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Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
On ruin probabilities for risk modelswith ordered claim arrivals
Claude Lefèvre
Université Libre de Bruxelles
7th Conference in Actuarial Science and Finance, Samos
Joint work with Philippe Picard, Université de Lyon
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 1/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
1 Ordered claim arrival processHomogeneous point processTime scale transformation
2 Ordered aggregate claim processi.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
3 Ordered insurance risk modelFinite-time ruin probability
4 Two particular risk modelsWhen Ft is linearWhen Ft is exponentialOver an infinite horizon
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 2/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
From the papers
Lefèvre,C. and Picard, P. (2011). Insurance : Mathematics andEconomics 49, 512-519.
Lefèvre,C. and Picard, P. (2012). Working paper.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 3/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
1. ORDERED CLAIM ARRIVAL PROCESS
De Vylder and Goovaerts (1999), (2000) proposed a claim arrivalprocess on a time interval [0, t], named homogeneous. We revisitthis model and propose a simple extension useful in insurance.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 4/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
HOMOGENEOUS POINT PROCESS
1) A point process on an interval [0, t] is constructed as follows.
Let N(t) be the total number of claims in [0, t]. The distribution ofN(t) may be chosen arbitrarily. If N(t) = n, the successive claiminstants form a sequence T1 < . . . < Tn of continuous r.v.
The associated densities πt,n(t1, . . . , tn) are defined by
πt,n(t1, . . . , tn)dt1 . . . dtn =
P[N(t) = n, T1 ∈ (t1, t1 + dt1), . . . ,Tn ∈ (tn, tn + dtn)],
on the set Dt,n = {0 < t1 < . . . < tn ≤ t}.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 5/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
Attention is then focused on the special case of a homogeneouspoint process on [0, t], in which all the densities πt,n are constant(i.e. independent of t1, . . . , tn).
Since the Lebesgue measure of Dt,n is equal to tn/n!,
πt,n = P[N(t) = n]n!
tn , n ≥ 0.
A point process on [0, t] defines a point process on anysubinterval [0, s] ⊆ [0, t]. In the homogeneous case, the restrictedprocess is again a homogeneous process. The associated densitiesπs,k have to satisfy the compatibility conditions
πs,k =∞∑
n=k
πt,n(t − s)n−k
(n − k)!, k ≥ 0.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 6/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
2) Consider the process Nt(s), 0 ≤ s ≤ t that counts thenumber of claims during (0, t). From above, one sees that
[Nt(s)|N(t) = n] =d Bin(n,
st
).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 7/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
Alternatively, the constant densities assumption implies that theconditional claim arrival times
[T1, . . . ,Tn|N(t) = n]
are distributed as the order statistics of a sample of n independent(0, t)-uniform random variables :
[U1:n(0, t), . . . ,Un:n(0, t)].
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 8/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
Examples
(a) Suppose that N(t) has a mixed Poisson law with mixing rv Xand parameter t. Then, N(s) has a mixed Poisson law with mixingrv X and parameter s.
(b) Suppose that N(t) has a negative binomial law with parameterst and p (P[N(t) = n] =
(t+n−1n
)(1− p)tpn). Then, N(s) has a
negative binomial law with parameters t andpt(s) ≡ ps/[(1− p)t + ps].
(c) Suppose that N(t) has a binomial law with parameters t and p.Then, N(s) has a binomial law with parameters t and ps/t.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 9/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
TIME SCALE TRANSFORMATION
An extended model is obtained just by making a transformationof the time axis.
Let Ft(s), 0 ≤ s <≤ t, be a nondecreasing continuous functionfrom Ft(0) = 0 to Ft(t) = 1.
The counting claim arrival process Nt(s), 0 ≤ s ≤ t is definedconditionally by
[Nt(s)|N(t) = n] =d Bin(n,Ft(s)).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 10/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
An equivalent representation is that the conditional claim arrivaltimes
[T1, . . . ,Tn|N(t) = n]
are distributed as the order statistics
[V1:n(0, t), . . . ,Vn:n(0, t)].
of a sample of n i.i.d. random variables on (0, t) with Ff (.) asdistribution function.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 11/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
Examples
(a) Suppose that N(t) has a mixed Poisson law with mixing rv Xand parameter t. Then, Nt(s) has a mixed Poisson law with mixingrv X and parameter tFt(s).
(b) Suppose that N(t) has a negative binomial law with parameterst and p. Then, Nt(s) has a negative binomial law with parameterst and pFt(s)/[1− p + pFt(s)].
(c) Suppose that N(t) has a binomial law with parameters t and p.Then, Nt(s) has a binomial law with parameters t and pFt(s).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 12/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
On [0, t∗] or R+
Point processes with such an order statistic structure on [0, t∗] orR+ have been studied and characterized (see e.g. Puri (1992)).
Here are three particular of special interest.
(a) Suppose that {N(t)} is a Poisson process with rate λ. Then,N(t) has a Poisson distribution with rate λt, and
Ft(s) = s/t.
This remains true for a mixed Poisson process with random rate Λ.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 13/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
(b) Suppose that {N(t)} is a linear birth process withimmigration with birth rate b and immigration rate a. Then, N(t)has a negative binomial distribution with parameters a/b and e−bt ,and
Ft(s) = (ebs − 1)/(ebt − 1).
In this model, a claim arrival increases the likelihood of furtherclaim arrivals (this translates an effect of contagion or clustering)
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 14/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Homogeneous point processTime scale transformation
(c) Suppose that {N(t)} is a linear death-counting process withdeath rate b and initial size m. Then, N(t) has a binomialdistribution with parameters m and 1− e−bt , and
Ft(s) = (1− e−bs)/(1− e−bt).
This arises, for instance, with a group life insurance covering aclosed group of m individuals.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 15/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
2. ORDERED AGGREGATE CLAIM PROCESS
Suppose that each j-th claim is of random severity Xjindependent of the claim arrival times. Consider a time interval(0, t). For s < t, denote by St(s) the aggregate claim amount untiltime s, i.e.
[St(s)|N(t) = n] =n∑
j=1
Xj I [Vj :n(0, t) < s]
=
[Nt(s)|N(t)=n]∑j=1
Xj .
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 16/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Our aim is to illustrate how the variability of N(t) can influencethe dispersion and the dependence structures of the processes
Nt(s), and St(s).
A similar question was discussed by Balakrishnan andKozubowski (2008) for a so-called weighted Poisson process.
We first suppose that the Xj ’s are i.i.d. Then, two cases withdependence are considered.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 17/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
I.I.D. CLAIM AMOUNTS
Suppose that the Xj ’s are i.i.d. (distributed as X ). As inBalakrishnan and Kozubowski (2008), we measure the relativedispersion of a r.v. Y trough its index of dispersionV (Y ) ≡ Var(Y )/E (Y ). This parameter takes the value 1 for aPoisson variable, so that a random variable Y is said to be over(under)-dispersed when V (Y ) > 1 (< 1).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 18/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Result 1
By usual conditional arguments, one gets
V [St(s)] = E (X ) + V (X ) + E (X )Ft(s){V [N(t)]− 1}.
Thus, if N(t) is over (under)-dispersed, then V [St(s)], the relativedispersion of St(s), increases (decreases) with time s. In particular,over (under)-dispersion of N(t) implies the same property for Nt(s).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 19/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Similarly, for 0 ≤ s1 ≤ s2,≤ t,
Cov [St(s1), St(s2)− St(s1)]
E [St(s1)]= E (X )[Ft(s2)−Ft(s1)]{V [N(t)]−1},
showing that two consecutive increments are positively (negatively)correlated if N(t) is over (under)-dispersed. Also,
Cov [St(s1), St(s2)]
E [St(s1)]= E (X ) + V (X ) + E (X )Ft(s2){V [N(t)]− 1}.
Thus, in case of over (under)-dispersion of N(t), the covarianceincreases (decreases) with the horizon s2 (for fixed s1).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 20/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Examples. These conclusions can be applied to the cases where
N(t) has a Poisson distribution, for which V [N(t)] = 1,
N(t) has a negative binomial distribution, for which V [N(t)] > 1,
N(t) has a binomial distribution, for which V [N(t)] < 1.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 21/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Result 2
An alternative approach to see the effect of the variability ofN(t) is by means of the convex order ≤cx .
Consider another similar aggregate claim process St(s), s < t,defined with the same claim amounts Xj but a total number ofclaims N(t) instead of N(t).
Then, one can show that
N(t) ≤cx N(t) implies
Nt(s) ≤cx Nt(s), St(s) ≤cx St(s).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 22/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Examples. It is known that N(t), a Poisson variable with parameterλt, ≤cx N(t), a negative binomial variable with parameters t andλ/(λ+ 1). Moreover, for λ < 1, N(t) ≥cx N∗(t), a binomialvariable with parameters t and λ.
Thus, among these three cases, Var [St(s)] will be the largest (resp.smallest) when N(t) has a negative binomial (resp. binomial)distribution.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 23/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
EXCHANGEABLE CLAIM AMOUNTS
Suppose that the Xj ’s are exchangeable ... The results are lesssimple ...
From the expression of Cov [St(s1), St(s2)], one sees that thiscovariance increases with s2 if
1 +Cov(X ,Y )
[E (X )]2>
E [N(t)]
E [N(t)] + V [N(t)]− 1.
So, when N(t) is overdispersed, this condition is satisfied ifCov(X ,Y ) is positive or, at least, not too much negative. WhenN(t) is underdispersed, Cov(X ,Y ) has to be sufficiently positive.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 24/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
EQUALIZED CLAIM AMOUNTS
De Vylder and Goovaerts (2000) proposed that, given N(t) = n,the n severities X1, . . . ,Xn are replaced by an equal amount thatcorresponds to their average Sn = Sn/n. This "equalized" modelprovides, in some sense, a smooth approximation to the originalone.
Let Set (s), s < t, be the associated aggregate claim process.
Here too, ... Result 1 ...
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 25/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts
Result 2
The equalized model is expected to be less variable than theinitial one. In fact, both models can be compared in the convexsense, namely
Set (s) ≤cx St(s).
This follows from the well-known property that for exchangeabler.v., the sequence of successive averages Sn is decreasing in theconvex sense.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 26/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
3. ORDERED INSURANCE RISK MODEL
Consider a risk model where
(i) the aggregate claim amounts are represented by the previousordered process St(s), for 0 < s < t ;
(ii) the claim amounts Xj are strictly positive random variables ;they are allowed to be discrete or continuous, independent orcorrelated, but are independent of the claim arrival process.Working with the partial sums Sj , let F (y1, . . . , yj) be the jointdistribution function of (S1, . . . , Sj) ;
(iii) the cumulated premiums, including the initial reserves, arerepresented by a function h(t), non-negative and non-decreasing,continuous or not, with h(0) = u and h(t)→∞ as t →∞.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 27/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
FINITE-TIME RUIN PROBABILITY
The surplus of the company at time s is given by
Ut(s) = h(s)− St(s), 0 < s < t.
The problem under concern is the evaluation of the non-ruinprobability during a time interval [0, t].
Ruin is said to occur at time T (> 0) as soon as the reservesbecome negative or null. So, the probability of non-ruin until time tis given by
φ(t) = P(T > t) = P[St(s) < h(s), for 0 < s < t].
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 28/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
Our central result provides a simple and compact formula forφ(t), or φ(t), in terms of a special family of Appell polynomials.Such a family is associated to some real sequence U = {ui , i ≥ 0} ;it is denoted
{An(t|U), n ≥ 0},
where An(t|U) is a polynomial of degree n in t, with A0 = 1.
Appell polynomials are well-known in mathematics.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 29/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
Their key property is that
A′n = An−1, with An(un−1) = 0, n ≥ 1.
They can be expressed under an integral form. This allows one tosee directly that
P[U1:n(0, 1) ≥ u1, . . . ,Un:n(0, 1) ≥ un, and Un:n(0, 1) ≤ x ]
= n! An(x |u1, . . . , un)
For their numerical evaluation, it is convenient to use the following(Taylor) expansion :
An(t|U) =n∑
j=0
An−j(0|U) t j/j!,
where the coefficients An−j(0|U) are obtained from the recursion
An(0|U) = −n∑
j=1
An−j(0|U) ujn−1/j!, n ≥ 1,
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 30/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
In ruin theory, Appell polynomials were first exploited in Picardand Lefèvre (1997). An extension of these polynomials in terms ofpseudopolynomials (Picard and Lefèvre (1996), (2003)) was alsoused. Various applications are in e.g. Picard, Lefèvre and Coulibaly(2003) and Lefèvre and Loisel (2009). Appell polynomials are alsoused in e.g. Ignatov and Kaishev (2000), Ignatov and Kaishev(2004), Kaishev and Dimitrova (2006). They are underlying in DeVylder and Goovaerts (2000).
For a different approach based on Laplace transforms, see e.g.Perry, Stadje and Zacks (1999), (2003) and Zacks (2005).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 31/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
Given any level y for the aggregate claim amount, define theinstant r(y) when the premium income becomes large enough toavoid ruin, i.e.
r(y) = inf{s : h(s) > y}, y ≥ 0.
In particular, if the premium rate is a constant c and the initialreserves are equal to u, then r(y) = 0 if y < u, otherwiseu + cr(y) = y , so that
r(y) = (y − u)+/c , y ≥ 0.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 32/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
Central result
For t ≥ 0, φ(t) is given by
E{N(t)! AN(t)
[1|Ft(r(S1)), . . . ,Ft(r(SN(t)))
]I[SN(t) < h(t)
]}.
This result generalizes formulas given in e.g. Picard and Lefèvre(1997), Ignatov and Kaishev (2000), De Vylder and Goovaerts(2000), Lefèvre and Loisel (2009).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 33/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
Proof.
(1)
φ(t) =∞∑
n=0
P[N(t) = n] P[T > t|N(t) = n]
(2)
P[(1)] =
∫Dh(t),n
P[T > t|N(t) = n, S1 = y1, . . . , Sn = yn] dF (y1, . . . , yn)
(3) Let T1, . . . ,Tn denote the n consecutive claim occurrencetimes :
P[(2)] = P[T1 ≥ r(y1), . . . ,Tn ≥ r(yn)|N(t) = n].
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 34/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
Finite-time ruin probability
(4) (T1, . . . ,Tn) is distributed, conditionally on [N(t) = n], as[F−1t (U1:n(0, 1)), . . . ,F−1t (Un:n(0, 1))] :
P[(3)] = P[U1:n(0, 1) ≥ Ft(r(y1)), . . . ,Un:n(0, 1) ≥ Ft(r(yn))].
(5)
P[U1:n(0, 1) ≥ u1, . . . ,Un:n(0, 1) ≥ un, and Un:n(0, 1) ≤ u]
= n! An(u|u1, . . . , un)
Thus,P[(4)] = n! An[1|Ft(r(y1)), . . . ,Ft(r(yn))].
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 35/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
TWO PARTICULAR RISK MODELS
In this part, it is assumed that
1) the premium rate is a constant c ,
2) and the d.f. Ft(s), 0 ≤ s ≤ t, is
(a) either of a linear form
st,
(b) or of an exponential form
ebs − 1ebt − 1
, b real.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 36/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
WHEN Ft IS OF LINEAR FORM
As for a (mixed) Poisson process.
Lemma
Let Sj = X1 + . . .+ Xj , j ≥ 1, where the Xj ’s are nonnegativeexchangeable random variables. Then,
E [An(x |S1, . . . , Sn) | Sn] = (x − Sn)xn−1
n!.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 37/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
From the general formula, we know that P(T ≥ t|N(t) = n),denoted by φ(t|n), is given by
φ(t|n) = n! E{An[1|(S1 − u)+/ct, . . . , (Sn − u)+/ct
]I [Sn < u + ct]
}.
Using the lemma and some properties of Appell polynomials, weobtain two different expressions for these probabilities.
Notation : let G be the d.f. of the Xj ’s (discrete or continuous),and denote
mk(y |t) =
∫[0,u+ct−y)
(1− z
u + ct − y
)G ∗k(dz),
bl (y |t, n) =
(nl
)(y − u
ct
)l (1− y − u
ct
)n−l
.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 38/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Proposition. For t > 0 and n ≥ 1,
φ(t|n) =∑
k+l=n
∫[0,u]
bl (y |t, n) mk(y |t) G ∗l (dy).
An alternative formula is
φ(t|n) = P(Sn < u+ct)−∑
k+l=n, l≥1
∫(u,u+ct)
bl (y |t, n) mk(y |t) G ∗l (dy).
Remark. If u = 0, this gives the well-known ballot type formula
φ(t|n) = E (1− Sn/ct)+.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 39/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Special cases
Consider the compound Poisson model with parameter λ andmeasure Q. Define for t > 0 and any Borel set A,
µt(A) ≡ P[S(t) ∈ A] =∞∑
n=0
e−λt (λt)n
n!Q∗n(A).
Observe that µt(A) may be considered as an analytic function forall reals t, and not only for t ≥ 0. Of course, µt(A) looses itsprobabilistic interpretation when t < 0. Nevertheless, one easilychecks that even when t < 0, the well-known Panjer algorithmallows us to determine the pseudo-distribution µt .
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 40/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Two compact formulas for φ(t) for the compound Poisson model :
Corollary. For t > 0,
φ(t) =
∫[0,u]
µ y−uc
(dy)
∫[0,u+ct−y)
(1− z
u + ct − y
)µt+ u−y
c(dz).
An alternative formula is
φ(t) = µt([0, u + ct))−∫(u,u+ct)
µ y−uc
(dy)
∫[0,u+ct−y)
(1− z
u + ct − y
)µt+ u−y
c(dz).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 41/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
The second formula is of Seal type and is traditionally presented inthe literature.
The first formula is much less standard. By comparison, it isespecially convenient since y is restricted to [0, u] (independently oft). Note that the time index in µ(y−u)/c is nonpositive, which isallowed as explained above.Different variants of this formula, under additional assumptions,were derived in Picard and Lefèvre (1997), De Vylder (1999), DeVylder and Goovaerts (1999), Lefèvre and Loisel (2009).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 42/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Consider the equalized risk model.
Corollary. For t > 0,
φ(t|n) =
∫[0,u+ct)
{ ∑k+l=n
(nl
)[−((l/n)y − u)−
ct]l [1− (l/n)y − u
ct]k−1
}
(1− y − uct
) G ∗n(dy).
An alternative formula is
φ(t|n) = P(Sn < u + ct) −∫(u,u+ct)
∑k+l=n, l≥1
(nl
)[((l/n)y − u)+
ct]l [1− (l/n)y − u
ct]k−1
(1− y − u
ct) G ∗n(dy).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 43/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Remark. If u = 0, φ(t|n) has the same value as in the initial i.i.d.model (ballot formula).
De Vylder and Goovaerts (2000) conjectured that when u > 0, thenon-ruin probability is larger in the equalized version ... this seemsplausible and is indeed observed through many numericalexperiments ... we are trying to prove it.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 44/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
WHEN Ft IS OF EXPONENTIAL FORM
As for a linear birth process with immigrationor a linear death-counting process.
Lemma
Let Πj = X1 . . .Xj , j ≥ 1, where the Xj ’s are nonnegative i.i.d.random variables (distributed as X ). For any reals α, x ,
E [(Πn)α An(x |Π1, . . . ,Πn)] = An[xE (Xα)|E (Xα+1), . . . ,E (Xα+n)],
provided E (Xα+i ), i ≥ 0, exist.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 45/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
In this exponential case, the general formula becomes
φ(t|n) = n! E{An
[1|(eβSi − eβu)+/(eβ(u+ct) − eβu), 1 ≤ i ≤ n
]I [Sn < u + ct]}.
Introducing some notation/definitions :
bl (y |t, n) = . . . ,
mk(y |t) = . . . ,
we get two formulas for conditional probabilities of non-ruin thathave the same structure as in the linear case :
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 46/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Proposition. For t > 0 and n ≥ 1,
φ(t|n) =∑
k+l=n
∫[0,u]
bl (y |t, n) mk(y |t) G ∗l (dy).
An alternative formula is
φ(t|n) = P(Sn < u+ct)−∑
k+l=n, l≥1
∫(u,u+ct)
bl (y |t, n) mk(y |t) G ∗l (dy).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 47/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
OVER AN INFINITE HORIZON
(a) N(t) is a Poisson process
Proposition
limt→∞
φ(t) = (1− λµ
c)+
∫[0,u]
µ y−uc
(dy),
An alternative formula is
limt→∞
φ(t) = I (c > λµ)− (1− λµ
c)+
∫(u,∞)
µ y−uc
(dy).
Here too, the first formula is non-standard, with a negative indexin µ(y−u)/c . For some special cases, see e.g. Gerber (1988), Shiu(1988) and Picard and Lefèvre (1997).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 48/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
(b) N(t) is a linear birth process with immigration
Proposition. Ruin is a.s.
This was expected : the premium income is linear while N(t) growsat an exponential rate as t →∞ (e.g. Resnick (1992)).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 49/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
(c) N(t) is a linear death-counting process
Proposition. limt→∞ φ(t) is given by
= (−1)mm∑
l=0
m!
l !Bm−l (∞)
∫[0,u]
e−mβ(u−y)[1− eβ(u−y)]lG ∗l (dy),
where Bn(∞) satisfy the recursion :
Bn(∞) = −n−1∑k=0
bn−kn
(n − k)!Bk(∞), n ≥ 1.
with B0 = 1 and bn = E (enβXj ).
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 50/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
An alternative formula is
= 1−m∑
l=1
(−1)m−l m!
l !Bm−l (∞)
∫(u,∞)
eβ(m−l)(y−u)[1−eβ(y−u)]lG ∗l (dy).
EXTENSIONS AND VARIANTS ... for a next time ...
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 51/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
From the papers
Lefèvre,C. and Picard, P. (2011). Insurance : Mathematics andEconomics 49, 512-519.
Lefèvre,C. and Picard, P. (2012). Working paper.
A few references -partial list-
Balakrishnan, N. and Kozubowski, T.J. (2008). Statistics andProbability Letters 78, 2346-2352.
De Vylder, F.E. and Goovaerts, M.J. (1999). Insurance :Mathematics and Economics 24, 249-271.
De Vylder, F.E. and Goovaerts, M.J. (2000). Insurance :Mathematics and Economics 26, 223-238.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 52/ 53
Ordered claim arrival processOrdered aggregate claim process
Ordered insurance risk modelTwo particular risk models
When Ft is linearWhen Ft is exponentialOver an infinite horizon
Ignatov, Z.G. and Kaishev, V.K. (2004). Journal of AppliedProbability 41, 570-578.
Kaishev, V.K. and Dimitrova, D.S. (2006). Insurance :Mathematics and Economics 39, 376-389.
Lefèvre, C. and Loisel, S. (2009). Methodology and Computing inApplied Probability 11, 425-441.
Picard, P. and Lefèvre, C. (1997). Scandinavian Actuarial Journal1, 58-69.
Picard, P. and Lefèvre, C. (2003). Stochastic Processes and theirApplications 104, 217-242.
Puri, P.S. (1982). Journal of Applied Probability 19, 39-51.
C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 53/ 53