Risk process with stochastic premiums
The Cramer-Lundberg model with stochastic premiums which is natural generalization of classical dynamic risk model is considered. Using martingale technique the Lundberg inequality for ruin probability is proved and characteristic equations for Lundberg coefficients are presented for certain classes...
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irk-123456789-45762009-12-08T12:00:33Z Risk process with stochastic premiums Zinchenko, N. Andrusiv, A. The Cramer-Lundberg model with stochastic premiums which is natural generalization of classical dynamic risk model is considered. Using martingale technique the Lundberg inequality for ruin probability is proved and characteristic equations for Lundberg coefficients are presented for certain classes of stochastic premiums and claims. The simple diffusion and de Vylder approximations for the ruin probability are introduced and investigated similarly to classical Cramer-Lundberg set-up. The weak and strong invariance principles for risk processes with stochastic premiums are discussed. Certain variants of the strong invariance principle for risk process are proved under various assumptions on claim size distributions. Obtained results are used for investigation the rate of growth of the risk process and its increments. Various modifications of the LIL and Erdos-Renyi-type SSLN are proved both for the cases of small and large claims. 2008 Article Risk process with stochastic premiums / N. Zinchenko, A. Andrusiv // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 189-208. — Бібліогр.: 36 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4576 en Інститут математики НАН України |
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The Cramer-Lundberg model with stochastic premiums which is natural generalization of classical dynamic risk model is considered. Using martingale technique the Lundberg inequality for ruin probability is proved and characteristic equations for Lundberg coefficients are presented for certain classes of stochastic premiums and claims. The simple diffusion and de Vylder approximations for the ruin probability are introduced and investigated similarly to classical Cramer-Lundberg set-up. The weak and strong invariance principles for risk processes with stochastic premiums are discussed. Certain variants of the strong invariance principle for risk process are proved under various assumptions on claim size distributions. Obtained results are used for investigation the rate of growth of the risk process and its increments. Various modifications of the LIL and Erdos-Renyi-type SSLN are proved both for the cases of small and large claims. |
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Zinchenko, N. Andrusiv, A. |
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Zinchenko, N. Andrusiv, A. Risk process with stochastic premiums |
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Zinchenko, N. Andrusiv, A. |
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Zinchenko, N. |
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Risk process with stochastic premiums |
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Risk process with stochastic premiums |
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Risk process with stochastic premiums |
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Risk process with stochastic premiums |
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Risk process with stochastic premiums |
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risk process with stochastic premiums |
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Інститут математики НАН України |
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2008 |
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Risk process with stochastic premiums / N. Zinchenko, A. Andrusiv // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 189-208. — Бібліогр.: 36 назв.— англ. |
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AT zinchenkon riskprocesswithstochasticpremiums AT andrusiva riskprocesswithstochasticpremiums |
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Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.189-208
NADIIA ZINCHENKO AND ANDRII ANDRUSIV
RISK PROCESS WITH STOCHASTIC PREMIUMS
The Cramér-Lundberg model with stochastic premiums which is na-
tural generalization of classical dynamic risk model is considered.
Using martingale technique the Lundberg inequality for ruin proba-
bility is proved and characteristic equations for Lundberg coefficients
are presented for certain classes of stochastic premiums and claims.
The simple diffusion and de Vylder approximations for the ruin pro-
bability are introduced and investigated similarly to classical Cramér-
Lundberg set-up. The weak and strong invariance principles for risk
processes with stochastic premiums are discussed. Certain variants
of the strong invariance principle for risk process are proved under
various assumptions on claim size distributions. Obtained results are
used for investigation the rate of growth of the risk process and its
increments. Various modifications of the LIL and Erdös-Renyi-type
SSLN are proved both for the cases of small and large claims.
0. Introduction. Definition of the model
Suppose that the following objects are defined on the same probability
space (Ω,F, P ):
– two independent Poisson processes N(t) and N1(t) with intensities
λ > 0 and λ1 > 0 (EN(t) = λt, N(0) = 0, EN1(t) = λ1t, N1(0) = 0);
– two sequences of i.i.d.r.v. (xi : i ≥ 1) and (yi : i ≥ 1) independent of
the Poisson processes and of each other with d.f.F (x) andG(x), respectively,
F (0) = 0, G(0) = 0.
Within the Cramér-Lundberg risk model with stochastic premiums the
risk process U(t), t ≥ 0, is defined as
U(t) = u+
N1(t)∑
i=1
yi −
N(t)∑
i=1
xi, (1)
Invited lecture.
2000 Mathematics Subject Classifications 60F17, 60F15, 60G52, 60G50.
Key words and phrases. Risk models, ruin probability; Lundberg’s inequality, ran-
dom sums, Lévy processes, stable processes, invariance principle, domain of attraction,
randomly stopped process, law of iterated logarithm, almost sure convergence.
189
190 NADIIA ZINCHENKO, ANDRII ANDRUSIV
where: u ≥ 0 is an initial capital; N(t) - the number of claims in the
time interval [0, t]; positive i.i.d.r.v. (xi : i ≥ 1) are claim sizes; N1(t) is
interpreted as a number of polices bought during [0, t]; (yi : i ≥ 1) stand for
sizes of premiums paid for corresponding polices.
So, in (1) not only total claim amount process S(t) =
N(t)∑
i=1
xi is a com-
pound Poisson process, but also the total premium amount process
Π(t) =
N1(t)∑
i=1
yi is a process of such type unlike the classical Cramér-Lundberg
risk model
U(t) = u+ ct−
N(t)∑
i=1
xi
with non-random linear continuous premium income function Π∗(t) = ct,
c > 0.
Due to properties of compound Poisson process Q(t) = Π(t) − S(t) is
again a compound Poisson process with intensity λ∗ = λ + λ1 and d.f. of
the jumps G∗(x) = λ1
λ∗G(x) + λ
λ∗F
∗(x), where F ∗(x) is a d.f. of the random
variable −x1. In the other words Q(t) admits the representation
Q(t) =
N∗(t)∑
i=1
ξi, (2)
where N∗(t) is homogeneous Poisson process with intensity λ∗ = λ+λ1 and
i.i.d.r.v. ξi have d.f. G∗(x).
Thus, moment generating function
E{exp(rQ(t))} = exp
(
λ∗
(
λ1
λ∗
My(r) +
λ
λ∗
Mx(−r) − 1
))
,
if moment generating function Mx(r) for claim size x1 and My(r) – for
premium y1 exist.
Denote by
τ = inf(t > 0 : U(t) < 0)
the ruin time, by
ψ(u, T ) = P ( inf
0<t≤T
U(t) < 0) = P (τ ≤ T |U(0) = u)
the probability of ruin in a finite time ( in time interval (0,T]) and by
ψ(u) = P (inf
t>0
U(t) < 0) = lim
T→∞
ψ(u, T )
the probability of ultimate ruin (the ruin probability in infinite time).
RISK PROCESS WITH STOCHASTIC PREMIUMS 191
We will also use notations ϕ(u) = 1 − ψ(u) and ϕ(u, T ) = 1 − ψ(u, T )
for non-ruin probabilities in infinite and finite time intervals, respectively.
It is obvious, that EQ(t) = t(λ1Ey1 − λEx1). In forthcoming we shall
suppose that EQ(t) > 0 for t > 0, i.e. that
λ1Ey1 > λEx1. (3)
This condition is analog of net profit condition in classical Cramér-
Lundberg model and, due to the SLLN, provides ψ(u) < 1.
Model (1) was studied by A. Boykov [6], L.Gilina [18], D. Gusak [20] and
in authors previous work [2]. V.Korolev, V.Bening, S.Shorgin [23] present
an interesting example of using (1) for modelling the speculative activity of
money exchange point and optimization of its profit.
In [6] Boykov proved that non-ruin probability ϕ(u) satisfy equation
(λ+ λ1)ϕ(u) =
λ
u∫
0
ϕ(u− x)dF (x) + λ1
∞∫
0
ϕ(u+ ν)dG(ν), (4)
and find its solution if:
(a) P (xi = 1) = P (yi = 1) = 1, in this case
ϕ(u) = ϕ([u]) = 1 −
(
λ
λ1
)[u]+1
, [u] − entire of x;
(b) premiums and claims have exponential distributions, i.e. G(x) =
1 − e−bt, F (x) = 1 − e−at, a, b > 0. In this situation
ψ(u) =
(a + b)λ
(λ+ λ1)a
exp
(
−λ1a− λb
λ+ λ1
u
)
. (5)
Note that in this case condition (3) is equivalent to λ1/b > λ/a.
In [6] it also proved that non-ruin probability ϕ(u, t) on finite interval
satisfy equation
∂ϕ(u, t)
∂t
+ (λ+ λ1)ϕ(u, t) =
λ
u∫
0
ϕ(u− x, t)dF (x) + λ1
∞∫
0
ϕ(u+ ν, t)dG(ν), (6)
In the case of exponential premiums and claims equation (6) can be reduced
to ∞∫
0
(1 − ϕ(u, t))e−tsdt = A(s)eα(s)u,
192 NADIIA ZINCHENKO, ANDRII ANDRUSIV
where
α(s) =
λb− λ1a+ s(b− a)
2(λ+ λ1 + s)
−
√
[λb− λ1a+ s(b− a)]2 + 4ab(λ + λ1 + s)s
2(λ+ λ1 + s)
,
A(s) =
λ
s(s+ λ+ λ1 − λ1b(b− α(s))−1)
.
Since one can obtain the explicit solution of equations (5), (6) and derive
the exact formulas for ruin/non-ruin probabilities only in a few exceptional
cases, the problem of finding practically useful estimates or approximations
for ψ(u) becomes rather important.
We start with Lundberg’s exponential inequality ( Section 1) and prove
it using standard martingale technique. Also we discussed two examples
of finding adjustment (Lundberg) coefficient for exponential premiums and
claims with gamma or mixture of exponential distributions.
In Section 2 we extend rather “simple” and “practically useful” de
Vylder approach to approximation of ruin probabilities for the models with
stochastic premiums.
Section 3 is devoted to diffusion approximation of ruin probabilities in
finite/infinite intervals. Unlike ad hoc de Vylder approximation, the diffu-
sion approximation has a solid theoretical base in functional limit theorems
(weak invariance principle) for Q(t) and risk process U(t).
In Section 4 we deal with other type of limit theorems, so called strong
invariance principle (SIP), which in certain sense is a bridge between weak
and a.s. convergence. Using rather general results due to SIP for random
sums, we proved SIP for risk processes with stochastic premiums. Special
cases, i.e. claims with finite second moment and large claims attracted to
α-stable law, are studied separately. In the cases of classical and Sparre
Andersen risk models SIP can be used to find approximations of ruin prob-
abilities on finite interval, see M.Csörgő, L. Horváth (1993). But in our
work we shall use SIP for other purpose: investigation of the rate of growth
of risk process and its increments.
Thus, in Section 5 we proved various modifications of the law of iterated
logarithm (LIL) and Erdös-Renyi-Révész-Csörgő-type strong law of large
numbers (SLLN) for risk processes. Cases of small claims, large claims
with finite variance and large claims attracted to asymmetric stable law are
discussed.
1. Martingale approach. Lundberg’s inequality
Similar to the case of classical risk process, martingale approach leads to
a simple, but useful exponential upper bound for ultimate ruin probability
for model (1).
RISK PROCESS WITH STOCHASTIC PREMIUMS 193
Consider the equation, which we traditionally will call “characteristic”
λ1(Ee
−Ry1 − 1) + λ(EeRx1 − 1) = 0 (7)
which is equivalent to
λ1My(−R) + λMx(R) = λ1 + λ.
if corresponding moment generating functions for y1 and x1 exist in some
neighborhood of zero.
Definition Lundberg coefficient(Lundberg exponent, adjustment coefficient)
for model (1) is a positive solution of equation (7), if such solution exists.
It is easy to check that the process exp{−R(Π(t)−S(t))} is a martingale
(relative to natural filtration), and ruin time τ is a stopping time, see [6].
Thus, standard considerations as in J. Grandell [19] give the possibility to
prove following statement.
Lemma 1. If R > 0 is a solution of (7), then
ψ(u) =
e−Ru
E{e−RU(τ)|τ <∞} .
More useful for applications is following upper bound.
Corollary 1(Lundberg’s inequality). Suppose that adjustment coeffi-
cient R exists, then
ψ(u) < e−Ru.
In the case of exponentially distributed premiums and claim sizes,i.e.
G(x) = 1−exp(−bx), F (x) = 1−exp(−ax), one can solve equation (7) and
obtain the explicit formula for Lundberg coefficient
R =
(λ1a− λb)u
λ+ λ1
.
We shall present another two examples when (7) has rather simple form.
Example 1. Assume that in model (1) claim sizes are distributed according
to mixture of exponential distributions with d.f. F (x) = p1F1(x)+ p2F2(x),
194 NADIIA ZINCHENKO, ANDRII ANDRUSIV
F1(x) = 1 − e−a1x, F2(x) = 1 − e−a2x, p1 + p2 = 1, while premiums are ex-
ponentially distributed with G(x) = 1− e−bx. Then characteristic equation
λ1
( ∞∫
0
e−RydG(y) − 1
)
+ λ
( ∞∫
0
eRxdF (x) − 1
)
= 0
implies
λ1
( ∞∫
0
be−(R+b)ydy − 1)
)
+
+λ
( ∞∫
0
p1a1e
−(a1−R)xdx+
∞∫
0
p2a2e
−(a2−R)xdx− 1
)
= 0 (8)
where R < min(a1, a2). Equation (8) is transformed to
λ1R
R + b
− λ
(
p1a2 + p2a1 −R
(a1 − R)(a2 −R)
)
= 0
Thus, Lundberg coefficient R is the positive solution of the equation
R2(λ1+λ)−R(λ1a1+λ1a2+λb−λp2a1−λp1a2)+λ1a1a2−λbp2a1−λbp1a2 = 0,
where
p1 + p2 = 1, a1 > 0, a2 > 0, λ > 0, λ1 > 0,
λ1
b
> λ
(p1
a1
+
p2
a2
)
.
Here the last inequality provides (3). Solution must satisfy additional con-
dition 0 < R < min(a1, a2).
Example 2. Now suppose that premiums have exponential distribution
with parameter b > 0 and claims have Γ(α, β)-distribution with density
βα
Γ(α)
xα−1e−βx.
In this case Lundberg coefficient R is a solution of the system⎧⎪⎪⎨⎪⎪⎩
λ1 + λ = λ1b
b+R
+ λβα
(β−R)α
0 < R < β
λ1
b
> λα
β
λ > 0, λ1 > 0, b > 0, α > 0, β > 0.
Martingale approach and Lundberg inequality are applicable in the case
of risk process with stochastic premiums perturbed by a Wiener process
RISK PROCESS WITH STOCHASTIC PREMIUMS 195
U1(t) = u+
N1(t)∑
i=1
yi −
N(t)∑
i=1
xi + ςW (t),
where W (t) is a standard Wiener process, ς > 0.
For such model Lundberg coefficient r is the positive solution of the
equation
E exp {−r(Π(t) − S(t) + ςW (t))} = 1
which leads to equation
λ1(Ee
−ry1 − 1) + λ(Eerx1 − 1) + ς2r2/2 = 0 (9)
if corresponding moment generating functions for x1 and y1 exist in some
neibourhood of zero.
Notice that the process exp{−r(Π(t) − S(t) + ςW (t))} is a martingale
(relative to natural filtration), and ruin time τ ∗ for perturbed risk model is
a stopping time. Thus, we have variant of Lundberg inequality:
Corollary 2. Suppose that adjustment coefficient r exists, then ruin proba-
bility ψ∗(u) for perturbed risk model satisfies
ψ∗(u) < e−ru.
The proof is standard:
1 = E exp{−r(Π(t) − S(t) + ςW (t))} =
= E exp{−r(Π(t ∧ τ ∗) − S(t ∧ τ ∗) + ςW (t ∧ τ ∗))} ≥
≥ E exp{−r(Π(t ∧ τ ∗) − S(t ∧ τ ∗)) +
(rς)2
2
(t ∧ τ ∗))}χ{τ∗≤t} ≥
≥ exp{ru}P (τ ∗ ≤ t) ⇒
⇒ P (τ ∗ ≤ t) = 1 − ϕ∗(u, t) ≤ exp{−ru}.
So,
ψ∗(u) ≤ exp{−ru}.
2. De Vilder approximation of ruin probability
Since the problem of derivation the exact formulas for ruin probabilities
for model (1) or even solution of characteristic equation (7) is more compli-
cated than for classical risk model, the methods which give rather “simple”
and practically applicable approximations for ψ(u) or ψ(u, T ) become very
196 NADIIA ZINCHENKO, ANDRII ANDRUSIV
actual. One of approaches to solution such a problem, similar to classical
case, is based on a simple idea to replace the origin risk process U(t) with an
approximating risk process Ũ(t) for which calculations of ruin probabilities,
i.e. distributions of infimum/supremum are easier. Some approximations
are chosen based on functional limit theorems (invariance principles for
random sums and risk processes), other – are ad hog procedures based on
considerations of simplicity for applications. De Vylder approximation be-
longs to the second group. Below we demonstrate its application to a model
with stochastic premiums.
We propose to replace the original risk process
U(t) = u+Q(t) = u+
N1(t)∑
i=1
xi −
N(t)∑
i=1
yi
with the risk process
Ũ(t) = u+ Q̃(t) = u+
Ñ1(t)∑
i=1
x̃i −
Ñ(t)∑
i=1
ỹi,
where premiums and claims are exponentially distributed, i.e. {x̃i} and
{ỹi} are independent sequences of i.i.d.r.v. exponentially distributed with
parameters α and β, relatively; Ñ1(t) and Ñ(t) - mutually independent
Poisson processes with intensities λ̃1 > 0 and λ̃ > 0, respectively. Parame-
ters α, β, λ̃1 and λ̃ are chosen in such a way that first four moments of Q̃(t)
and Q(t) are equal. More precise we shall demand that
EQ(t) = EQ̃(t), E(Q(t) −EQ(t))k = E(Q̃(t) − Q̃(t))k, k = 2, 3, 4.
Denote by mk = Eyk1 , μk = Exk1, k ≥ 1. Keeping in mind that
M1(t) = EQ(t) = (λ1m1 − λμ1)t,
Mk(t) = E(Q(t) − EQ(t))k = (λ1mk − λμk)t, k = 2, 3, 4.
we get following system for determining parameters α, β, λ̃1 and λ̃
λ1m1 − λμ1 =
λ̃1
α
− λ̃
β
, λ1m2 + λμ2 =
2λ̃1
α2
+
2λ̃
β2
λ1m3 − λμ3 =
6λ̃1
α3
− 6λ̃
β3
, λ1m4 + λμ4 =
24λ̃1
α4
+
24λ̃
β4
. (10)
Finally, let us input obtained values of parameters in the formula (5), i.e.
formula for ruin probability in the case of exponential claims and premiums.
Thus, we obtain de Vylderm approximation of the ultimate ruin probability
ψ(u) ≈ ψDV (u) =
α + β
α
λ̃1
λ̃1 + λ̃
exp
{
λ̃β − λ̃1α
λ̃1 + λ̃
u
}
, (11)
RISK PROCESS WITH STOCHASTIC PREMIUMS 197
De Vylder approximation can be applied in the case when theoretical dis-
tributions of {xi} and {yi} are unknown, but there are experience data
about premiums (ŷ1, ..., ŷn1) and claim sizes (x̂1, ..., x̂n1) in time interval
[0, t]. Then, using sample moments instead of theoretical ones, put
λ̂1 =
n1
t
, λ̂ =
n
t
,
m̂k =
n∑
i=1
ŷi/n μ̂k =
n∑
i=1
x̂i/n
in the mentioned above system (10) and calculate corresponding parameters
α, β, λ̃1, λ̃. Formula (11) again provides de Vylder approximation ψDV (u).
3. Diffusion approximation
On the first glance diffusion approximation is similar to de Vylder ap-
proach, it can be considered as if origin risk process U(t) is replaced by
a Wiener process with a drift in such a way that first two moments coin-
cide. Then well known expressions for distribution of infimum of the Wiener
process on the finite/infinite interval provide approximations for ruin proba-
bilities.
Denote by Wã,σ̃2(t) the Wiener process with the mean ãt, variance - σ̃2t.
Random process Wã,σ̃2(t) is stochastically equivalent to ãt+ σ̃W (t), where
W (t) is a standard Wiener process.
According to diffusion approximation risk process U(t) = u + Q(t) =
u + Π(T ) − S(t) is replaced by process u+Wã,σ̃2(t) such that EQ(t) = ãt,
V arQ(t) = σ̃2t. Thus we have
ã = λ1m1 − λμ1, σ̃2 = λ1m2 + λμ2. (12)
Therefore diffusion approximation for ultimate ruin probability for model
(1) is given by
ψ(u) ≈ ψD(u) = P{inf
t>0
Wã,σ̃2(t) < −u} = exp(−2uã/σ̃2) (13)
and diffusion approximation for ruin probability on finite interval is deter-
mined by expression
ψ(u, T ) ≈ ψD(u, T ) = P{ inf
0<t≤T
Wã,σ̃2(t) < −u} =
= 1 − Φ(
ãT + u
σ̃
√
T
) + exp(−2uã/σ̃2)Φ(
ãT − u
σ̃
√
T
), (14)
198 NADIIA ZINCHENKO, ANDRII ANDRUSIV
where Φ(.) is d.f. of standard normal distribution, values of ã, σ̃ are pre-
sented in (12).
Unlike De Vylder approach diffusion approximation has a solid theo-
retical base in functional limit theorems (weak invariance principles) for
randomly stopped sums. A number of general results can be find, for ex-
ample, in P. Billingsley [1968, p.17], A. Gut [1989 ,ch.5], P. Embrechts et
al. [1997, p.2.5], W. Whitt [2002], D. Silvestrov [2004], V. Korolev et al.
[2007 , ch.7], applications to diffusion approximation for classical model -
in J.Grandell [1991]. Weak convergence of risk processes to α-stable Lévy
process was studied in [15, 16].
Following theorem from P.Embrechts et al. [1997, p.2.5] will serve as an
auxiliary result for further conclusions.
Theorem A1 (Invariance principle for randomly stopped sums).
Let {ξi} be a sequence of i.i.d.r.v., Eξ1 = M1, V arξ1 = σ2
ξ < ∞, partial
sum process S(n) =
∑n
i=1 ξi, S(0) = 0. Assume that the renewal counting
process N(t) = sup{n ≥ 1 :
∑n
i=1 ηi ≤ t}, t ≥ 0 and {ξi} are independent,
Eη1 = 1/λ, V arη1 = σ2
η <∞. Then as n→ ∞(
(σ2
ξ + (M1λση)
2)λn
)−1/2(
S(N(nt)) − λM1nt
)
⇒W (t), (15)
where W (t), t ≥ 0 is a standard Wiener process, ⇒ means weak convergence
in Skorokhod space D[0,∞) equipped with J1-metric.
Reminding that Poisson process is renewal counting process with {ηi}
exponentially distributed with parameter λ > 0, Eη1 = 1/λ, V arη1 = 1/λ2
we have
Corollary 2 (Functional CLT for Poisson sums). Let {ξi} and S(n)
be as in Theorem A1, N(t) - Poisson process with intensity λ > 0, S(N(t))
- compound Poisson process. Then as n→ ∞(
M2λn
)−1/2(
S(N(nt)) − λM1nt
)
⇒W (t), (16)
in Skorokhod J1-metric, M2 = Eξ2
1 = σ2
ξ +M2
1 .
Corollary 3 (Functional CLT for risk process with stochastic pre-
miums). Let Q(t) = Π(t) − S(t) be a risk process (1) with Exk1 = μk,
Eyk1 = mk, k = 1, 2, λ1 and λ be parameters of corresponding Poisson
processes N1(t) and N(t), then(
σ̃2n
)−1/2(
Q(nt) − (λ1m1 − λμ1)nt
)
⇒W (t), (17)
RISK PROCESS WITH STOCHASTIC PREMIUMS 199
where W (t), t ≥ 0 is a standard Wiener process, σ̃2 = λ1m2 + λμ2.
Proof obviously follows from Corollary 2 due to the fact that Q(t) is a
compound Poisson process, see (2), with intensity λ∗ = λ+λ1, whose jumps
have mean ã
λ∗ = λ1
λ∗m1 − λ
λ∗μ1 and second moment σ̃2
λ∗ = λ1
λ∗m2 + λ
λ∗μ2.
Corollary 3 once more underline the reasonability of proposed diffusion
approximation for model (1). Detail discussion of diffusion approximation
for classical Cramér-Lundberg model is presented by J.Grandell [19]; his
conclusions in whole can be extended on model (1) with stochastic premi-
ums.
Note that in the case, when claims are so heavy-tailed that Ex2
1 = ∞,
more precise {xi} belong to the domain of attraction of α-stable law, 0 <
α < 2, appropriate approximating process is α-stable Lévy process.
4. Strong invariance principle for risk process with
stochastic premiums
The other type of limit theorems is strong invariance principle (SIP),
which occurs to be a bridge between weak and a.s. convergence.
Strong invariance principle (almost sure approximation) is a class of limit
theorems that provide sufficient ( or necessary and sufficient) conditions
for the possibility to construct the i.i.d.r.v. {ξi, i ≥ 1} and Lévy process
{Y (t), t ≥ 0} on the same probability space in such a way that a.s.∣∣∣∣∣∣
[t]∑
i=1
ξi −mt− Y (t)
∣∣∣∣∣∣ = o(r(t)), (20)
where [a] is entire of a > 0, m = Eξ1, r(t) - non-random function - approxi-
mation error (error term), depending on additional assumptions posed on
{Xi, i ≥ 1}. Concrete assumptions on {ξi, i ≥ 1} clear up the type of
Y (t) and the form of r(.). Since we deal with i.i.d.r.v. it is natural to
consider Wiener process W (t) or α-stable Lévy process Yα(t), t ≥ 0, as an
approximation process Y (t) in (20).
While usual invariance principle deals with convergence of distributions
of functionals of Sn, SIP tells how “small” can be difference between sample
pathes of Sn and limiting process Y (t).
It is obvious that using (20) with appropriate error term one can easily
(almost without the proof) transfer the results about the asymptotic behav-
ior of Lévy process Y (t) or its increments on the rate of growth of partial
sums and corresponding increments.
In forthcoming we will use the concept of SIP in a wider sense, and say
that a random process ξ(t) admits the a.s. approximation by the random
200 NADIIA ZINCHENKO, ANDRII ANDRUSIV
process η(t) if ξ(t) (or stochastically equivalent process {ξ′
(t), t ≥ 0}) can be
constructed on the rich enough probability space together with η(t), t ≥ 0,
in such a way that a.s.
|ξ(t) − η(t)| = o(r1(t)) ∨O(r1(t)), (21)
where r1(.) is again a non-random function.
The investigations of SIP for partial sum process was originated by pi-
oneer works of A.Skorokhod (1961) ( Skorokhod embedding scheme) and
W.Strassen (1965, 1967) in the middle of 60-th; for detail bibliography see
M.Csörgő, P.Révész (1981) and more recent M.Csörgő, L. Horváth (1993),
M.Alex and J. Steinebach (1994), also survey by N.Zinchenko (2000).
Summarizing all known results for i.i.d.r.v. with finite variance we have
as in M.Csörgő, L. Horváth (1993)
Theorem A2. I.i.d.r.v. {ξi, i ≥ 1} with Eξ1 = M1 can be defined on the
same probability space together with standard Wiener process {W (t), t ≥ 0}
in such a way that a.s.
sup
0≤t≤T
|S(t) −M1t−W (t)| = o(r(T )), (22)
where
- r(T ) = T 1/p iff E |ξ1|p <∞, p > 2 ;
- r(T ) = (T ln lnT )1/2 iff E |ξ1|2 <∞ ;
- right hand side of (22) is O(lnT ) iff E exp(uX) <∞ for u ∈ (0, u0).
Now consider the case Eξ2
i = ∞ . More precise assume that {ξi, i ≥ 1}
are in domain of normal attraction of Gα,β ( notation {ξi} ∈ DNA(Gα,β) ).
This means weak convergence
S∗
n = n−1/α(S(n) − an) ⇒ Gα,β,
where
an =
⎧⎨⎩
nEξ1 if 1 < α < 2,
0 if 0 < α < 1,
(2/π)β lnn if α = 1.
It is well known that {ξi} ∈ DNA(Gα,β) iff for large x the tails of its d.f.F (x)
satisfy
1 − F (x) = c1x
−α + d1(x)x
−α, F (−x) = c2x
−α + d2(x)x
−α,
where c1 > 0 , c2 > 0 , d1(x) → 0 , d2(x) → 0 as x → ∞. Thus, for
X ∈ DNA(Gα,β), E |X|p <∞, ∀p < α, but E |X|p = ∞ for any p > α .
RISK PROCESS WITH STOCHASTIC PREMIUMS 201
It occurs that the fact {ξi} ∈ DNA(Gα,β) is not enough to obtain “good”
error term in (20), thus, certain additional assumptions are needed. We for-
mulate them in terms of ch.f. (see Zinchenko [31, 33, 34], Berkes et al. [4,5],
Mijnheer [26]).
Assumption (C) : there are a1 > 0, a2 > 0 and l > α such that for
|u| < a1
|f(u) − gα,β(u)| < a2|u|l
where f(u) = e−iuM1ϕ(u) is a ch.f. of (ξ1 − Eξ1) if 1 < α < 2 and
f(u) = ϕ(u), i.e. ch.f. of ξ1 if 0 < α ≤ 1.
Assumption (C) not only provides the weak convergence Sn ⇒ Gα,β,
but also determines the rate of convergence.
As in [33] we have
Theorem A3. Let M1 = Eξ1 for 1 < α < 2 and M1 = 0 for 0 < α ≤ 1.
Under assumption (C) it is possible to define α-stable process Yα,β(t), t ≥ 0,
such that a.s.
sup
0≤t≤T
|S(t) −M1t− Yα,β(t)| = o(T 1/α−ρ1), (23)
for all 0 < ρ1 < �0), �0 = (l − α)/80α.
In our work we shall focus on SIP for randomly stopped sums
D(t) = S(N(t)) =
N(t)∑
i=1
ξi, (24)
where N(t) is a renewal (counting) process
N(t) = inf{x > 0 : Z(x) > t}
associated with the sums of i.i.d.r.v. Z(n) =
n∑
i=1
ηi , 0 < Eη1 < 1/λ < ∞,
{Zi, i ≥ 1} are independent of {ξi, i ≥ 1}.
A number of results concerning SIP for N(t), D(t) and wider classes of
inverse processes and superposition of the processes can be find in M.Csörgő,
L. Horváth [8] for the case of a.s. approximation with Wiener process,
particularly, for {ξi} and {ηi} with finite variance. Case of heavy-tailed
summands attracted to α-stable law was studied by N.Zinchenko [35] , see
also N.Zinchenko and M.Safonova [36].
We will need following results:
202 NADIIA ZINCHENKO, ANDRII ANDRUSIV
Theorem A4 [8]. Denote by V arξ1 = σ2, V arη1 = τ 2, ν2 = λσ2+λ3M2
1 τ
2.
(i) Suppose that E|ξ1|p <∞, E|η1|p <∞, p > 2, then {ξi} and N(t) can
be constructed on the same probability space together with Wiener process
{W (t), t ≥ 0} in such a way that a.s.
sup
0≤t≤T
|D(t) − λM1t− νW (t)| = o(T 1/p), (25)
(ii) If E exp(uξ1) < ∞ and E exp(uη1) < ∞ for all u ∈ (0, uo) then
right side of (25) is O(lnT ).
Corollary (SIP for Poisson sums, summands with finite variance).
Let N(t) be Poisson process with intensity λ > 0 ( {ηi} are exponentially
distributed with parameter λ) then under condition E|ξ1|p <∞, p > 2, there
is a Wiener process {W (t), t ≥ 0} such that a.s.
sup
0≤t≤T
∣∣∣∣D(t) − λM1t−
√
λ(τ 2 +M2
1 )W (t)
∣∣∣∣ = o(T 1/p), (26)
and for {ξi} with light tails, i.e. with finite moment generating function
E exp(uξ1) <∞, u ∈ (0, uo), right side of (26) is O(lnT ).
As a next step assume that {ξi} are attracted to α-stable law Gα,β with
1 < α < 2, |β| ≤ 1 (condition α > 1 needed to have a finite mean). SIP for
randomly stopped sums in this case was studied in [35,36].
Theorem A5. Let {ξi, i ≥ 1} satisfy (C) with 1 < α < 2, β ∈ [−1, 1],
Eη2
1 < ∞. Then {ξi, i ≥ 1}, {ηi, i ≥ 1}, N(t) can be defined together with
α-stable process Yα(t) = Yα,β(t), t ≥ 0 so that a.s.
|S(N(t)) −M1λt− Yα,β(λt)| = o(t1/α−�2), ρ2 ∈ (0, ρ∗0), (27)
for some �∗0 = �∗0(α, l) > 0.
Corollary 4 (SIP for Poisson sums with summands attracted to
α-stable law). Theorem A5 holds if N(t) is a Poisson process with inten-
sity λ > 0.
We will use such general results to investigate the possibility of a.s. ap-
proximation of risk process Q(t) with stochastic premiums.
Corollary 5 (SIP for risk process with stochastic premiums, finite
variance case).
(I) If in model (1) both premiums {yi} and claims {xi} have moments
of order p > 2, then there is a Wiener process {Wã,σ̃2(t), t ≥ 0} with
RISK PROCESS WITH STOCHASTIC PREMIUMS 203
ã = λ1m1 − λμ1, σ̃
2 = λ1m2 + λμ2 such that a.s.
sup
0≤t≤T
|Q(t) −Wã,σ̃2(t)| = o(T 1/p). (26)
(II) If premiums {yi} and claims {xi} are light-tailed with finite moment
generating function in some positive neighborhood of zero, then a.s.
sup
0≤t≤T
|Q(t) −Wã,σ̃2(t)| = O(logT ), (27)
Proof immediately follows from Corollary 4 since Q(t) is a compound
Poisson process (see (2)) with intensity λ∗ = λ + λ1, whose jumps have
mean ã
λ∗ = λ1
λ∗m1 − λ
λ∗μ1, and second moment σ̃2
λ∗ = λ1
λ∗m2 + λ
λ∗μ2.
The other way to prove (26) or (27) is a.s. approximation of Π(t) and
S(t) separately by (λ1m2)
1/2W1(t) and (λμ2)
1/2W2(t) with corresponding
error terms, where W1(t) and W2(t) are independent standard Wiener pro-
cesses.
Remark. In model (1) it is natural to suppose that premiums have dis-
tributions with light tails or tails which are lighter than for claim sizes.
Therefore moment conditions, which determine the error term in SIP, are
in fact conditions on claim sizes.
Now consider the case when claims are so large that they have infinity
variance.
Corollary 6 (SIP for risk process with stochastic premiums and
large claims attracted to α-stable law). Suppose that claim sizes {xi}
satisfy (C) with 1 < α < 2, β ∈ [−1, 1], premiums {yi} are i.i.d.r.v. with
finite moments of order p > 2, then a.s.
|Q(t) − (λ1m1 − λμ1)t− λ1/αYα,−β(t)| = o(t1/α−�2), ρ2 ∈ (0, ρ∗0), (28)
for some �∗0 = �∗0(α, l) > 0.
Proof can be derived with the help of a.s. approximation of Π(t) by
(λ1m2)
1/2W1(t) with the error term o(t1/p) (Theorem A4) and −S(t) =
−∑n
i=1 xi – by independent of W1(t) α-stable process λ1/αYα,−β(t) with
error term o(t1/α−�2), 1 < α < 2 (Theorem A5) and application of LIL for
standard Wiener process. Note that Yα,−β(t) = −Yα,β(t).
The other way of the proof – application of Theorem A5 to process Q(t)
in form (2) taking in consideration the form of intensity and scale parameter.
204 NADIIA ZINCHENKO, ANDRII ANDRUSIV
5. The rate of growth of risk process with stochastic
premiums and magnitude of its increments
Now we shall apply the results of previous section about SIP to investi-
gation the rate of growth of risk process Q(t) as t→ ∞ and its increments
Q(t + at) − Q(t) on intervals whose length at grows but not faster than
t. Such question about the order of magnitude of increments for classical
risk process and renewal model was asked in Embrechts et al. [10], partial
answers on this question can be found in [7-9], [11-14], [24,25].
We will follow the approach developed in Zinchenko and Safonova [36]
for investigation the rate of growth of random sums. The key moments
are application of SIP for Q(t) with appropriate error term and known
results about asymptotic behavior of the Wiener and stable processes and
their increments, namely, various modifications of the LIL and Erdös-Rényi-
Csörgő-Révész type SLLN [7,8,11].
Theorem 6 ( LIL for risk process with stochastic premiums). If
in model (1) both premiums {yi} and claims {xi} have moments of order
p > 2, then
lim sup
t→∞
|Q(t) − ãt|√
2t ln ln t
= σ̃, where ã = λ1m1 − λμ1, σ̃2 = λ1m2 + λμ2. (29)
Proof follows from classical LIL for standard Wiener process and SIP
for Q(T ) with error term o(t1/p) (Corollary 5).
Notice that Theorem 6 covers not only the case of small claims, but also
the case of large claims with finite variance.
Next result deals with the case of large claims with infinite variance.
More precise, we shall consider the case when r.v. {xi, i ≥ 1} are attracted
to an asymmetric stable law Gα,1, but premiums have Eyp1 <∞.
Theorem 7. Let {xi, i ≥ 1} satisfy (C) with 1 < α < 2, β = 1 and
Eyp1 <∞, p > 2 . Then a.s.
lim sup
t→∞
Q(t) − (λ1m1 − λμ1)t
(λt)1/α(B−1 ln ln t)1/θ
= 1, (30)
where
B = B(α) = (α− 1)α−θ| cos(πα/2)|1/(α−1), θ = α/(α− 1). (31)
RISK PROCESS WITH STOCHASTIC PREMIUMS 205
Proof. Condition (C) provides normal attraction of {xi} to a stable
law with 1 < α < 2, β = 1 and, therefore, Q(t) − (λ1m1 − λμ1)t can be
a.s. approximated by the stable Lévy process λ1/αYα,−1(t) via Corollary
6. Stable process Yα,−1(t) has only negative jumps and obeys the following
modification of the LIL [17, 32]
lim sup
t→∞
Yα,−1(t)
t1/α(B−1 ln ln t)1/θ
= 1. (32)
Combining (32) and order of error term in SIP (28), we obtain (30).
Investigation of the asymptotic of increments of Q(t) we shall also carry
out step by step.
We shall start with the light-tailed case.
Theorem 8. Let claims {xi, i ≥ 1} and premiums {yi, i ≥ 1} be indepen-
dent sequences of i.i.d.r.v. with Ey1 = m1, Ey
2
1 = m2, Ex1 = μ1, Ex
2
1 = μ2,
and finite moment generating functions
E exp(rx1) <∞, E exp(ry1) <∞ as |r| < r0, r0 > 0. (33)
Assume that function aT , T ≥ 0, satisfies following conditions:
(i) 0 < aT < T ,
(ii) T/aT does not decrease in T .
Also let
aT/ lnT → ∞ as T → ∞. (34)
Then a.s.
lim sup
T→∞
|Q(T + aT ) −Q(T ) − aT (λ1m1 − λμ1)|
γ(T )
= σ̃, (35)
where
γ(T ) = {2aT (ln lnT + lnT/aT )}1/2, σ̃2 = λ1m2 + λμ2.
Theorem 9. Let {xi, i ≥ 1}, {yi, i ≥ 1} and aT satisfy all conditions of
Theorem 8 with following assumption used instead of (34)
Exp1 <∞, Eyp1 <∞, p > 2.
Then (35) is true if aT > c1T
2/p/ lnT for some c1 > 0.
206 NADIIA ZINCHENKO, ANDRII ANDRUSIV
On the second step we assume that i.i.d.r.v. {xi, i ≥ 1} are attracted to
an asymmetric α-stable law. Denote by
dT = a
1/α
T {B−1(ln lnT + lnT/aT )}1/θ,
where
B1 = B(α) = (1 − α)α−θ |cos(πα/2)|1/(α−1) , θ = α/(α− 1).
Theorem 10. Suppose that {xi, i ≥ 1} satisfy (C) with 1 < α < 2, β = 1
and Eyp1 <∞, p > 2, Ex1 = μ1, Ey1 = m1. Function aT is non-decreasing,
0 < aT < T , T/aT is also non-decreasing and provides dT
−1T 1/α−�2 → 0 as
T → ∞ for certain �2 > 0 determined by error term in SIP. Then a.s.
lim sup
T→∞
Q(T + aT ) −Q(T ) − (λ1m1 − λμ1)aT
dT
= λ1/α. (36)
Proof. Az in theorem 7, Q(t)− (λ1m1 −λμ1)t can be a.s. approximated
by asymmetric stable Lévy process λ1/αYα,−1(t) via Corollary 6. Thus, in
the proof we use Erdös-Rényi-Csörgő-Révész type limit theorem for α-stable
process Yα,−1 ( see [32] )
lim sup
T→∞
d−1
T (Yα,−1(T + aT ) − Yα,−1(T )) = 1
and error term in strong invariance principle for Q(T ) from Corollary 6.
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: znm@univ.kiev.ua
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