On the Ruin Probability of the Generalised Ornstein-Uhlenbeck Process in the Cramér Case

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by  Damien Bankowski

Applied Probability Trust (6 January 2011)
ON THE RUIN PROBABILITY OF THE GENERALISED
ORNSTEIN-UHLENBECK PROCESS IN THE CRAME´R CASE†
1 DAMIEN BANKOVSKY,∗ Australian National University
1
0
2 CLAUDIAKLU¨PPELBERG,∗∗ Technische Universita¨t Mu¨nchen

n
a ROSSMALLER,∗∗∗ Australian National University
J

5
Abstract

]
R For a bivariate L´evy process (ξt,ηt)t≥0 and initial value V0 define the
P Generalised Ornstein-Uhlenbeck(GOU) process
.
h t
at Vt :=eξt(cid:16)V0+Z0 e−ξs−dηs(cid:17), t≥0,
m
and theassociated stochastic integral process
[
t
1 Zt :=Z e−ξs−dηs, t≥0.
v 0
4 Let Tz :=inf{t>0:Vt <0|V0=z}andψ(z):=P(Tz <∞)forz≥0bethe
3
ruintimeandinfinitehorizon ruinprobabilityoftheGOU.Ourresultsextend
0
previous work of Nyrhinen (2001) and others to give asymptotic estimates for
1
. ψ(z)andthedistributionofTz asz→∞,underverygeneral,easilycheckable,
1 assumptions, when ξ satisfies a Cram´er condition.
0
Keywords: exponential functionals of L´evy processes; generalised Ornstein-
1
Uhlenbeck process; ruin probability; stochastic recurrence equation
1
: 2000 Mathematics Subject Classification: Primary 60H30;60J25;91B30
v
Secondary 60H25;91B28
i
X
1. Introduction
r
a
Let(ξ,η)=(ξ ,η ) beabivariateL´evyprocessonafilteredcompleteprobability
t t t≥0
space (Ω,F,F,P) and define a generalised Ornstein-Uhlenbeck (GOU) process by
t
V :=eξt V + e−ξs−dη , t≥0, (1.1)
t 0 s
(cid:16) Z0 (cid:17)
and the associated stochastic integral process Z =(Z ) by
t t≥0
t
Z := e−ξs−dη . (1.2)
t s
Z
0
†THISRESEARCHWASPARTIALLYSUPPORTEDBYARCGRANTDP1092502.
∗Postal address: Mathematical Sciences Institute, Australian National University, Canberra,
Australia,email: [email protected]
∗∗Postal address: Center for Mathematical Sciences, and Institute for Advanced Study, Technische
Universit¨atMu¨nchen,85747Garching,Germany,email: [email protected]
∗∗∗Postal address: Mathematical Sciences Institute, and School of Finance and Applied Statistics,
AustralianNationalUniversity,Canberra,Australia,email: [email protected]
1
2 Damien Bankowski, Claudia Klu¨ppelberg and Ross Maller
V is a random variable (r.v.), not necessarily independent of (V ) . To avoid
0 t t>0
trivialities, assume that neither ξ nor η are identically zero.
Such processes have attracted attention over the last decade as continuous time
analogues of solutions to stochastic recurrence equations (SRE); cf. Carmona, Petit
and Yor [7, 8], Erickson and Maller [13]. The link between SREs and the GOU was
made in de Haan and Karandikar[11]. GOU processes turn up naturally in stochastic
volatility models (e.g., the continuous time GARCH model of Klu¨ppelberg, Lindner
and Maller [22]), but most prominently as insurance risk models for perpetuities in
life insurance or when the insurance company receives some stochastic return on
investment; such investigations started with Dufresne [12] and Paulsen [29]. More
references are given later.
This paper is intended to fill a gap left between Bankovsky [2] and Bankovsky and
Sly [3], where more details on the insurance background can be found. Define
T :=inf{t>0:V <0|V =z}, z ≥0,
z t 0
(with the convention throughout that inf∅=∞), and let
ψ(z):=P infV <0|V =z =P infZ <−z =P (T <∞), z ≥0, (1.3)
t 0 t z
(cid:16)t>0 (cid:17) (cid:16)t>0 (cid:17)
be theinfinite horizon ruin probability fortheGOU.Notethatψ(z)isanonincreasing
function of z, and we can ask how fast it decreases as z →∞.
Ourmainresult,Theorem2.1,providesaverygeneralasymptoticresultforψ(z)as
z →∞forthecasewhenlim Z existsasana.s. finiter.v. andshowsthat,undera
t→∞ t
Cram´er-likecondition on ξ, ψ(z) decreases approximatelylike a power law. This is an
extensionofasimilarasymptoticresultofNyrhinen[28],who,likeus,utilisesadiscrete
time result of Goldie [16] for proof. We use more recent developments in the theory of
discretetime perpetuities andthe continuoustime GOUto update Nyrhinen’s results.
InSection3weprovidesomeexampleswhichcannotbedealtwithbythe priorresults
but satisfy the conditions of our theorem.
To conclude this introduction, we describe some previous literature relating to the
GOU and its ruin probability, beginning with those papers which examine the GOU
in its full generality. The process appears implicitly in the work of de Haan and
Karandikar [11] as a continuous generalisation of an SRE. Basic properties are given
by Carmona et al. [8]. A general survey of the GOU and its applications is given by
Maller, Mu¨ller and Szimayer [26]. Exact conditions for no ruin (ψ(z) = 0 for some
z ≥0) are givenby Bankovskyand Sly [3] whilst conditions for certain ruin (ψ(z)=1
for some z ≥0) are examined by Bankovsky [2].
The study of the GOU is closely related to the study of integrals of the form Z,
definedin(1.2). ItisshowninLindnerandMaller[25]thatstationarityofV isrelated
to convergence of a stochastic integral constructed from (ξ,η) in a similar way to Z.
Amongthe few papersdealingwithZ inits fullgenerality,EricksonandMaller[13]
give necessary and sufficient conditions for the almost sure convergence of Z to a r.v.
t
Z as t → ∞, and Bertoin, Lindner and Maller [4] present necessary and sufficient
∞
conditionsforthecontinuityofthedistributionofZ ,whenitexists. Fasen[14],using
∞
point process methods, gives an account of the extremal behaviour of a GOU process.
There are a larger number of papers dealing with V and Z when (ξ,η) is subject
to restrictions. We discuss a selection of those papers which are relevant to ruin
Ruin probability of the genOU process 3
probability. Harrison [18] presents results on the ruin probability of V when ξ is a
lineardeterministicfunctionandη isaL´evyprocesswithfinitevariance. Hisapproach
is based on an exponential martingale argument, which corresponds to the Cram´er
case. The heavy-tailed case is investigated in Klu¨ppelberg and Stadtmu¨ller [23] and
extended by Asmussen [1]. See also Maulik and Zwart [27] and Konstantinides and
Mikosch [24].
Paulsen[29]generalisesHarrison’sresults,andpresentsnewruinprobabilityresults
for V, when ξ and η are independent with finite activities. This independent case is
also treated in Kalashnikov and Norberg [20] and Paulsen [30, 31]. Chiu and Yin [9]
generalise some of Paulsen’s results to the case in which η is a jump-diffusion process.
Cai [6] and Yuen et al. [36] present results when η is a compound Poisson process.
Most relevant works containing restrictions on (ξ,η) focus on the case when Z
t
converges to Z as t → ∞; cf. Yor [35] and Carmona et al. [7]. Gjessing and
∞
Paulsen [15] study the distribution of Z when ξ and η are independent with finite
∞
activity, and obtain exact distributions in some special cases. Hove and Paulsen [19]
use MarkovchainMonte Carlomethods to find the distributionof Z in some special
∞
cases. Klu¨ppelberg and Kostadinova [21] and Brokate et al. [5] provide results on
the tail of the distribution of Z when η is a compound Poisson process plus drift,
∞
independent of ξ.
2. Main Results
Our main results apply under a Cram´er-likecondition on ξ: assume that
Ee−wξ1 =1 for some w >0. (2.1)
Thefollowingconsequencesof (2.1)arewellknownandeasilyverified. Condition(2.1)
implies that Eξ is well defined, with Eξ− <∞, Eξ+ ∈(0,∞], and Eξ ∈(0,∞], and
1 1 1 1
so limt→∞ξt = ∞ a.s. Further, Ee−αξ1 is finite and nonzero for all α ∈ [0,w], and
c(α) := lnEe−αξ1 is finite at least for all α ∈ [0,w). The derivatives c′(α) and c′′(α)
are finite at leastfor all α∈[0,w), andc′′(α)∈(0,∞] for all α≥0. So c(α) is strictly
convex for α∈[0,∞) and µ∗ :=c′(w)=−E[ξ1e−wξ1]∈(0,∞].
We will need the Fenchel-Legendre transform of c, defined as
c∗(v):=sup{αv−c(α):α∈R}, v ∈R. (2.2)
Next, let
α :=sup{α∈R:c(α)<∞,E|Z |α <∞}∈[0,∞], (2.3)
0 1
and define the constant
x := lim (1/c′(α))∈[0,∞]. (2.4)
0
α→α0−
Adistributionisspread out ifithasaconvolutionpowerwithanabsolutelycontinuous
component.
Theorem 2.1. Suppose that the following conditions hold:
Condition A: ψ(z)>0 for all z ≥0,
Condition B: there exists w>0 such that Ee−wξ1 =1 (i.e. (2.1) holds),
Condition C: there exist ε>0 and p,q >1 with 1/p+1/q=1 such that
E[e−max{1,w+ε}pξ1]<∞ and E[|η |max{1,w+ε}q]<∞. (2.5)
1
4 Damien Bankowski, Claudia Klu¨ppelberg and Ross Maller
Then 0≤x <1/µ∗ <∞, the function
0
xc∗(1/x) for x∈(x ,1/µ∗),
R(x):= 0
(cid:26) w for x≥1/µ∗,
is finite and continuous on (x ,∞) and strictly decreasing on (x ,1/µ∗), and we have
0 0
lim (lnz)−1lnP(T ≤xlnz)=−R(x) (2.6)
z
z→∞
for every x>x . In addition,
0
lim (lnz)−1lnψ(z)=−w. (2.7)
z→∞
If, further, the distribution of ξ is spread out, then there exist constants C >0 and
1 −
κ>0 such that
zwψ(z)=C +o(z−κ) as z →∞. (2.8)
−
Remark 2.1. (i)ψ(z)>0forallz ≥0isofcoursealogicalassumptiontomakeinthe
contextofTheorem2.1, thoughnotnecessarilyeasyto verify. Necessaryandsufficient
conditions for it in terms of the L´evy measure of (ξ,η) are given in [3]. The moment
conditions in Theorem 2.1 are also easily expressed in terms of the L´evy measure
of (ξ,η), cf. Sato [33], p. 159. They imply that E[sup |Z |max{1,w+ε}] < ∞
0≤t≤1 t
(see Lemma 5.1 below). We also have E[ln(max{1,|η |}] < ∞ in Theorem 2.1, and
1
lim ξ =∞ a.s., so Z converges a.s. to a finite r.v. Z as t→∞ by Proposition
t→∞ t t ∞
2.4 of [25] or Theorem 2 of [13].
(ii) Let Z :=Z −inf Z be the process reflected in its minimum, and set
t t 0≤s≤t s
(M,Q,L):= e−ξ1,Z ,−eξ1Z . (2.9)
1 1
(cid:0) (cid:1)
Then the value C in (2.8) is given by the formula in (2.19) of Goldie [16], namely
−
1 − w − w
C = E Q+Mmin L,infZ − M infZ . (2.10)
− wµ∗ h(cid:16) (cid:8) t>0 t(cid:9) (cid:17) (cid:16)(cid:0) t>0 t(cid:1) (cid:17) i
When ξ and η are independent, it was pointed out by Paulsen [31] that this constant
can be written in a slightly different form, which, by Theorem 4 of [3], is also true in
the dependent case. Namely, let G(z):=P(Z ≤z), h(z):=E[G(−V )|T <∞]∈
∞ Tz z
[0,1], and h:=lim h(z). Then
z→∞
1 w w
C = E (Q+MZ )− − (MZ )− .
− wµ∗h ∞ ∞
h(cid:16) (cid:17) (cid:16) (cid:17) i
(iii) The requirement that ξ is spread out can be replaced with the less restrictive
1
requirement that ξ be spread out, where T is uniformly distributed on [0,1] and
T
independent of ξ. We omit details of this, which can be carried out as in [31].
3. Examples
In this section we provide examples of L´evy processes for which Conditions A, B
and C of Theorem 2.1 are satisfied. Note that conditions B and C only involve the
marginalprocesses ξ and η and they apply to all examples treated in the literature so
Ruin probability of the genOU process 5
far; cf. Klu¨ppelberg and Kostadinova [21] for detailed references. The only condition
which may involve dependence between ξ and η is Condition A.
We denote the characteristictriplet of(ξ,η) by ((γ˜ ,γ˜ ),Σ ,Π ). The character-
ξ η ξ,η ξ,η
istic triplet of the marginal process ξ is denoted by (γ ,σ2,Π ), where
ξ ξ ξ
γ =γ˜ + xΠ (d(x,y)), (3.1)
ξ ξ ξ,η
Z{|x|<1}∩{x2+y2≥1}
and σ2 is the upper left entry in the matrix Σ . Similarly for η. The random jump
ξ ξ,η
measure and Brownian motion components of (ξ,η) will be denoted respectively by
N and (B ,B ); see Section 1.1 of [3] for further details.
ξ,η ξ η
Example 1. [Bivariate compound Poisson process with drift]
Let(Nt)t≥0beaPoissonprocesswithintensityλ>0,and,independentofit,(Xi,Yi)i∈N
an iid sequence of random 2-vectors. For γ ,γ ∈R set
ξ η
Nt
(ξ ,η ):=(γ ,γ )t+ (X ,Y ), t≥0,
t t ξ η i i
Xi=1
with E|X |<∞ and λ, γ and EX such that γ +λEX >0. For this process,
1 ξ 1 ξ 1
c(α)=lnEe−αξ1 =−αγ −λ 1−Ee−αX1 <∞
ξ
(cid:0) (cid:1)
for α∈R such that Ee−αX1 is finite, with c′(0)=−γξ−λEX1 <0.
Weconsiderthespecialcasewhere(X ,Y )isbivariateGaussianwithmean(m ,m )
1 1 X Y
and positive definite covariance matrix
σ2 σ
Σ := X X,Y .
X,Y (cid:18) σX,Y σY2 (cid:19)
Then Condition C obviously holds. For Condition B, note that
c(α)=−αγξ−λ 1−e−mXα+σX2α2/2 →∞ as α→∞. (3.2)
(cid:0) (cid:1)
Consequently, a Lundberg coefficient exists and Condition B is satisfied. To establish
ConditionAwe notethat (ξ,η) isa finite variationprocessandinvokeRemark 2(2)of
[3], also using the notation from that paper. In fact, by that Remark 2(2), ψ(z) = 0
for some z >0 would imply that P (A )=P(X ≤0,Y ≤0)=0, which obviously
X,Y 3 1 1
is not the case. So Condition A holds.
Example 2. A Brownian motion with drift, i.e., with
(ξ ,η )=(γ ,γ )t+(B ,B ), t≥0,
t t ξ η ξ,t η,t
where γ > 0 and (B ,B ) is bivariate Brownian motion with mean 0 and positive
ξ ξ η t
definite covariance matrix, is easily seen to satisfy Conditions A, B, C.
Example 3. [Jump diffusion ξ and Brownian motion η]
Let (B ) be Brownian motion with mean zero and variance σ2, (N ) a Poisson
t t≥0 t t≥0
process with intensity λ>0, and (Xi)i∈N iid r.v.s, all independent. Set
Nt
(ξ ,η )=(γ ,γ )t+ B + X ,B , t≥0,
t t ξ η t i t
(cid:0) Xi=1 (cid:1)
6 Damien Bankowski, Claudia Klu¨ppelberg and Ross Maller
whereγ >0,andassumethatγ +λEX >0. ConditionAholds,sincethe Gaussian
ξ ξ 1
covariance matrix of (ξ,η) is of the form
σ2+λEX2 1
Σ := 1 , (3.3)
ξ,η (cid:18) 1 σ2 (cid:19)
and, hence, is not of the form excluded by Theorem 1 of [3]. Moreover, c(α) is the
sameasin(3.2)withtheadditionofatermα2σ2/2,soagainc′(0)=−γ −λEX <0.
ξ 1
(a) Now assume that X is, as in the Merton model, normally distributed with mean
1
m and variance σ . Then Conditions B and C are satisfied just as in Example 1.
X X
(b) ThepicturechangesslightlywhenweconsiderLaplacedistributedX withdensity
f(x) = ρe−ρ|x|/2 for x ∈ R, ρ > 0. Then Ee−αX = ρ (ρ+α)−1+(ρ−α)−1 /2 for
−ρ<α<ρ with singularities at −ρ and ρ. Moreover, (cid:0) (cid:1)
ρ 1 1
c′(α)=−γ +ασ2+λ − ,
ξ 2 (ρ−α)2 (ρ+α)2
(cid:16) (cid:17)
implying that c′(0) = −γ < 0. So a Lundberg coefficient w > 0 exists. Since the
ξ
normalr.v. B hasabsolutemomentsofeveryorder,forConditionCtoholditsuffices
1
that w<ρ, which is guaranteed, since ρ is a singularity of c.
Example 4. [Subordinated Brownian motion ξ and spectrally positive η]
Let (B ) be a standard Brownian motion and (S ) a driftless subordinator with
t t≥0 t t≥0
Π {R}=∞. For constants µ, γ , γ , define
S ξ η
(ξ ,η )=(γ ,γ )t+(B(S )+µS ,S ), t≥0.
t t ξ η t t t
SubordinatedBrownianmotionsplayanimportantroleinfinancialmodeling;cf. Cont
and Tankov [10], Ch. 4. The bivariate process above has joint Laplace transform
e(α1γξ+α2γη)tE[eα1(B(St)+µSt)+α2St] = e(α1γξ+α2γη)tE[eΨB(α1)St+(α1µ+α2)St]
= et[ΨS(ΨB(α1)+α1µ+α2)+α1γξ+α2γη],
where Ψ andΨ are the Laplaceexponents of B and S,respectively. Thus Ψ (α)=
B S B
−α2/2. By setting α =0 and t=1 we obtain
2
c(α)=lnEe−αξ1 =Ψ (Ψ (−α)−αµ)−αγ =Ψ (−α2/2−αµ)−αγ .
S B ξ S ξ
Consider the variance gamma model with parameters c,λ > 0, where S is a gamma
subordinator with L´evy density ρ(x) = cx−1e−λx for x > 0 and Laplace transform
Ee−uSt =(1+u/λ)−ct. Assume γξ+cµ/λ>0 and γη ≤ 0. Now, ΨS(u)=−cln(1−
u/λ), giving
αµ α2
c(α)=−αγ −cln 1+ − .
ξ
λ 2λ
(cid:16) (cid:17)
c(α) is well defined for α ∈ (µ − µ2+2λ,µ + µ2+2λ), which includes 0, and
c′(0)=−γ −cµ/λ<0. Then, sincpe c(µ+ µ2+p2λ)= ∞, the Lundberg coefficient
ξ
w exists. p
In order to check Condition A, we have, in the notation of Theorem 1 of [3],
Π (A ) = Π (A ) = 0, since η has only positive jumps, and θ = 0. Now with
ξ,η 2 ξ,η 3 2
Ruin probability of the genOU process 7
u ≥ 0, Au = {x ≤ 0,y ≥ 0 : y < u(e−x−1)} = {x ≥ 0,y ≥ 0 : y < u(ex−1)}. Since
4
Π (R) = ∞, η has jumps arbitrarily close to 0, and we have Π (Au) > 0 for u > 0,
η ξ,η 4
while Π (A0)= 0. Thus θ := inf{u≥ 0:Π (Au)>0}= 0. There is no Gaussian
ξ,η 4 4 ξ,η 4
component, so σ2 =0, which puts us in the situation of the seconditem ofTheorem 1
ξ
of [3], and to verify that ψ(z)>0 for all z ≥0 we only need (since θ =θ =0)
2 4
g(0)=γ − yΠ (dx,dy)<0. (3.4)
η ξ,η
Z
x2+y2≤1
e
But by (3.1),
γ =γ − yΠ (dx,dy)≤γ ,
η η ξ,η η
Z0≤y≤1,x2+y2>1
e
thus g(0)<γ ≤0, since we chose γ ≤0. Hence Condition A holds in this model.
η η
4. Discrete Time Background and Preliminaries
Our continuous time asymptotic results will be transferred across from discrete
time versions, and our first task in the present section is to show how (V ) can be
t t≥0
expressedas a solutionof one of two SREs, and give the associateddiscrete stochastic
series for (Z ) . Earlier papers in this area also adopted this approach and we will
t t≥0
tap into some of their results in proving Theorem 2.1.
We beginbydescribingthe discretetime setupwe use. Forn∈N considerthe SRE
Y =A Y +B , (4.1)
n n n−1 n
where (An,Bn)n∈N is an iid sequence of R2-valued random vectors independent of an
initial r.v. Y . The recursion in (4.1) can be solved in the form
0
n n n
Y =Y A + A B (4.2)
n 0 j j i
jY=1 Xi=1j=Yi+1
(with n =1). From (1.1) we can write, for n∈N
j=n+1
Q
n−1 n
V =eξn−ξn−1 eξn−1 V + e−ξs−dη +eξn e−ξs−dη . (4.3)
n 0 s s
(cid:16) (cid:0) Z0 (cid:1)(cid:17) Z(n−1)+
Thus, if we let Y =V and define the R2-valued random vectors
0 0
n
(A ,B ):= eξn−ξn−1,eξn e−ξs−dη , (4.4)
n n s
(cid:16) Z(n−1)+ (cid:17)
then V satisfies (4.1). An alternative formulation considers for n∈N the SRE
n
Y =C Y +C D , (4.5)
n n n−1 n n
where (Cn,Dn)n∈N is an iid sequence independent of Y0. The solution is
n n n
Y =Y C + C D . (4.6)
n 0 i j i
jY=1 Xi=1Yj=i
8 Damien Bankowski, Claudia Klu¨ppelberg and Ross Maller
Using (4.3) it is clear that V is a solution of (4.5) if we let V =Y and define
n 0 0
n
(C ,D ):= eξn−ξn−1,eξn−1 e−ξs−dη . (4.7)
n n s
(cid:16) Z(n−1)+ (cid:17)
Then it is easily verified that
n i−1
Z = C−1D (4.8)
n j i
Xi=1jY=1
(with 0 =1). Note that even when ξ and η are independent, the r.v.s A and B
j=1 n n
may bQe dependent, and similarly for Cn and Dn. But we have
Lemma 4.1. (An,Bn)n∈N and (Cn,Dn)n∈N are iid sequences.
Proof. We begin by proving that the sequence (Cn,Dn)n∈N is iid. Fix n ∈ N and
define the new L´evy process (ξ¯,η¯ ) := (ξ − ξ ,η − η ) for s ≥ 0.
s s n−1+s n−1 n−1+s n−1
Thus (ξ¯s,η¯s)s≥0 =D (ξs,ηs)s≥0. Note that we can bring the term eξn−1 through the
integral sign in (4.7) and write Dn = (nn−1)+e−(ξs−−ξn−1)dηs. (ξ,η) has independent
increments, so (C ,D ) is independentRof (C ,D ) for every n6=m. Now
n n m m
n
(C ,D ) = eξn−ξn−1, e−(ξs−−ξn−1)dη
n n s
(cid:16) Z(n−1)+ (cid:17)
1 1
= eξ¯1, e−ξ¯s−dη¯ = eξ1, e−ξs−dη =(C ,D ).
s D s 1 1
(cid:16) Z0+ (cid:17) (cid:16) Z0+ (cid:17)
Thuswehaveprovedthat(Cn,Dn)n∈Nisaniidsequence. Thisimpliesthat(Cn,CnDn)
is also an iid sequence, and then (An,Bn)n∈N is also an iid sequence since
n
(Cn,CnDn)= eξn−ξn−1,eξn e−ξs−dηs =(An,Bn). (cid:3)
(cid:16) Z(n−1)+ (cid:17)
Inordertodirectlyaccessparticularresultsfrompreviouspapers,whendiscretizing
V we will use the approach via the recursion (4.1) and the sequence (4.2), whereas
when discretizing Z we will use the approach via the series (4.8). There has been
significantattention paid to sequences of the form (4.2) and (4.8), and they are linked
via the fixed point of the same SRE, see Vervaat [34] and Goldie and Maller [17].
Next we describe two important papers relating to the GOU and its ruin time. In
them, ξ andη aregeneralL´evyprocesses,possiblydependent. Therelevantpapersare
Nyrhinen [28] and Paulsen [31], which are very closely related to Theorem 2.1.
Nyrhinen [28] contains asymptotic ruin probability results for the GOU, in which
(ξ,η)isallowedtobeanarbitrarybivariateL´evyprocess. Hediscretizesthestochastic
integralprocess Z and deduces asymptotic results in the continuous time setting from
similar discrete time results. We describe Nyrhinen’s results in some detail, and then
make some comments.
Let (Mn,Qn,Ln)n∈N be iid random vectors with P(M > 0) = 1 and (M,Q,L) ≡
(M1,Q1,L1). Define the sequence (Xn)n∈N by
n i−1 n
X = M Q + M L , with X =0. (4.9)
n j i j n 0
Xi=1jY=1 jY=1
Ruin probability of the genOU process 9
For u > 0 define the passage time τX := inf{n ∈ N : X > u} and the function
u n
c (α):=lnEMα. Assume there is a w+ >0 such that EMw+ =1. Define
M
α+ :=sup α∈R:c (α)<∞, E|Q|α <∞, E(ML+)α <∞ ∈[0,∞]. (4.10)
0 M
(cid:8) (cid:9)
Also let
y¯:=sup y ∈R:P supX >y >0 ∈(−∞,∞]. (4.11)
n
n (cid:0)n∈N (cid:1) o
Nyrhinen provides asymptotic results for X under the following
n
Hypothesis H: Suppose that 0<w+ <α+ ≤∞ and y¯=∞.
0
Under Hypothesis H, and assuming that P(M > 1) > 0, the following quantities
are well-defined: µ+ := 1/c′ (w) ∈ (0,∞) and x+ := lim (1/c′ (t)) ∈ [0,∞).
M 0 t→α+− M
0
Let c∗ (v) be the Fenchel-Legendre transform of c as in (2.2). Define the function
M M
R:(x+,∞)→R∪{±∞} by
0
xc∗ (1/x) for x∈(x+,1/µ+),
R(x):= M 0
(cid:26) w for x≥1/µ+.
In our situation, R is finite and continuous on (x+,∞) and strictly decreasing on
0
(x+,1/µ+).
0
Proposition 4.1. [Nyrhinen’s main discrete results, [28], Theorems 2 and 3]
Assume Hypothesis H. Then the following hold.
(i) For every x>x ,
0
lim (lnu)−1lnP(τX ≤xlnu)=−R(x) (4.12)
u
u→∞
and
lim (lnu)−1lnP(τX <∞)=−w. (4.13)
u
u→∞
(ii) If the distribution of lnM is spread out, there are constants C > 0 and κ > 0
+
such that
uw+P(τX <∞)=C +o(u−κ), as u→∞. (4.14)
u +
C can be obtained from the formula in Theorem 6.2 and (2.18) of Goldie [16].
+
NyrhinencontinuesinhisTheorem3togiveequivalencesfortheconditiony¯=∞,but
they are difficult to verify, as he admits. We discuss these more fully later.
Nyrhinen’s continuousresult is obtained by applying his discrete results to the case
n
(M ,Q ) = e−(ξn−ξn−1),eξn−1 e−ξs−dη =(C−1,D ) (cf. (4.7)),
n n (cid:16) Z(n−1)+ s(cid:17) n n
t n
and L : = eξn sup e−ξs−dη − e−ξs−dη . (4.15)
n s s
(cid:16)n−1<t≤nZ(n−1)+ Z(n−1)+ (cid:17)
(Mn,Qn,Ln)n∈N is an iid sequence, as follows by an easy extension of our proof of
Lemma 4.1. With these allocations Z can be written via (4.8) in the form
n
n i−1 n
Z = M Q =X −L M . (4.16)
n j i n n j
Xi=1jY=1 jY=1
Nyrhinen proves the following result with equality in distribution:
10 Damien Bankowski, Claudia Klu¨ppelberg and Ross Maller
Proposition 4.2. Let (M ,Q ,L ) and Z be as defined in (4.15), (??) and (4.16).
n n n n
Define X as in (4.9). Then
n
sup Z =X and sup Z = max X .
t n t m
n−1<t≤n 0≤t≤n m=1,...,n
Proof. For n∈N we have
t
sup Z = Z + sup e−ξs−dη
t n−1 s
Z
n−1<t≤n n−1<t≤n (n−1)+
n
= Z + e−ξs−dη +e−ξnL
n−1 s n
Z
(n−1)+
n
= X − M L +e−ξnL = X .
n j n n n
jY=1
This further implies that sup Z =max X . (cid:3)
0≤t≤n t m=1,...,n m
Define the first passage time of Z above u >0 by τZ :=inf{t≥ 0:Z >u}. Then
u t
Proposition 4.2 implies that for all t>0,
P(τZ ≤t)=P(τX ≤t) and P(τZ <∞)=P(τX <∞).
u u u u
So (4.12) and (4.13) hold with τX replaced by τZ, when Hypothesis H is satisfied for
u u
the associated values of (M ,Q ,L ). If, further, the distribution of lnM is spread
n n n
out, then (4.14) holds with τX replaced by τZ. This is the content of Theorem 4 and
u u
Corollary 5 of [28].
Remark 4.1. We make some comments on Nyrhinen [28].
(i) We begin with the discrete results. Firstly, the sequence X defined in (4.9)
n
converges as n → ∞ a.s. to a finite r.v. under Hypothesis H. To see this, note that
if we choose L = L then X is the inner iteration sequence I (L) for the random
n n n
equation φ(t) = Mt+Q. Goldie and Maller [17] prove that I (L) converges a.s. to
n
a finite r.v. iff n M → 0 a.s. as n → ∞ and I < ∞, where I is an
j=1 j M,Q M,Q
integral involvingQthe marginal distributions of M and Q. Since these conditions have
no dependence on the distribution of L, it is clear that they are precisely those under
which X converges a.s. for iid (M ,Q ,L ). We now show that these conditions are
n n n n
infactsatisfiedunderHypothesisH,andthusthe sequencesX and n i−1M Q
n i=1 j=1 j i
converge a.s., and to the same finite r.v.. P Q
Under Hypothesis H and our assumption P(M = 0) = 0, ElnM is well-defined
and ElnM ∈[−∞,0). Hence the randomwalk S := n (−lnM )=−ln n M
n j=1 j j=1 j
drifts to ∞ a.s., and it follows that n M → 0 aP.s. as n → ∞. SinceQα+ > 0
j=1 j 0
there exists s > 0 such that E|Q|s <Q∞, thus Eln+|Q| < ∞. Hence Corollary 4.1
of [17] implies that the integral condition I < ∞ is satisfied and the sequence
M,Q
n i−1M Q converges a.s.
i=1 j=1 j i
P(ii)QNyrhinentransfershisdiscreteresultsintocontinuoustime,butthecorrespond-
ing results are difficult to apply in general. The most problematic assumption is his
condition y¯ = ∞ (see (4.11)). In our notation, this is equivalent to the condition
ψ(z) > 0 for all z ≥ 0. Theorem 1 of [3] gives necessary and sufficient conditions