teddis ountedsumoftaxpaymentsoverthelifetimeoftheriskpro ess. In this paper we will embed this taxmodelinto a general Lévyframework. Utilizingex ursiontheoryandexploitingthestru tureofthemodel,we will establishthesimplerelationbetweenruinprobabilitieswithandwithouttax inthis more general lassofmodels. Furthermore, expressionsfor arbitrary momentsofdis ountedtax paymentsuntil ruinwillbederived. Itturns out that lose onne tionsofthedistributionoftaxpaymentstothedistribution of dividend payments a ording to a horizontal barrier strategy, that were observed intheCramér-Lundberg model, arryover to theLévysetup. The paper is organized as follows. In Se tion 2, we will review some pre-liminaries on spe trally negative Lévy pro esses that will be needed later on. Se tion 3 introdu esthe taxmodel under onsiderationand derivesthe ruin probability aswell asmoments of dis ountedtax payments until ruin. Finally,inSe tion4 theproblemofanoptimal hoi e ofathresholdsurplus levelfor startingtaxation to maximizethe expe ted tax in omewill be ad-dressed.

2. Preliminaries on spe trally negative Lévy pro esses Let

X = (X(t))

t≥0

beaspe trallynegativeLévypro essor,inotherwords, a Lévypro ess withno positive jumps (toavoidtrivialities, we ex ludethe ase where

X

is a negative subordinator or a deterministi drift). The law of

X

su hthat

X(0) = u

≥ 0

will be denoted by

P

u

and the orresponding expe tation by

E

u

(for a general introdu tion to Lévypro esses we refer to Bertoin [3℄or Kyprianou [14 ℄).

Asthe Lévypro ess

X

hasnopositivejumps,its Lapla etransformisgiven by

E

0

h

e

λX(t)

i

= e

tψ(λ)

for

λ

≥ 0

and

t

≥ 0

, where

ψ(λ) =

−Ψ(iλ)

. In this ase, the Lapla e exponent

ψ

is stri tly onvex and

lim

λ→∞

ψ(λ) =

∞

. Thus, there exists a fun tion

Φ : [0,

∞) → [0, ∞)

su hthat

ψ(Φ(λ)) = λ,

λ

≥ 0.

We now dene the so- alled s ale fun tions

{W

q

; q

≥ 0}

of the pro ess

X

as in Bertoin [4℄. For ea h

q

,

W

q

: [0,

∞) → [0, ∞)

is the unique, stri tly in reasing and ontinuousfun tion withLapla e transform

Z

∞

0

e

−λz

W

q

(z) dz =

1

ψ(λ)

− q

,

for

λ > Φ(q)

.

2.1. Two-sided exit problem. S ale fun tions arise naturally when on-sidering two-sided exitproblems for spe trally negative Lévypro esses. In-deed, let

a

bea positive real numberand dene

T

(0,a)

= inf

{t ≥ 0 | X(t) /

∈ (0, a)} .

When the pro ess

X

startswithin the interval (i.e.

X(0) = u

∈ (0, a)

), the randomtime

T

(0,a)

istherstexittimeof

X

fromthisinterval. Sin e

X

has no positive jumps, it will hit the point

a

when exiting above, but it might jump below zero when exiting below. Its Lapla e transform on the event where the pro ess

X

leaves the intervalat theupperboundary is given by (1)

E

u

e

−qT

(0,a)

_{; X(T}

(0,a)

) = a =

W

q

(u)

W

q

(a)

,

q

≥ 0.

Consequently,when

q = 0

, (2)

P

u

X(T

(0,a)

) = a =

W

0

(u)

W

0

(a)

.

If

X

hasapositivemean, we have that (3)

P

u



inf

t≥0

X(t)

≥ 0



= ψ

′

(0+)W

0

(u).

in-2.2. Smoothness ofthe s ale fun tions. Atseveralpla es inthispaper, dierentiabilityofthes alefun tionswillberequired. Ifthesamplepathsof

X

areofunbounded variation,thenthes ale fun tions

W

q

are ontinuously dierentiable. When thesample paths of

X

areof bounded variation, then the s ale fun tions are ontinuously dierentiable if and only if

Π

has no atoms, or in other words if

{x < 0 | Π({x}) > 0} = ∅

. Note that if

X

has a Gaussian omponent, then its sample paths are of unbounded variation and,moreover,its s ale fun tionsareeventwi e ontinuouslydierentiable. Further, if the Lévymeasure

Π

has a density, then the s ale fun tions are always dierentiable (see Doney [7℄ or Chan and Kyprianou [5℄ for more details).

3. The model

Let

X

be the underlying Lévy risk pro ess with dierentiable s ale fun -tions. Let

S

X

_{= (S}

X

_(t))

t≥0

denotetherunningmaximumof

X

,i.e.

S

X

_{(t) =}

max

0≤s≤t

X(s)

. Thispro essis ontinuousand,of ourse,in reasing. Clearly,

S

X

_{(0) = u}

as

X(0) = u

. For

0

≤ γ ≤ 1

,dene apro ess

U

γ

= (U

γ

(t))

t≥0

by

U

γ

(t) = X(t)

− γ(S

X

(t)

− X(0)).

One an think of

U

γ

as the surplus pro ess of an insuran e ompany that paysout taxesataxedrate

γ

wheneveritisinaprotablesituation (or,in otherwords,wheneverthesurplusis ata runningmaximum). When

γ = 1

, thisamountsto thesituation where the ompanypays outasdividends any apitalaboveits initial value.

3.1. A u tuation identity. Thefollowing theoremgeneralizesboth The-orem VII.8inBertoin [3 ℄and Equation(1) .

Theorem 3.1. For any

0 < u < a

, let

τ

+

a

= inf

{t > 0: U

γ

(t) > a

}

and

τ

₀

−

= inf

_{{t > 0: U}

γ

(t) < 0

}

withthe onvention

inf

∅ = ∞

. If

γ < 1

, then (4)

E

u

h

e

−qτ

a

+

_I

{τ

a

+

<τ

₀

−

}

i

=

 W

q

(u)

W

q

(a)



1/1−γ

.

Proof. Weonly have to onsider the ase when

X

driftsto innity(indeed, if

X

has no drift, then, akin to the proof of Theorem VII.8 in Bertoin [3℄, we an usean approximation by adding a small positive drift and if

X

has negativedrift,thenwe anintrodu e anewprobabilitymeasureunderwhi h

X

hasa positive drift). It is well-known that

S

X

is a lo al time at

0

for

S

X

_{− X}

. Then, let

ǫ

be the ex ursion pro ess of

S

X

_{− X}

away from

0

, let

ǫ

¯

be the ex ursion height pro ess, and let

n

be the ex ursion measure. If

X

drifts to innity, then

ǫ

isa Poisson point pro ess and

ǫ

¯

isalso a Poisson point pro ess with hara teristi measure

ν

givenby

ν(x,

∞) = W

′

0

(x)/W

0

(x)

. Bythedenition of anex ursion, the event

{τ

+

a

< τ

0

−

}

isthesame as

Then, bythe denitionof aPoisson point pro ess,wehave that

P

u

{τ

a

+

< τ

0

−

} = P{N = 0}

= exp

(

−

Z

a−u

1−γ

0

W

₀

′

(u + (1

− γ)s)

W

0

(u + (1

− γ)s)

ds

)

= exp



−

₁

1

− γ

Z

a−u

0

W

₀

′

(u + s)

W

0

(u + s)

ds



=

 W

0

(u)

W

0

(a)



1/1−γ

.

where

N

is aPoissondistributedrandom variablewithparameter

Z

a−u

1−γ

0

n (¯

ǫ

s

≥ u + (1 − γ)s) ds

that ountsthe number ofPoissonpoints

(s, ¯

ǫ

s

)

in

{(x, y) ∈ R

2

| 0 ≤ x ≤ (a − u)/(1 − γ), u + (1 − γ)x ≤ y}.

When

q > 0

,we an dene themeasure

P

Φ(q)

u

on

F

τ

a

+

withRadon-Nikodym derivative

dP

Φ(q)

u

dP

= e

Φ(q)(X(τ

a

+

)−u)−qτ

a

+

_,

where

(

F

t

)

t≥0

denotestheltrationgeneratedby

X

. Under

P

Φ(q)

u

,

X

isstilla spe trallynegative Lévypro ess,butnowwith

W

Φ(q)

0

asitss ale fun tions, whi h are given by

e

Φ(q)x

_W

Φ(q)

0

(x) = W

q

(x)

; see Chapter 8 of Kyprianou [14℄ for details. Observe that

X(τ

+

a

) = S

X

(τ

a

+

)

for

τ

+

a

<

∞

. Sin e

a = U

γ

(τ

a

+

) = X(τ

a

+

)

− γ(S

X

(τ

a

+

)

− u)

for

τ

+

a

<

∞

,we have

X(τ

_a

+

)I

{τ

a

+

<∞}

=

a

_{− γu}

1

− γ

I

{τ

a

+

<∞}

.

We thenfurther getthat

E

u

h

e

−qτ

a

+

_I

{τ

a

+

<τ

₀

−

}

i

= P

Φ(q)

u

τ

a

+

< τ

0

−

exp



−Φ(q)

 a − γu

₁

− γ

− u



=

W

Φ(q)

0

(u)

W

₀

Φ(q)

(a)

!

1/1−γ

exp



−

Φ(q)(a

₁

− u)

− γ



=

e

−Φ(q)u

_W

q

(u)

e

−Φ(q)a

_W

q

(a)

!

1/1−γ

exp



−

Φ(q)(a

− u)

1

_{− γ}



.

3.2. The survival probability. Let

φ

γ

(u) = P

u



inf

t≥0

U

γ

(t)

≥ 0



denotethe survivalprobabilityintherisk modelwithtaxrate

γ

andinitial surplus

u

. Hen e,

φ

0

(u)

isthesurvivalprobabilityintheriskmodelwithout tax. For the ompound Poisson risk model, Albre her and Hipp [1℄ estab-lishedasimplerelationbetween thesurvivalprobabilityofariskmodelwith and without tax. We will now utilize Theorem 3.1 to generalize this result to spe trallynegative Lévyrisk pro esses.

Corollary 3.1. If

γ < 1

, then

φ

γ

(u) = (φ

0

(u))

1/1−γ

.

Proof. From Theorem 3.1,we have that

φ

γ

(u) = ψ

′

(0+)W

0

(u)

1/1−γ

,

sin e

lim

a→∞

W

0

(a) = (ψ

′

₍₀₊₎₎

−1

. The result follows from Equation (3).



Note that

φ

γ

(u) > 0

if and only if

φ

0

(u) = ψ

′

_{(0+) > 0}

, whi h is the ase underthe netprot ondition

E

0

[X(1)] > 0

. However,theexpe tation need not be nite.

3.3. The dis ountedtaxpayments. Letusfromnowonassumethatthe netprot ondition isfullled, i.e.

E

u

[X(1)] > 0

.

Let

τ

γ

be the timeofruin ofthe risk pro esswith tax,i.e.

τ

γ

= inf

{t ≥ 0 | U

γ

(t) < 0

} .

Letfurther

T (γ) = γ

Z

τ

γ

0

e

−δt

dD(t),

denotethepresent valueofalltaxpaymentsuntil thetimeofruin

τ

γ

,where

D(t) = S

X

(t)

− X(0)

and

δ

≥ 0

an beinterpreted asthefor eof interest. Re allfromZhou [19 ℄ that

(5)

V

1

(u, u) =

W

δ

(u)

W

_δ

′

(u)

,

where

V

1

(u, u)

is the expe tation of thepresent value of all dividends paid until ruin when a horizontal barrier is at level

u

. Utilizing a methodology fromZhou[19℄forhorizontalbarriermodels,we willnow ompute

v

(γ)

1

(u) =

E(T (γ))

. Notethat

v

(1)

Theorem 3.2. If

γ < 1

and

δ > 0

, then the expe ted dis ounted sum of tax payments until ruinisgiven by

(6)

v

(γ)

1

(u) =

γ

1

_{− γ}

Z

∞

u

 W

δ

(u)

W

δ

(s)



1/(1−γ)

ds.

Proof. For ea h

n

≥ 1

,dene anexit time

T

n

by

T

n

= inf

{t ≥ 0 | X(t) /

∈ (γ/n, u + 1/n)} .

As

X

hasno positivejumps,we have

v

(γ)

₁

(u)

_{≥ E}

u

[T (γ); X(T

n

) = u + 1/n] .

T

n

is stri tly lessthan

τ

γ

on the event

{X(T

n

) = u + 1/n

}

, using the inte-gration by parts formula and the strong Markov property at time

T

n

, we get

E

u

[T (γ); X(T

n

) = u + 1/n]

≥ (γ/n)E

u

h

e

−δT

n

_{; X(T}

n

) = u + 1/n

i

+ v

₁

(γ)

(u + (1

− γ)/n)E

u

h

e

−δT

n

_{; X(T}

n

) = u + 1/n

i

.

Hen e,

v

₁

(γ)

(u)

≥ γ

W

δ

(u

− γ/n)

nW

δ

(u + (1

− γ)/n)

+ v

₁

(γ)

(u + (1

− γ)/n)

W

δ

(u

− γ/n)

W

δ

(u + (1

− γ)/n)

.

Infa t, one an showthat

v

(γ)

₁

(u) =

γ

W

δ

(u

− γ/n)

nW

δ

(u + (1

− γ)/n)

+ v

(γ)

₁

(u + (1

_{− γ)/n)}

W

δ

(u

− γ/n)

W

δ

(u + (1

− γ)/n)

+ o(1/n),

when

n

goesto innity. Indeed, introdu ing, for ea h

n

≥ 1

, the exittime

T

′

n

dened by

T

_n

′

= inf

{t ≥ 0 | X(t) /

∈ (0, u + 1/n)} ,

we getthat

v

₁

(γ)

(u) = E

u

T (γ); X(T

n

′

)

≤ 0 + E

u

T (γ); X(T

n

′

) = u + 1/n



≤ E

u

"

γ

Z

T

n

′

0

e

−δt

dD(t); X(T

_n

′

)

_{≤ 0}

#

+ E

u

"

γ

Z

T

_n

′

0

e

−δt

dD(t); X(T

_n

′

) = u + 1/n

#

+ E

u

"

γ

Z

τ

γ

∨T

n

′

T

′

n

e

−δt

dD(t); X(T

_n

′

) = u + 1/n

#

≤ γ

_nW

W

δ

(u)

δ

(u + 1/n)

+ v

(γ)

₁

(u + (1

_{− γ)/n)}

W

δ

(u)

W

δ

(u + 1/n)

+ o(1/n),

where we have again used the integration by parts formula, the strong Markovproperty andthefollowing two fa ts( f. Zhou[19℄):

E

u

"

Z

T

_n

′

0

e

−δt

dD(t); X(T

_n

′

)

≤ 0

#

= o(1/n);

E

u

"

Z

T

_n

′

0

e

−δt

D(t) dt; X(T

_n

′

) = u + 1/n

#

= o(1/n),

when

n

goesto innity.

Consequently,usingthe ontinuityandthedierentiabilityofthes ale fun -tions,wegetthat

v

(γ)

₁

(u) lim

n→∞

1

₋

W

δ

(u−γ/n)

W

δ

(u+(1−γ)/n)

γ/n

− lim

n→∞

W

δ

(u

− γ/n)

W

δ

(u + (1

− γ)/n)

=

1

− γ

γ

n→∞

lim

v

(γ)

₁

(u + (1

− γ)/n) − v

1

(γ)

(u)

(1

_{− γ)/n}

W

δ

(u

− γ/n)

W

δ

(u + (1

− γ)/n)

and further (7)

(v

(γ)

1

)

′

(u) =

γ

1

_{− γ}

 W

′

δ

(u)

γW

δ

(u)

v

₁

(γ)

(u)

− 1



.

ThisistheanalogueofEquation(14)intheProofofTheorem2inAlbre her andHipp[1 ℄. Using theintegratingfa tor te hnique forordinarydierential equations,wegetthat itssolution isgiven by

v

(γ)

₁

(u) =



C

−

γ

1

_{− γ}

U

2

(u)



e

U

1

(u)/(1−γ)

_,

for some onstant

C

,where

U

1

(u) =

Z

u

0

W

_δ

′

(s)

W

δ

(s)

ds ,

U

2

(u) =

Z

u

0

e

−U

1

(s)/(1−γ)

_ds.

We have that

W

′

δ

(s)/W

δ

(s)

≥ 0

and

lim

s→∞

W

_δ

′

(s)

W

δ

(s)

= Φ(δ).

The latter result an be found in Avramet al. [2℄ or in Zhou [20℄. Hen e,

U

1

isunbounded be ause

Φ(δ) > 0

for

δ > 0

. Also,sin e

τ

γ

→ ∞

as

u

→ ∞

(forany

γ

), with(5)we havethat

lim

u→∞

v

(γ)

1

(u) <

∞

. Thus,

lim

u→∞

U

2

(u) =

1

_{− γ}

γ

C

and then (8)

v

(γ)

1

(u) =

γ

1

− γ

e

(1−γ)

−1

R

u

0

W ′

_{δ (s)}

Wδ(s)

ds

Z

∞

u

e

−(1−γ)

−1

R

s

0

W ′

_{δ (t)}

Wδ(t)

dt

_ds.

Remark 3.1. If

X

has a negative drift (i.e.

E

u

[X(1)] < 0

), then (6) also holdsfor

δ = 0

.

Remark 3.2. Using Equation (8),we an also write (9)

v

(γ)

1

(u) =

γ

1

− γ

e

(1−γ)

−1

R

u

0

(V

1

(s,s))

−1

ds

Z

∞

u

e

−(1−γ)

−1

R

0

s

(V

1

(t,t))

−1

dt

_ds,

re overing Theorem 2 of Albre her and Hipp [1 ℄ in our more general Lévy setting.

Remark 3.3. Using L'Hpital's rule, we re over the following interesting relation:

(10)

lim

u→∞

v

(γ)

1

(u) = γ lim

_u→∞

V

1

(u, u).

A dire t probabilisti reasoning to obtain this identity goes as follows: in the absen e of ruin, the only dieren e for the al ulation of

v

(γ)

1

(u)

and

V

1

(u, u)

is that, whenever tax (dividend) payments start and last until the nextdeviationfromtherunningmaximum,inthetax aseonlytheproportion

γ

of the in ome ispaidwhereas inthe horizontal barrier ase allthe in ome is paid. The only further dieren e is then that the surplus level at the next payment stream isdierent, but the latter does notmatter if the distan e to the ruin boundary does not matter, whi h in the limit

u

→ ∞

is the ase. Hen e we immediately arrive at (10).

3.4. Higher moments. We will now investigate higher moments of

T (γ)

. Let

v

(γ)

k

(u)

bethe

k

-thmomentof

T (γ)

whentheinitialsurplusisequalto

u

. Re allfromRenaudandZhou[17 ℄,andalsofromKyprianouandPalmowski [15℄,that (11)

V

k

(u, u) = k!

k

Y

i=1

W

iδ

(u)

W

′

iδ

(u)

,

where

V

k

(u, u)

isthe

k

-thmoment ofthepresent valueofall dividendspaid until ruin when the horizontal barrier is at level

u

. Note that

v

(1)

k

(u) =

V

k

(u, u)

. Sowe only need to addressthe ase

γ < 1

:

Theorem 3.3. If

γ < 1

and

δ > 0

, then the

k

-th moment of the present value of tax payments until ruinisrelated to the

(k

− 1)

-th moment by (12)

v

(γ)

k

(u) =

kγ

1

− γ

Z

∞

u

v

_k−1

(γ)

(s)

 W

kδ

(u)

W

kδ

(s)



1/(1−γ)

ds.

Proof. First,pro eedingasintheprooffor Theorem3.2andusingestimates fromthe proof of Proposition 1in Renaud andZhou [17 ℄,we have that

v

(γ)

_k

(u) = kv

_k−1

(γ)



u + (1

_{− γ)/n}



γ

n

W

kδ

(u

− γ/n)

W

kδ

(u + (1

− γ)/n)

+ v

(γ)

_k

(u + (1

_{− γ)/n)}

W

kδ

(u

− γ/n)

W

kδ

(u + (1

− γ)/n)

+ o(1/n).

Further,weget that

(v

(γ)

_k

)

′

(u) =

γ

1

_{− γ}

 W

′

δ

(u)

γW

δ

(u)

v

(γ)

_k

(u)

− kv

k−1

(γ)

(u)



.

Solvingthis ordinary dierential equationleads to

v

(γ)

_k

(u) =

kγ

1

_{− γ}

e

(1−γ)

−1

R

u

0

W ′

_{kδ (s)}

Wkδ(s)

ds

Z

∞

u

v

(γ)

_k−1

(s) e

−(1−γ)

−1

R

s

0

W ′

_{kδ (t)}

Wkδ(t)

dt

_ds.

Nowthe statement follows fromsimple algebrai manipulations.



Remark 3.4. From (12 ), we get by L'Hpital's rule

lim

u→∞

v

(γ)

k

(u) = kγ lim

_u→∞

v

(γ)

k−1

(u)

W

kδ

(u)

W

_kδ

′

(u)

.

With (11) we an hen e generalize the asymptoti relation (10) to arbitrary momentsof tax and dividend payments, respe tively:

(13)

lim

u→∞

v

(γ)

k

(u) = γ

k

_u→∞

lim

V

k

(u, u).

The alternative probabilisti argument from Remark (3.3 ) also arries over to explainformula (13 ).

3.5. Examples.

3.5.1. Cramér-Lundbergpro esswithexponential laims. If

X

isa ompound Poissonpro ess withexponential jumps(with Poisson parameter

λ

and ex-ponential parameter

α

),then thes ale fun tionsaregiven by

W

δ

(x) =

(α + ρ)e

ρx

(1

_{− η(x))}

c(ρ

_{− r)}

(see e.g. Kyprianou &Palmowski [15℄),where

η(x) =

α + r

α + ρ

e

(r−ρ)x

_,

and

ρ

and

r

arethepositiveandnegative,respe tively,solutionofthe equa-tion

cR

2

+ (cα

_{− λ − δ)R − αδ = 0.}

Pluggingthisexpressionintoformula(6) ,weeventuallyarriveattheexpli it formula

v

(γ)

₁

(u) =

γ

ρ

(1

− η(u))

1/1−γ

×

2

F

1



1

1

_{− γ}

,

ρ

(ρ

_{− r)(1 − γ)}

,

ρ

(ρ

_{− r)(1 − γ)}

+ 1; η(u)



,

whi h wasalreadyderived inAlbre herand Hipp[1℄. Here

2

F

1

(a, b, c; z) =

Γ(c)

Γ(b)Γ(c

_{− b)}

Z

1

0

t

b−1

(1

_{− t)}

c−b−1

(1

_{− zt)}

−a

dt

3.5.2. Brownian motionwith drift. Let

X(t) = mt + σB(t)

be a Brownian motion with drift (with

m

6= 0

and

σ > 0

). As in this ase

ψ(λ) = mλ +

(1/2)σ

2

λ

2

and

Φ(α) =

−ω + θ

α

,one an verifythat

W

δ

(x) =

1

σ

2

_θ

δ



e

(−ω+θ

δ

)x

_{− e}

−(ω+θ

δ

)x



,

where

θ

δ

=

√

m

2

_{+ 2δσ}

2

_/σ

2

and

ω = m/σ

2

(see also Avram et al. [2 ℄). In parti ular, we have

W

0

(x) =

1

m



1

− e

−

2m

σ2

x



.

Thus,

v

₁

(1)

(u) = V

1

(u, u) =

σ

2

2m



e

2m

σ2

u

− 1



,

whi h re overs Equation(2.20) inGerberandShiu[10 ℄. Also,if

γ < 1

and if

δ > 0

,thenone obtains

v

(γ)

₁

(u) =

γ

1

− γ

h

e

(θ

δ

−ω)u

₍₁

_{− e}

−2θ

δ

u

₎

i

1/1−γ

×

Z

∞

u

h

e

(θ

δ

−ω)s

₍₁

_{− e}

−2θ

δ

s

₎

i

−1/1−γ

ds.

Sin e

θ

δ

> ω

when

σ > 0

and

δ > 0

, letting

r =

e

−2θδ s

e

−2θδ u

inthe integral and simplifyingyields

v

(γ)

₁

(u) =

γ

1

_{− γ}

"

(1

− e

−2θ

δ

u

₎

1/1−γ

θ

δ

− ω

#

×

2

F

1



(1

_{− γ)}

−1

,

θ

δ

− ω

2θ

δ

,

3θ

δ

− ω

2θ

δ

; e

−2θ

δ

u



.

4. Optimality of the tax barrier

As tax payments stop at ruin, it is natural to ask whether the expe ted dis ounted tax payments over the lifetime of the pro ess an be optimized when taxpaymentsare only startedafter the surplus hasrea hed a ertain level

M

(seeAlbre herandHipp[1℄fora orrespondingstudyinthe Cramér-Lundbergframework). Dueto the strongMarkov propertywe learly have

(14)

v

(γ)

1,M

(u) =

W

δ

(u)

W

δ

(M )

v

(γ)

₁

(M )

for

u < M

and

v

(γ)

1,M

(u) = v

(γ)

1

(u)

for

u

≥ M

(as then tax payments start right away). Hen ethe goalis to maximize(14 ) withrespe tto

M

.

Assumption 4.1. In what follows, we assume that ea h s ale fun tion is three timesdierentiableandthat itsrstderivative isastri tly onvex fun -tion(sothat

W

′′

Assumption4.1isforinstan efullledifthe Lévymeasurehasa ompletely monotone density (seeLoeen [16 ℄ for thestri t onvexityof

W

′

δ

and Chan and Kyprianou [5℄ for innite dierentiability). Among parti ular exam-ples fullling Assumption4.1 areGammapro ess and theinverseGaussian pro ess (formore examples,see Loeen [16 ℄).

Dierentiating Equation (14) with respe t to

M

, one nds that

M

0

is a riti alpoint of

M

7→ v

(γ)

1,M

(u)

if (15)

v

(γ)

1

(M

0

) = V

1

(M

0

, M

0

)

or equivalently

(v

(γ)

1

)

′

(M

0

) = 1,

where (7 ) wasused for thelatter equivalen e. To spe ify thenature of this riti alpoint,we usethese ond derivative:

(16)

∂

2

v

_1,M

(γ)

(u)

∂M

2









M =M

0

=

γ

1

_{− γ}

W

δ

(u)

(W

δ

(M

0

))

2

v

₁

(γ)

(M

0

)W

δ

′′

(M

0

).

Clearly,sin e

lim

M →∞

v

(γ)

1,M

(u) = 0

for any

u

,there is a point

M

⋆

_{∈ [0, ∞)}

where the fun tion

M

7→ v

(γ)

1,M

(u)

rea hesits global maximum. Remark 4.1. Note that

M

7→ v

(γ)

1,M

(u)

an not have a lo al minimum in

[0,

_∞)

. Indeed,ifthereexistedalo alminimum,thenbyvirtueof

lim

M →∞

v

(γ)

1,M

(u) =

0

, there would have to exist a lo al maximum for a larger value

M

. But in viewof (16) andthe stri t onvexity of

W

′

δ

, this an not o ur.

Similarly, we dedu e that after a potential saddlepoint there an not be a lo almaximum. Re allthat

V

₁

′

(s, s) = 1

−

W

δ

(s)W

′′

δ

(s)

(W

_δ

′

(s))

2

and fromRemark3.3 that (17)

lim

u→∞

v

(γ)

1

(u) < lim

_u→∞

V

1

(u, u).

Remark4.2. Fromtheabove,itfollowsthat

M

7→ v

(γ)

1,M

(u)

also annothave a saddlepoint

M

0

in

[0,

∞)

. Indeed, otherwise from

v

(γ)

1

(M

0

) = V

1

(M

0

, M

0

)

and

W

′′

δ

(M

0

) = 0

, one an observe that

V

₁

′′

(M

0

, M

0

) =

−W

δ

(M

0

)W

_δ

′′′

(M

0

)

(W

′

δ

(M

0

))

2

and

(v

(γ)

1

)

′′

(M

0

) = 0

. Hen e, the fun tion

s

7→ v

(γ)

1

(s)

− V

1

(s, s)

rea hes a lo al minimum value of

0

atthis point

M

0

(as

W

′′′

δ

(M

0

) > 0

), implying that

v

₁

(γ)

is greater than

V

1

in a neighbourhood of

M

0

, so that this saddlepoint wouldhave tobefollowed by amaximum oranothersaddlepoint,whi hitself isex luded by the onvexity of

W

′

δ

(u)

.

As a onsequen e, Equation (15) hasat most one positive solution

M

0

. If

V

1

(0, 0)

≤ v

(γ)

1

(0)

,thendue to(17 )su h asolution

M

0

> 0

existsandisthe point ofglobal maximum,i.e.

M

⋆

_{= M}

0

. If

V

1

(0, 0) > v

(γ)

1

(0)

,then

M

⋆

_{= 0}

(i.e. taxpaymentsstart immediately), as a solutionof (15) , by (17), wouldhave to bea ompanied byase ond one, whi h annot be the ase.

Notethat

M

⋆

isindependent of theinitial surplus

u

.

From the above dis ussion, we get the following nal result whi h extends Theorem 3inAlbre herand Hipp[1℄.

Theorem 4.1. Suppose that the s ale fun tions of

X

are three times dif-ferentiable and that their rst derivatives are stri tly onvex fun tions. If

V

1

(0, 0) > v

1

(γ)

(0)

, then the optimal height

M

⋆

is equal to

0

. If

V

1

(0, 0)

≤

v

₁

(γ)

(0)

, then the optimal height

M

⋆

is the unique positive solutionof Equa-tion (15 ). The maximum valueis thus given by

(18)

v

(γ)

1,M

⋆

(u) =

(

V

1

(u, M

⋆

),

if

u < M

⋆

;

v

₁

(γ)

(u),

if

u

≥ M

⋆

. Proof. If

u < M

⋆

,then

v

(γ)

_1,M

⋆

(u) =

W

δ

(u)

W

δ

(M

⋆

)

v

(γ)

₁

(M

⋆

) =

V

1

(u, M

⋆

₎

V

1

(M

⋆

, M

⋆

)

v

₁

(γ)

(M

⋆

) = V

1

(u, M

⋆

).

Otherwise,westart to paytaxesright awayand

v

(γ)

1,M

⋆

(u) = v

(γ)

1

(u)

.



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[2℄ F.Avram,Z.Palmowski,andM. R.Pistorius.Ontheoptimaldividendproblemfor aspe trallynegativeLévypro ess.Ann.Appl.Probab.,17(1):156180, 2007. [3℄ J.Bertoin.Lévy pro esses.CambridgeUniversityPress,1996.

[4℄ J.Bertoin.Exponentialde ayandergodi ityof ompletelyasymmetri Lévypro esses inaniteinterval.Ann.Appl.Probab.,7(1):156169,1997.

[5℄ T.ChanandA.E.Kyprianou.Smoothnessofs alefun tionsforspe trallynegative Lévypro esses.Submitted,2007.

[6℄ S. N. Chiu and C. Yin.Passage times for a spe trally negative Lévy pro ess with appli ationstorisktheory.Bernoulli,11(3):511522, 2005.

[7℄ R.A.Doney.Someex ursion al ulationsforspe trallyone-sidedLévypro esses.In Séminaire de Probabilités XXXVIII,volume1857 ofLe ture Notesin Math., pages 515. Springer,2005.

[8℄ H. Furrer. Risk pro esses perturbed by

α

-stable Lévy motion. S and. A tuar. J., (1):5974, 1998.

[9℄ J. Garridoand M. Morales. Onthe expe teddis ounted penalty fun tionfor Lévy riskpro esses. N.Am.A tuar.J.,10(4):196218,2006.

[11℄ M. Huzak, M. Perman,H. ’iki¢,and Z. Vondra£ek.Ruinprobabilities and de om-positions forgeneralperturbedriskpro esses.Ann.Appl.Probab., 14(3):13781397, 2004.

[12℄ C.KlüppelbergandA. E.Kyprianou.OnextremeruinousbehaviourofLévy insur-an eriskpro esses.J.Appl.Probab.,43:594598,2006.

[13℄ C.Klüppelberg,A.E.Kyprianou,andR.A.Maller.Ruinprobabilitiesandovershoots forgeneralLévyinsuran eriskpro esses.Ann.Appl.Probab.,14(4):17661801, 2004. [14℄ A. E.Kyprianou.Introdu tory le tures onu tuationsof Lévypro esses with

appli- ations.Springer-Verlag,Berlin,2006.

[15℄ A. E. Kyprianou and Z. Palmowski. Distributional study of de Finetti's dividend problem for ageneralLévyinsuran eriskpro ess. J.Appl.Probab., 44(2):428443, 2007.

[16℄ R. Loeen. Optimality of the barrier strategyinde Finetti's dividend problem for spe trallynegativeLévypro esses. Preprint,2007.

[17℄ J.-F.RenaudandX.Zhou.Distributionofthepresentvalueofdividendpaymentsin aLévyriskmodel.J.Appl.Probab.,44(2):420427,2007.

[18℄ H. Yangand L.Zhang. Spe trallynegativeLévypro esseswithappli ationsinrisk theory.Adv.inAppl.Probab.,33(1):281291,2001.

[19℄ X. Zhou.Dis ussion on: Onoptimal dividend strategiesin the ompoundPoisson model, by H. Gerberand E.Shiu [N. Am.A tuar.J. 10 (2006), no. 2, 7693℄. N. Am.A tuar. J.,10(3):7984,2006.

[20℄ X.Zhou.Exitproblemsforspe trallynegativeLévypro essesree tedateitherthe supremumortheinmum.ToappearinJ.Appl.Probab., 2007.

H.Albre her,JohannRadonInstituteforComputationalandApplied Math-emati s (RICAM), Austrian A ademy of S ien es and, University of Linz, Altenbergerstrasse 69,A-4040Linz, Austria

E-mailaddress: hansjoerg.albre heroeaw.a .at

J.-F.Renaud,JohannRadonInstituteforComputationalandApplied Math-emati s(RICAM),AustrianA ademyofS ien es,Altenbergerstrasse69, A-4040Linz, Austria

E-mailaddress: jean-fran ois.renaudoeaw.a .a t

X. Zhou, Department of Mathemati sand Statisti s, Con ordiaUniversity, 1455deMaisonneuve BlvdW., Montréal(Québe ),H3G1M8,Canada E-mailaddress: xzhoumathstat. on ordia. a