WOBABILlTY
,
AND
MATHEMATICAL STATISTICS
Val. 25, Faac. 1 (ZWS], pp. 173-181
ON APPROXIMATIONS OF RISK PROCESS
WITH RENEWAL ARRIVALS IN a-STABLE DBMAKN
2.MICBNA (W~octaw)
Abstract. In this paper we approximate risk process by an
a-stable U v y motion (1 < a < 2). We consider two conditions imposed on the value of the premium rate. The first one assumes that the
premiums exceed only slightly the expected claims (heavy traffic) and
the second one assumes that the premiums are much larger than the
average claims (light tr&c). We consider the distribution of claim sues
belonging to the domain of attraction 01an a-stable law and the process counting claims is a renewal process constructed from random
variables belonging to the domain of attraction of an ol'-stable law.
Comparing rr and a' we obtain three different asymptotic risk processes. In the classical model we get a Brownian diffusion approximation
which fits fist two moments. If a' > a, we get Mittag-Lemer distribution for the i n f i t e time ruin probability, and if a'< a, we obtain
exponential distribution for the infinite time ruin probability.
AMS Subject Classikation: Primary 91B30; Secondary 60F17,
60G52.
Key words and phrases: Risk process, heavy traffic, light traffic,
a-stable LBvy motion.
Collective risk theory is concerned with random fluctuations of the total net assets - the capital of an insurance company. Consider a company
which only writes ordinary insurance policies such as accident, disability,
health and whole life. The policyhoIders pay premiums regularly and at certain
random times make claims to the company. A policyholder's premium, the
gross risk premium, is a positive amount composed of two components. The
net risk premium is the component calculated to cover the payments of claims
on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the
average and also allows an accumulation of capital. So the risk process has the
Cramk-Lundberg form R (t) = u + p t
&, where u > 0 is the initial risk
-I::,
12
- PAMS 25.1
174
Z. Michna
reserve of the company and the policyholders pay a gross risk premium of
p > 0 per a unit time. The successive claims (5)are assumed to form a sequence
of i.i.d. random variables with mean EY1 = p and claims occur at jumps of
a renewal counting process N t , t 2 0.
The ruin time T is defined as the first time the company has a negative risk
reserve and one of the key problems of collective risk theory concerns calculating the ultimate ruin probability 'Y = P ( T < oo ( R(0) = u), i.e. the probability that the risk process ever becomes negative. On the other hand, an
insurance company will typically be interested in the probability that ruin
occursbefore time <i.e. Y (t)= P (T < t IR (0) = u). Many of the results are in
the forrd of complicated analytic expressions. For a comprehensive treatment
of the theory the reader should consult Asmussen [I], Embrechts et al. [5] or
Rolski et al. [15].
Diffusion approximation of random walks via Donsker's theorem is a classical topic of probability theory. The first application in risk theory is the paper
by Iglehart [lo], and two further standard references in the area are Grandell
[8] and [9]. For claims with infinite variance the paper by Furrer et al. [ 7 ]
suggested an approximation by an a-stable L6vy motion. The idea is to let the
number of claims grow in a unit time interval and to make the claim sizes
smaller in such a way that the risk process converges weakly to an a-stable
Lkvy motion with a linear drift.
In this paper we consider two settings of the premium with respect to the
average claims, that is heavy and light tr&c. The terms 'light' and 'heavy
traffic' come from queueing theory and have an obvious interpretation. Light
trafic means that the premium is much larger than the average amount of
claims per unit time and heauy tra& describes the case when the premium rate
exceeds only slightly the average amount of claims per unit time. Thus we
approximate a risk process with renewal arrivals by an a-stable LCvy motion in
these two settings. The stability parameter a depends on the tails of the arrival
-and claim distributions and the scale parameter depends on the expected values
of claims and arrivals. This work extends the results of Furrer et al. 17). The
- fits first
classical risk process is approximated by a Brownian diffusion which
two moments.
For a similar treatment in the theory of queueing systems see Szczotka
and Woyczynski [19].
2. WEAK CONVERGENCE OF STOCHASTIC PROCESSES
Let us consider the claim surplus process
Risk process with renaval arrivaIs
175
where {K) are i.i.d. random variables, EYl = p, N , is a renewal process constructed from the sequence of i.i.d. random variables (Tk) with ET, = l/A > 0.
Moreover, we assume that p-Ap > 0 and, in general, the process N , which
counts claims does not have to be independent of the sequence of the claim
sizes (E;,) as assumed in many risk modeIs. Then
R It) = u - StP)( t )
is a risk process. If N , is Poisson process independent of the sequence of the
claim sizes-(&),- we get--the classical risk model.
Assume that the sequence {Y,) satisfies
when n + m, where EY, = p and Z,(1) is an a-stable random variable with
1 < ac: < 2, scale parameter a (for ac: = 2 we put a2 = Var Yl),skewness parameter P and #(n) = nl/" L(n), where L(n) is a slowly varying function at infinity.
Then
when c -+ 03 in the Skorokhod J1 topology (see Prokhorov [14]). Moreover,
we assume that the i.i.d. random variables {T,) fulfill
when n -+ oo, where ET, = 1/d > 0 and Zb,(l) is an a'-stable random variable
with 1 < a' < 2, scale parameter d (for a' = 2 we put gr2= Var TI), skewness
parameter p' and & (n) = nila' E (n),where L: (n) is a slowly varying function at
infinity. Then
when c -+ cc in the Skorokhod MI topology (see Bingham [4] or Michna
C131)PROPOSITION
1. Assume that the sequences {Y,) and
and that a < a'. Then
1
-(
Nc*
C
4(c)k=l
I$ - Apct) * Z , (At)
when c -, co in the Skorokhod J1 topology.
(5)sutiSfY (1) and (3)
176
Z . Michna
Proof. We have
1
---(z
1 Net
Net
4(c) k = l
&-APct)
.
C (Y,-P)+P
#(~)k=1
=-
N,, - k t
$(c) * 2, Cat)
because
(note that N,/c
over -
3
At as c + CQ and use Theorem 3.1 of Whitt [22]) and more.-
in the Skorokhod M1 topology, and 4'(c)/# (c) + 0 when c + co, which implies
N,, - Act
4 (c)
-0.
Ei
Similarly we can show the following proposition.
PROPOSZTION
2. Assume that the sequences I&) and
and that a' < a. Then
{x)satisfY (1)and (3)
when c -, co in the Skorokhod M I topology.
In the case a = a' the problem is more complicated but it can be solved.
with
PROPOSITION
3. Assume that the sequences {&) and
4 (n) = #(n) a d me independent. Then
(z)satiSfy (1) and (3)
when c + m in the Skorokhod M1 topology.
P r o of. The assumption
4 (n) #' (n) implies
that
ol = a',
and thus
N,, - Act
#(4
,Illaz, (t) - p l l 11"Z&(t)
+
because
1
NCt
C (%-p)=s-ZZ,(At) and
4(~)k=1
---
1
Na-Act_ - ~ I + I I ~a(Zt
dJ(c)
when c + co in the Skorokhod M1 topology. Thus, using Theorem 5.1 of Whitt
[22], we complete the proof.
177
Risk ~rocesswith renewal arrivals
3. WEAK APPROXIMATION OF C L A M SURPLUS' PROCTSS
In this section we wiU approximate the claim surplus process by an
a-stable Levy motion in two settings: heavy and light traffic. Now we are in
a position to rewrite Propositions 1, 2 and 3 in terms of the claim surplus
process. For the clear presentation let us introduce the following notation:
c, ( p ) = Ip -;IC1)-a1(apl)
(5)
and similarly
.--
d
car
(p -J p ) -a'/(=* - 1)
We first consider heavy traffic approximation, that is, the p r e ~ u m sexceed
only slightly the expected claims on the average.
THEOREM
4. Assume that the sequences {&) and (G}belong to the domain
ofnorm1 attraction of a stable Jaw, that is, # ( n ) = nl/" and # ( n ) = nl/"'.Let the
functions ca(p)and c , ( p ) be deJined in (5) and (6), respectively. Then, as p l l p ,
(6)
L
Y
(c. ( p ) )- ' l a S"' (tc. ( p ) ) * ,I1" 2. ( I ) - t
-
in the Skorokhod J1 topology when a <a', and
(c@*( p ) )- lla'
S l P ) (tc..
(p))
-fil
+
l"'
in the Skorokhod M I topology when a' < a, and
( C a ( p ) ) - lIa S ( P ) (tea( p ) ) * A'" Z, (t)- , d l
in the Skorokhod MI topology when a'
independent.
=a
2'a, (tl- t
+
liaZ'a(t)-t
and the sequences {&) and {T,) are
P r o of. Using the identity
5
Y ,-Apct
= S(P)(ct)+ct
(p-rlp)
k=l
and substituting c = ca(p)= ( ~ - I , U ) - " / ( " -in
~ )Proposition 1 for a _<a', we get
the desired convergence. Similarly, using Propositions 2 and 3 we show
the
convergence for a' < a and or = a'. rn
In the light traffic approximation we assume that the premiums are much
larger than the expected claims on the average.
T ~ R E 5.
M Assume that the sequences {G) and {G)belong to*the domain
of normal attraction o f a stable law, that is, # (n) = nl/" and @(n) = nlja'. Let the
functions ca(p)and c,, ( p ) be deJined in (5) and (6), respectively. Then, as p + co,
+
(c. (p))lCS(P)(t/c. ( p ) ) ( p- Apl2 t * A"' Z . ( t )
in the Skorokhod J 1 topology when u < or', and
(c., (p))'"
b)
(tica,( p ) )+( p- 1 ~t *) -~
pA1
+
Z h, 0)
liar
178
Z. Michna
in the Skorokhod M I topology when a' < a, and
(c.
+ (p-ip)=
t =Ilia Za(t)-pL1 +
( P ) ) l l u S ( P ) (t/c. (p))
&(t)
l@
in the Skorokhod M 1 topology when a' = ol and the sequences (&} and
independent.
{G)are
Proof. As before, using the identity
b -t
-
-. .
.
c t
=
+
(ct) ct ( p- Ap)
k=l
and su&tituting c = l / c , ( p ) = ( p -Ap)"i(a-l) in Proposition 1 for a < u', we get
the desired co&ergence. ~ i r n i l ~ rusing
l ~ , Propositions 2 and 3 we show the
convergence for or' < a and a = a'. PP
Now we want to take advantage of the above results, and therefore we
present the crucial theorems. For the finite time ruin probability we need the
foIIowing result.
THEOREM
6. Under the assumptions of Theorem 4, us ppllp,
a'] (sea (p))> u)
P (sup (ca( p # - '"
+
P (sup ( X (s) -s) > u)
sdt
s<t
for every t > 0 when a < a' and
x(s) =
if
E
< a',
q a=d.
(::~2~~-m1+11az6(3
Similarly, as pplllp,
P (sup (ca#( p ) )- lia's(" (sc.. ( p ) ) > u) + P (SUP (x(s)- s) > u )
sdi
s6t
for every t > 0 when a' < a and
x (3) = -pJ1
(8)
.
+
lIa' Z'a,(s)
-
-
-
Proof. In the Gaussian case, that is, ol = a' = 2, the assertion follows from
Proposition 5 of Michna 1121, and in other cases from Theorem 2 of Furrer
et al.
The convergence of infinite time ruin probabilities is much more cumbersome because the functional sup,,, x (s) is not continuous in Skorokhod topologies.
En.
THEOREM
7 . Under the assumptions of Theorem 4, as pJA,u,
179
Risk process with renewal arrivals
when a < E' and the process X is dejned in (7). Similarly, as pJAp,
P (sup (c., (p))- liU's(p)
(scar (P)) > U)
o
sr
P (sup (x(s) -s) > U)
s=-o
when a' < a and the process X i s dejined in (8).
P r o of. In the Gaussian case, that is, a = d = 2 the assertion follows from
Theorem 7.1 in Asmussen [2] and in the ol-stable case, that is, 1 < a < 2, from
Theorem 3 in Szczotka and Woyczynski [I81 (using duality of ruin probability
and distribution bf statiihiry waiting time for G/G/l queues, see e.g. Aimussen
C11). la
Thus let us consider heavy traffic approximation of a claim surplus process and approximate ruin probability. For a < a' and p J l p we obtain
Y (t) = P (sup S'P)(s) >
4=P(
s
sst
sup SIP) (su,(p)) > u)
~Ic~P)
where the process X is defined in (7), in the second last equality we have used
Theorem 6 and in the last equality the self-similarity property of the process X.
Similarly we proceed for a' < ol and for infinite time ruin probabilities using
Theorem 7. Thus the infinite time ruin probability for pJhp can be approximated in the following way:
Y
= P (sup S(P' (3)
s>o
> u) w P (sup (X(s) -( p- Ap) s) > u )
s>o
and the finite time ruin probability as
Y (t) = ~ ( s u p S @ ) (>
s )u) x P(SUP(X(S)-(p-I&)
s St
> u),
s<r
where the process X is defined in (7) and (8).
Now let us focus on the case a' = a = 2 which includes the classical risk
model. Then the infinite time ruin probability can be approximated in the
following way:
180
Z. Michna
where B (sj is the standard Brownian motion, o2 = V~arY, and ar2= Var TI.
For the finite time horizon we get
Y ( t )= P (supS(P)(s) > u) W 1- @
s<t
{d(;c::p--y:'2)]
-
where -@ (x)is the standard normal distribution. In the classical case, when N , is
a ~oissGnprocess with intensity L (Var Ti = a'' = l/A2), we obtain
and
Y (t) = P (sup SIP' (s) > 24)
s41
Interesting results can be obtained in the case a' < a. We do not assume
that the interarrivals {T,) are positive random variables, only their expectation
has to be positive. But if we assume that (T,) are positive random variables,
then the limit process has the skewness parameter equal to - 1. Thus, as pJAp,
the infinite time ruin probability has the following form:
where a = (p -Rpj (pa')-a'R-"'-i cos 171: (a' -2)/2).
In the case a < a', if the claim sizes {&I
are positive random variables, as
-plAp, we obtain the following approximation for the infinite time-ruin probability:
where a = (p -2p) a-" A-1 cos {x (a -21/21 (see Furrer [6] or Szczotka and
Woyczy6ski [18]).
In the case ar= a' and 1 < a < 2 it is possible to obtain in the limit symmetric a-stable Lkvy motion, and the exact form of the Laplace-Stieltjes transform
of the infinite time ruin distribution (G(u) = 1-Y (u)) is given in Szczotka
and Woyczyriski [18j.
Risk process with renewal 'arrivals
181
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[I]
[2]
[3]
[4]
Department of Mathematics
Wroclaw University of Economics
53-345 Wroclaw, Poland
E-mail: [email protected]
Received on 21.1.2005;
revised version on 5.7.2005