Ruin probabilities in tough times - Part 1 - Heavy-traffic approximation for fractionally integrated random walks in the domain of attraction of a nonGaussian stable distribution

0.35 MB·English

by  Ph. Barbe

RUIN PROBABILITIES IN TOUGH TIMES
Pgggart 1
1
HEAVY-TRAFFIC APPROXIMATION FOR FRACTIONALLY
1
0 INTEGRATEDRANDOM WALKS INTHE DOMAIN OF ATTRACTION
2 OF A NONGAUSSIANSTABLE DISTRIBUTION

n Ph. Barbe(1) and W.P. McCormick(2)
a
J (1)CNRS (UMR8088), (2)University of Georgia

4
2 Motivatedby applicationsto insurancemathematics,we prove
Asobmsterah
eta.vy-traffic limit theorems for process which encompass the frac-

] tionallyintegratedrandomwalkaswellassomeFARIMAprocesses,when
R the innovations are in the domain of attraction of a nonGaussian stable
P distribution.
.
h
Primary: 60F99; Secondary: 60G52,
at A60MGS222,06100KS2u5b,j6e2
Mt C10la,s6s0i(cid:12)G
7a0ti,o6n2sP: 20.
m
[ heavy traffic, ruin probability, fractional random walk,
KFAeyRwIMorAds:process,fractional L´evy stable process.

1
v
7
3 1. Introduction and main result. The purpose of this paper is
4 to study ruin probability when the claim process is nonstationary,
4
has long range dependence, innovations in the domain of attraction
.
1
of a stable distribution and when the premiums can barely cover the
0
1 claims; hence the title.
1 The motivation for such a study, beyond the development of
:
v some of the mathematics needed to build more realistic models for
i
X some insurance companies, are manifold; this introduction seeks to
r describethem,relatingthecontentof thispaperto variousproblems
a
considered before.
To start with, the claim processes we are interested in encom-
pass many nonstationnary fractional autoregressive moving average
(FARIMA) processes without prehistorical influence. As such, they
also include the usual partial sum process. From an applied per-
spective, these processes are of interest because they are part of the
standard models in time series, and their ontological justification as
aggregationof simpler processes(Granger,1980)has some appealin
economics and econometrics. From a theoretical perspective, their
interest lies in the fact that they are not Markovian, may not have
stationary solutions, may exhibit long range dependence, so may
1
not be amenable to the classical techniques, and yet are tractable.
Consequently, a technical understanding of these models yields a
greater understanding of the underlying stochastic phenomenon in-
volved in simpler models. Indeed, as less technical tools become
available, we have to resort to more fundamental aspects of the pro-
cess involved. In that respect, the main contribution of this paper
is threefold: firstly, it reveals the role of extreme values in fractional
randomwalks with innovations in the domain of attractionof a non-
Gaussian distribution; secondly, it gives a method of proof which,
unlike all those we are aware of for the classical random walk, does
not rely on either some form of Kolmogorov’s maximal inequality
or the Wiener-Hopf factorization — see a discussion of the classi-
cal proofs in Shneer and Wachtal (2009); thirdly, it shows that the
so-called exponential representation of uniform order statistics may
be used in sequential problems, even though this representation is
nonsequential in nature.
A certain number of results known for the partial sum process
have been extended to its fractionally integrated version, and, more
generally, to FARIMAones. Forinstance,motivatedby applications
in econometrics, Donsker’s (1951) invariance principle, asserting the
convergenceof the rescaled partial sum process to a Wiener process,
has been extended to some FARIMA processes by Philipps (1987)
and Akonom and Gouri´eroux(1987); the latter authors showed that
a fractional integral of the Wiener process, that is, a fractional
Brownian motion, may arise as limiting process — see Wu and
Shao (2006) for extensions and further references. In a similar
spirit, Barbe and Broniatowski (1998) extented Varadhan’s (1966)
large deviation result for partial sums to some FARIMA processes
— see also Ghosh and Samorodnitsky (2009) for related results
in the stationary case. In that extension, the derivative involved in
Varadhan’sactionfunctionalwas replacedby a fractionalderivative.
The classical ruin estimate of Cram´er has been partially extended
by Barbe and McCormick (2008b); and its heavy tail analogue,
Veraverbeke’s(1977)Theorem2,hasbeenalsostudiedin thesetting
of FARIMA processes by Barbe and McCormick (2008a). The
general thrust of these works is to understand how classical results
for partial sums extend to their fractional analogue, somewhat
paralleling the developments related to fractional Brownian motion
and fractional L´evy processes in probability.
In the classical applied probability area of queueing theory, a well
2
studied topic is that of the so-called heavy traffic approximations,
referring to an asymptotic analysis of the behavior of queues with
traffic intensity near one. In an insurance mathematics setting,
this translates into studying a risk process when the premium can
barely cover the claims. For the partial sum process, heavy traffic
approximationsarewellunderstoodandapresentationin bookform
may be found in Resnick (2007) — see also Shneer and Wachtel
(2009) as well as Kosin´ski, Boxma and Zwart (2010) for further
referencesandresults,andWhitt(2002)forexamplesofapplications
in queueing theory. One of the purposes of this paper is to present
an analogous result in the setting of FARIMA processes.
In yet another direction which we will not pursue, heavy traffic
approximation can be interpreted in terms of moving boundary
crossingprobability. Inparticular,ourresultcangivetheprobability
thata fractionalrandomwalk, or more generaly, a FARIMA process
without prehistorical influence, crosses a moving curved boundary.
Throughout this paper we use the letter c for a generic constant
whose value may change from one occurence to another.
We use the symbol . between two sequences, as in say a . b ,
n n
to signify that a 6 b 1+o(1) as n tends to infinity.
n n
(cid:0) (cid:1)
2. Main result. The processes which we will be dealing with are
defined through an analytic function g on ( 1,1) and a distribution
−
functionF ontherealline. Thesetwopiecesofdataallowustobuild
a so-called (g,F)-process as follows. Consider a sequence (X ) of
i i>1
independentrandomvariables,allhavingF fordistributionfunction.
We consider the series expansion of g,
g(x)= g xi.
i
Xi>0
A (g,F)-process (S ) is defined by S = 0 and
n n>0 0
S = g X , n > 1.
n i n−i
06Xi<n
Setting X = 0 if i is negative, and writing B for the backward shift
i
operator acting on sequences, that is BX = X , we see that the
i i−1
above expression for S amounts to
n
S = g(B)X .
n n
3
If g(x) = 1/(1 x), then (S ) is the partial sum process of the
n
−
X . If g is a rational function continuous on [ 1,1], then a (g,F)-
i
−
process is an ARMA one. If g is (1 Id)−d times a rational function
−
continuous on [ 1,1], then a (g,F)-process is a FARIMA one.
−
Introducing the notation
g = g ,
[0,n) i
06Xi<n
we see that if F has finite mean, then S = g(B)(X EX ) +
n n n
−
g EX . If the sequence (g ) is ultimately positive and not
[0,n) 1 n
summable, then S drifts to + if EX is positive and if
n 1
∞ −∞
EX is negative. Our heavytrafficapproximationyields the limiting
1
behavior of max S as the expected value of X tends to 0 from
n>0 n 1
below. This amounts to assuming that the innovations are centered
and seek the asymptotic behavior of max (S ag ) as a
n>0 n [0,n)
−
tends to 0 from above. As mentioned in the introduction, this can
be interpreted as a problem on moving boundary crossing, for the
inequality max (S ag ) > x is equivalent to the process S
n>0 n [0,n) n
−
crossing the boundary x+ag .
[0,n)
A simple examination of known heavy traffic approximation re-
sults for the classical random walk shows that different asymptotic
behaviors are to be expected according to the tail behavior of the
innovation, and, in particular,accordingtothe finitenessof the vari-
ance. The invariance principle of Barbe and McCormick (2010) also
suggests that one should distinguish the cases where the sequence
(g ) diverges to infinity and that where it tends to 0, correspond-
n
ing respectively to a fractional integration and differentiation of the
random walk. In this paper, we will concentrate on the fractional
integration; a companion paper deals with the fractional differentia-
tion. This discussion, as well as technical requirements for the proof
lead us to introduce some assumptions.
The summability of the g is related to the behavior of g near
i
1 and 1, and bears on the long range dependenceproperties of the
−
process. We will restrictourselvestowhatGranger(1988)calledthe
generalized integrated processes, assuming that
g(1 1/Id) is regularly varying of positive index γ. (2.1)
−
Karamata’stheoremforpowerseriesassertsthatif(g )is asymp-
n
totically equivalent to a monotone sequence, then (2.1) is equiva-
lent to (g ) being regularly varying of index γ 1. In this case,
n
−
4
g /g (i/n)γ−1 as n tends to infinity and i/n stays bounded away
i n
∼
for the origin and infinity. We will assume more; firstly, that
(g ) is normalized regularly varying, (2.2)
n
meaning that (see Bingham, Goldie and Teugels, 1989, Theorem
1.9.8)
g γ 1
n+1
= 1+ − 1+o(1)
g n
n (cid:0) (cid:1)
as n tends to infinity, and, secondly, that there exists a positive δ
such that
g i γ−1
lim nδ sup i = 0. (2.3)
n→∞ n−δ6i/n61(cid:12)gn −(cid:16)n(cid:17) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We will prove that assumptions (2.2) and (2.3) hold for FARIMA
processes; therefore, they are not overwhelmingly restrictive in
applications. Note that (2.3) implies (2.1) and that there is no loss
of generality to assume that δ < 1/2; indeed, if (2.3) holds for some
δ, then it holds for any smaller one. Furthermore,(2.2) implies (2.1)
as well.
Asfarastheinnovationsareconcerned,weassumethattheyhave
a mean but no variance, and, more precisely, that
F is centered and in the domain of attraction of a
stable distribution of index α in (1,2). (2.4)
Whenever G is a cummulative distribution function, we write
G for 1 G. We write F for the distribution function of X .
∗ 1
− | |
Assumption (2.4) implies that one of the tails of F is regularly
varying of index α and that F is tail balanced, meaning the
−
following. Write M F for the distribution function of X. Then
−1
−
F coincides with F + M F on the positive half-line when F
∗ −1
is continuous. The tail balance condition is that F pF and
∗
∼
M F qF at infinity where p and q are nonnegative numbers
−1 ∗
∼
which add to 1. For simplicity, we consider throughout the paper
and without mentioning it any further that p does not vanish.
Writing
F←(u)= inf x : F(x)> u
{ }
for the ca`gla`d quantile function associated to F, (2.4) implies that
F←(1 1/Id) is regularly varying of index 1/α; if q does not vanish,
−
5
(M F)←(1 1/Id) is regularly varying of the same index 1/α.
−1
−
Paralleling (2.3), we assume that for some positive κ,
F←(1 λ/t)
lim tκ sup − λ−1/α = 0. (2.5)
t→∞ t−κ6λ6tκ(cid:12)F←(1 1/t) − (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
Ifthisassumptionholdsforsomeκ,thenitholdsforanysmallerone.
While this assumption is stated in a form convenient for our usage,
itsmeaningismademoreexplicitinthefollowingresult,whoseproof
is deferred to section 4.
Proposition 2.1. Let κ be a positive real number less than both
1 and 2/(α+1). The following are equivalent as t tends to infinity:
F←(1 λ/t)
(i) tκ sup − λ−1/α = o(1).
t−κ6λ6tκ(cid:12)F←(1 1/t) − (cid:12)
(cid:12) − (cid:12)
(ii) F←(1 1/(cid:12)t)= ct1/α 1+o(t−κ(1+(cid:12)1/α)) .
−
(cid:0) (cid:1)
Moreover, if F is continuous and increasing, it is also equivalent to
(iii) F(t)= (c/t)α 1+o(t−κ(α+1)) .
(cid:0) (cid:1)
It is likely that our need for (2.5) is an artifact of the technique
used in the proof, and that our result holds in a much greater
generality. This issue is discussed after the proof, in section 3.6.
Considering (1/g ), Proposition 2.1 implies that condition (2.3)
n
is equivalent to the existence of a positive ǫ such that
g = cnγ−1 1+o(n−ǫ)
n
(cid:0) (cid:1)
as n tends to infinity.
Concerning the lower tail we will assume either an analogue of
(2.5), namely
(M F)←(1 λ/t)
lim tκ sup −1 − λ−1/α = 0, (2.6)
t→∞ t−κ6λ6tκ(cid:12)(M−1F)←(1 1/t) − (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
an assumption which is relevant when q doesnot vanish and in some
cases when q vanishes, or, forcing q to vanish,
M F(t)6 cF(tlogt)logt ultimately. (2.7)
−1
Note thatwhile assumptions(2.6)and (2.7)do notcover all possible
distributions, it is not much more restrictive than (2.5) in practical
6
applications; indeed all classical distributions which satisfy (2.5)
satisfy either (2.6) or (2.7).
The distribution function F yields the L´evy measure ν, defined
by its density with respect to the Lebesgue measure λ,
dν
(x) = pαx−α−1 (x)+qα( x)−α−1 (x).
(0,∞) (−∞,0)
dλ 1 − 1
It induces a L´evy stable process L with L´evy measure ν, that is
0
a process with selfsimilar and independent increments, such that,
under (2.4),
EeitL0(1) = exp (eitx 1 itx)dν(x) .
(cid:16)Z − − (cid:17)
The subscript 0 is to indicate that this process is centered. A
fractional L´evy stable process is defined through the Riemann-
Liouville integral
t
L(γ−1)(t)= γ (t u)γ−1dL (u).
0 Z − 0
0
We will use the function
Id
k = .
F←(1 1/Id)
∗ −
It is regularly varying of positive index 1 1/α.
−
Our main result is the following.
Theorem 2.2. Assume that γ is greater than 1 and that (2.2),
(2.3), (2.4) and (2.5) hold. If either (2.6) or (2.7) hold, then
1
lim sup(S ag )=d sup L(γ−1)(t) tγ .
a→0 ag 1−1/k←(1/a) n>1 n− [0,n) t>0(cid:0) 0 − (cid:1)
(cid:0) (cid:1)
Moreover, the random variable sup L(γ−1)(t) tγ is almost
t>0 0 −
surely finite. (cid:0) (cid:1)
Example. We consider a FARIMA process without prehistorical
influence defined as follows. Let Θ and Φ be two real polynomials,
the roots of Φ being outside the complex unit disk and Θ not
vanishing at 1. The FARIMA process
(1 B)γΦ(B)S = Θ(B)X
n n
−
7
is a (g,F)-process with g = (1 Id)−γΘ/Φ. We assume that γ is
−
greaterthan1. Thisprocessis thenafractionallyintegratedrandom
walk.
Lemma 6.1 in Barbeand McCormick(2010)implies thatassump-
tion (2.2) holds. Checking that (2.3) holds is easy, because Lemmas
6.1 and 6.2 in Barbe and McCormick (2010) show that when g is
(1 Id)−γΘ/Φ, there exists a converging sequence (a ) such that
n
−
a
g = cnγ−1 1+ n .
n
n
(cid:16) (cid:17)
This implies that whenever i and n tend to infinity with i at most
n,
g i γ−1 i γ−1 1+a /i
nδ i = nδ i 1
(cid:12)gn −(cid:16)n(cid:17) (cid:12) (cid:16)n(cid:17) (cid:12)1+an/n − (cid:12)
(cid:12) (cid:12) a a(cid:12) (cid:12)
(cid:12) (cid:12). nδ i n(cid:12) . (cid:12) (2.8)
(cid:12) i − n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
If i is in the range [n1−δ,n], then nδ/i tends to 0 whenever δ is less
than1/2,and, similarly, nδ/n tendsto0 as n tendstoinfinity. Since
(a ) converges, we see that (2.8) implies (2.3).
n
To fix the ideas, consider a distribution function F such that
F(x) cx−α. Then F←(1 1/t) (ct)1/α as t tends to infinity.
∼ − ∼
Since F←(1 1/t) p−1/αF←(1 1/t), we obtain
∗ − ∼ −
k(t) (p/c)1/αt(α−1)/α
∼
as t tends to infinity. It follows that
k←(1/a) (c/p)1/(α−1)a−α/(α−1)
∼
as a tends to 0. Since g(1 1/x) xγΘ(1)/Φ(1) as x tends to
− ∼
infinity, we have
1 Θ(1)
ag 1 (c/p)γ/(α−1)a(α−1−αγ)/(α−1)
− k←(1/a) ∼ Φ(1)
(cid:16) (cid:17)
as a tends to 0. Hence, assuming that (2.5) holds, we obtain that
Θ(1)
sup(S ag ) (c/p)γ/(α−1) a1−γα/(α−1)sup L(γ−1)(t) tγ
n− [0,n) ∼ Φ(1) 0 −
n>0 t>0(cid:0) (cid:1)
8
as a tends to 0. In particular, the left hand side grows like
a1−γα/(α−1). It is interesting to note that the exponent involved
depends on α and γ only through γα/(α 1), that is γ times the
−
conjugate exponent of α.
3. Proof of Theorem 2.2. For the classical random walk, there
exists two ways of proving a heavy traffix approximation: one based
ontheWiener-HopffactorizationproposedbyKingman(1961,1962,
1965),onebasedonafunctionallimittheoremproposedbyProhorov
(1963). We follow Prohorov’s approach.
Throughout the proof we will use many times the following form
of Karamata’s theorem for power series (see Bingham, Goldie and
Teugels,1989,Corollary1.7.3). If(g )isregularlyvaryingofpositive
n
index γ 1, it is asymptoticallyequivalent to an increasing sequence
−
and
γg g(1 1/n)
[0,n)
g − (3.1)
n
∼ n ∼ nΓ(γ)
as n tends to infinity.
Toproceedwiththeproof,uptoanasymptoticequivalence,define
Λ = Λ(1/a) by the relation
ak(Λ) 1 (3.2)
∼
as a tends to 0. It follows from Barbe and McCormick’s (2010)
Theorem 5.2 that, in the sense of weak convergence of distribution
∗
of stochastic processes in D[0, ) endowed with the topology of
∞
uniform convergence on compactas,
k(Λ)
d (γ−1)
S L
g ⌊ΛId⌋ −→ 0
[0,Λ)
as Λ tends to infinity. Since (3.1) and (2.3) imply that (g ) is a
[0,n)
regularly varying sequenceof index γ, (3.2) implies that we have the
convergence of stochastic processes
k(Λ)
(S ag ) d L(γ−1) Idγ. (3.3)
g ⌊ΛId⌋− [0,ΛId) −→ 0 −
[0,Λ)
Consequently, for any positive T,
k(Λ)
sup (S ag ) d sup L(γ−1)(t) tγ
g n− [0,n) −→ 0 −
[0,Λ) 06n6ΛT 06t6T(cid:0) (cid:1)
9
as a tends to 0.
Sincesup L(γ−1)(t) tγ isnondecreasinginT,itconverges
06t6T 0 −
almost surely as T(cid:0) tends to infin(cid:1)ity, possibly to infinity. Hence, to
prove Theorem 2.2, it suffices to show that
lim limsupP n > ΛT : S > ag = 0 (3.4)
n [0,n)
T→∞ a→0 {∃ }
and sup L(γ−1)(t) tγ is almost surely finite.
t>0 0 −
As point(cid:0)ed out by Shn(cid:1)eer and Wachtel (2009), or differently in
Szczotka and Woykzyn´ski (2003), the main difficulty in proving a
heavy traffic approximation for sums is to show that the maximum
of the process does not occur too far in time, that is, in our case
proving (3.4). In the context of (g,F)-processesthis task is far more
involved thanfor ordinaryrandomwalks, mostlybecausethereis no
analogue of Kolmogorov’s maximal inequality. In order to explain
our proof, that is, the remainder of this paper, we need to make a
preliminary study of (3.4).
Given (3.2), we see that (3.4) is equivalent to
lim limsupP n> ΛT : S > g /k(Λ) = 0. (3.5)
n [0,n)
T→∞ Λ→∞ {∃ }
Substituting Λ for ΛT, using that
k(Λ/T) T(1/α)−1k(Λ)
∼
as Λtendstoinfinity, andusing that1 1/α is positive, substituting
−
T for T1−1/α, we obtain that (3.5) is equivalent to
lim limsupP n> Λ : S > Tg /k(Λ) = 0. (3.6)
n [0,n)
T→∞ Λ→∞ {∃ }
WecannowexplainhowtoproveTheorem2.2. Theproofhasfour
main steps. The first two aim at showing that instead of considering
all n exceeding Λ in (3.6), we can reduce the range to all n between
Λ and Λ1+ǫ, where ǫ is positive but can be chosen as one wishes.
This is achieved by showing in the first step that the innovations
coming from the central part of the distribution can be ignored. In
the second step, a simple bound on the contribution of the largest
innovations permits us to show that if the event involving S in
n
(3.6) occurs, it is very likely that n is less than Λ1+ǫ. Being able to
concentrate on the range of n between Λ and Λ1+ǫ, the third step
10