Directory UMM :Data Elmu:jurnal:I:Insurance Mathematics And Economics:Vol26.Issue2-3.2000:

Time stochastic

s

-convexity of claim processes

Michel Denuit

1

Université Libre de Bruxelles, Bruxelles, Belgium

Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999

Abstract

The purpose of this paper is to study the conditions on a stochastic process under which thes-convex ordering and the s-increasing convex stochastic ordering between two random instants is transformed into a stochastic ordering of the same type between the states occupied by this process at these moments. In this respect, the present work develops a previous study by Shaked and Wong (1995) [Probability in the Engineering and Informational Sciences 9, 563–580]. As an illustration, we show that the binomial and the Poisson processes, commonly used in actuarial sciences to model the occurrence of insured claims, possess this remarkable property. © 2000 Elsevier Science B.V. All rights reserved.

Keywords:s-Convex stochastic orderings;s-Increasing convex stochastic orderings; Stochastics-convexity; Stochastics-increasing convexity; Claim processes

1. Introduction

Recently, Lefèvre and Utev (1996), Denuit and Lefèvre (1997), Denuit et al. (1999a,b,c,d) introduced and stud-ied broad classes of stochastic order relations among real-valued random variables; they call them thes-convex,

s-increasing convex,s-concave ands-increasing concave orderings. An original feature of these relations lies in the fact that they take into account the particular structure of the respective supports of the random variables to be compared. This leads so to stronger comparison results than those obtained by considering all the random variables as valued in the real line. Without going into details, we mention that these stochastic orderings can be seen as particular cases of the general Tchebycheff-type order relations introduced by Denuit et al. (1999d).

The problem investigated in this paper is as follows. Consider a stochastic processX = {Xt, t ∈R+}describing for instance the total amount of claims affecting an insurance company during [0, t], or the number of insured claims occurring in [0, t], for instance. LetT1andT2be two random instants and letbe some stochastic order relation.

We wonder whether you can maintain that

T1T2⇒XT1 XT2,

i.e. that an-ordering between two random instantsT₁andT₂is transformed into an-ordering between the states

XT1 andXT2occupied by the processX at these moments. We investigate here this problem by considering for the

E-mail address:denuit@stat.ucl.ac.be (M. Denuit)

1_{Present address: Institut de Statistique, Universit´e Catholique de Louvain, Lo Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium.}

0167-6687/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 ( 9 9 ) 0 0 0 4 9 - 9

relationthes-increasing convex ordering (also called the stop-loss order of degrees−1 in the actuarial literature) and thes-convex ordering. Applications of the results in risk theory are provided at the end of the article.

The paper is organized as follows. First, in Section 2, we present thes-convex and the s-increasing convex stochastic orderings. Then, in Section 3, we introduce the notion of time stochastics-convexity ands-increasing convexity for stochastic processes. We mention that a similar problem has been previously discussed by Shaked and Wong (1995), though these authors restrict their study to point processes and usual stochastic order relations, such as the stop-loss order, the convex order or the stochastic dominance, for example. In Section 4, we show that the processes commonly used in actuarial sciences to model, on the one hand the occurrence of insured claims, and on the other hand the total claim amount (namely the binomial and the Poisson processes, as well as their compound versions), possess this remarkable property. Finally, in Section 5, we show hows-convex order lower bounds on the aggregate claim amount affecting an insurance company before ruin occurs (in the classical risk model) can be obtained.

2. Orderings of convex type

Many stochastic order relationsS_∗ commonly used to compare two random variablesXandYvalued in a subset

Sof the real lineRare defined (or, at least, can be characterized) by reference to a classU∗Sof measurable functions

with domainS(usually a convex cone) as follows:Xis said to be smaller thanY in theS_∗-sense, which is denoted asXS_∗ Y, when

Eφ (X)≤Eφ (Y ) ∀φ∈U∗S, (2.1)

provided that the expectations exist. Whitt (1986) introduced the term ofintegral stochastic ordering generated by

US

∗ for

S

∗ defined through (2.1). Such relations have been extensively studied by Marshall (1991) and Müller (1997).

In decision making context, the functionφinvolved in the definition (2.1) typically represents a utility function. The preferences shared by all the decision-makers whose utility function satisfies certain reasonable conditions (i.e. belongs toUS

∗) constitute therefore a partial order

S

∗ of all risks.

Among standard integral stochastic orderings used in actuarial sciences, the stochastic dominance (usually denoted byst) is obtained whenU∗S =Ust, the class of the non-decreasing functions;sttranslates therefore the common preferences of all the actuaries thinking that more money is better (in the sense thatXst Y means the financial loss modeled by the random variableXis preferred over the one modeled byY all the profit-seeking insurers). The stop-loss order, also known in the statistical literature as the increasing convex order (usually denoted bysℓ or

icx) is obtained whenU∗S =Uicx the class of the non-decreasing and convex functions;sℓtranslates therefore the common preferences shared by all the risk-averse profit-seeking actuaries. Finally, the stop-loss order with equal means, better known in the statistical literature as the convex order (usually denoted bysℓ,=orcx) is obtained

whenU∗S=Ucx, the class of the convex functions. For more details about these standard stochastic order relations, the reader is referred, e.g., to Goovaerts et al. (1990) and Kaas et al. (1994). See also Denuit (1997) and Shaked and Shanthikumar (1994).

We are concerned here with thes-convex ands-increasing convex stochastic orderings. The latter are the integral stochastic orderings generated by the cones of thes-convex and of thes-increasing convex functions, respectively. More precisely, Popoviciu (1933) defined thes-convexity as follows: a real-valued functionφdefined onS⊆R_is said to bes-convex onSif and only if

[x0, x1, . . . , xs]φ≥0

for all choices ofs+1 distinct pointsx0, x1, . . . , xsinS, where the divided difference [. . .]φis defined recursively by

[x₀, x₁, . . . , xk]φ=[x1, x2, . . . , xk]φ−[x0, x1, . . . , xk−1]φ

xk−x0

starting from [xk]φ=φ (xk),k=0,1, . . . , s. In addition,φis said to bes-increasing convex when it is simultane-ouslyk-convex fork=1,2, . . . , s. We denote byUsS−cx(resp.U

S

s−icx) the class of thes-convex (resp.s-increasing convex) functions onS; obviously,

UsS−icx = ∩sk=1U

S

k−cx.

For more details about divided differences and convexity of higher degree, we refer the reader, e.g., to Roberts and Varberg (1973) (see also Chapters I and II in Denuit, 1997).

Now, given two random variablesXandY valued inS,Xis said to be smaller thanY in thes-convex (resp.

s-increasing convex) sense, denoted byXS_s−cx Y (resp.X S_s−icx Y) when (2.1) holds withU∗S =U

S

s−cx (resp.

U∗S=U

S

s−icx). We mention that the 1-increasing convex and the 1-convex orders reduce to the standard stochastic dominance, while the 2-increasing convex order is the stop-loss order and the 2-convex order is the convex order, i.e.

XR₁₋+_icxY ⇔X₁₋R+_cx Y ⇔Xst Y, X R+

2−icxY ⇔XsℓY, X R+

2−cx Y ⇔Xsℓ,=Y.

It is to be noted that, fors=1 and 2, the supportSof the risksXandY to be compared has no influence on the resulting ranking, i.e.

XS_s−(i)cxY ⇔XR_s−+_(i)cxY fors=1 and 2,and whateverS⊆R+is.

The above equivalence comes from the integral definition (2.1) of the stochastic orderingsS_s−cxandS_s−icxtogether with the fact that any non-decreasing or convex function onScan always be continued as a function with the same shape on the half-positive real lineR+. Quite surprisingly, as soon ass≥3, the influence ofSonS_s−cxandS_s−icx

is of primordial importance. Indeed, if instead of consideringXandY as valued inS, we see them as valued in a larger subsetV of the real line, it is easily seen that ifS ⊂Vthen

XS_s−cx Y ⇒XV_s−cx Y, XS_s−icx Y ⇒XV_s−icx Y, (2.2) but the reciprocal implication in (2.2) is not necessarily true. This particularity of thes-convex ands-increasing convex orderings has been extensively studied by Denuit et al. (1999c), where numerous counter examples are provided for the reciprocal implication in (2.2).

When the supportS of the random variables to be compared possesses some structure (when it is a continuum or an arithmetic grid, for example),S_s−cxandS_s−icx can be characterized through (2.1) with forU∗Smore usual

classes of functions. For instance, whenS =[a, b],bpossibly infinite, it suffices, in order to characterize[_sa,b_−cx], to consider forU∗[a,b]the class of the regulars-convex functions given by

Us[−a,bcx]∩Cs([a, b])=

φ: [a, b]→R|φs/≥0 on [a, b] , (2.3) whereCs_{([a, b])denotes the class of the functionsφ: [a, b]→Rpossessing a continuoussth derivativeφ}s/ _on [a, b]. In fact,U_s[a,b_−cx]∩Cs_{([a, b])is weakly dense inU}[a,b]

s−cx. In order to characterize

[a,b]

s−icx, it suffices to consider forU∗[a,b]the class of the regulars-increasing convex functions given by

U_s[₋a,b_icx] ∩Cs([a, b])=nφ: [a, b]→R|φk/≥0 on [a, b] fork=1,2, . . . , so. (2.4) On the other hand, whenS =Dn ≡ {0,1, . . . , n},n∈N0≡{1,2,3,. . .}, it can be shown (see, e.g., Denuit et al.,

1999c) that

UDn

s−cx =

φ:Dn→R|1sφ≥0 onDn−s , (2.5)

where1kdenotes the usualkth degree forward difference operator defined recursively by

with the convention that10φ≡φ. Moreover,

UDn

s−icx = n

φ:Dn→R|1kφ≥0 onDn−kfork=1,2, . . . , s o

. (2.6)

The orderingsDn

s−cxand

D_n

s−icxhave been studied in detail by Denuit and Lefèvre (1997) and Denuit et al. (1999b). To end with, let us mention that thes-convex and thes-increasing convex orderings are closely related. More precisely, given two risksXandY valued inS, the following equivalence holds:

XS_s−cx Y ⇔

_EXk _=EYk _{fork=1,2, . . . , s−1,}

XS_s−icx Y. (2.7)

Thes-convex ordering can thus be seen as a strengthening of thes-increasing convex ordering obtained by requiring in addition that the firsts−1 moments of the random variables to be compared are equal. We mention that the aforementioned strengthening has been previously studied in the actuarial literature.

The reader interested in a deep study of thes-convex ands-increasing convex orderings is referred to Denuit (1997).

3. Time stochasticsss-convexity

The idea underlying the time stochastics-convexity is very similar to the philosophy of the stochastics-convexity. The latter concept has been recently introduced and investigated in details by Denuit and Lefèvre (1998) and Denuit et al. (1999d). We begin with a brief description of it.

Let us consider a family of random variables{Xθ, θ ∈ 2}valued in a subset S ⊆ Rindexed by a single parameterθ ∈2⊆R. Now, given a functionφ:S→R, let us construct the new functionφ∗defined as

φ∗:2→R, θ7→φ∗(θ )≡Eφ (Xθ), (3.1) provided that the expectation exists. A natural question is to which extent some properties of the functionφcan be transmitted to the functionφ∗. Such a question is rather general and has already been discussed in probability and statistics, for instance by Shaked and Shanthikumar (1988); see also Chapter VI of the book of Shaked and Shanthikumar (1994). Denuit et al. (1999d) defined the stochastic s-convexity and the stochastics-increasing convexity as follows.

Definition 3.1. The family{Xθ, θ ∈ 2} of random variables valued inS and indexed byθ ∈ 2is said to be stochasticallys-convex when, given any functionφ ∈ US

s−cx, we have thatφ∗ ∈ Us2−cx, with φ∗ defined in (3.1). Similarly, the family{Xθ, θ ∈2}is said to be stochasticallys-increasing convex when, given any function

φ∈U_sS_−icx, we have thatφ∗∈U2 s−icx.

Most parametric families of probability distributions possess the stochastics-convexity ands-increasing convexity properties, as Denuit and Lefèvre (1998) and Denuit et al. (1999d) showed, so that this notion possesses many applications, namely in relation with mixture models and compound sums.

Now, letX = {Xt, t ∈ T}be a stochastic process with time spaceT ⊆ R+and state space S ⊆ R, say. Analogous to (3.1), let us define forφ:S→Rthe associated functionφ∗as

φ∗:T →R, t 7→φ∗(t )≡Eφ (Xt), (3.2) provided that the expectation exists. The time stochastics-convexity ands-increasing convexity can then be defined as follows.

Definition 3.2. A stochastic processX = {Xt, ∈T}with time spaceT and state spaceS is said to possess the time stochastics-convexity property when, given any functionφ∈US

s−cx, we have thatφ∗∈U

T

in (3.2). Similarly,Xis said to possess the time stochastics-increasing convexity property when, given any function

φ∈U_sS_−icx, we have thatφ∗∈U_sT_−icx.

Let us now examplify the practical use of this notion. LetT₁andT₂be two random variables valued inT and independent ofX;XT1 andXT2 are the states occupied by X at the instantsT1 andT2, respectively. We study

here conditions onX under which ans-convex ordering or ans-increasing convex ordering betweenT1andT2is

transformed into an ordering of the same type betweenXT1 andXT2, i.e. under which

T1Ts−cxT2⇒XT1

S

s−cxXT2, T1

T

s−icx T2⇒XT1

S

s−icx XT2. (3.3)

Let us examine thes-convex case, for instance. It is easily seen that

Eφ (XTi)=Eφ

∗_(T

i) fori=1,2.

From the integral definition (2.1) of thes-convex ordering, we then deduce that a sufficient condition for (3.3) to hold is that

φ∈U_sS_−cx ⇒φ∗∈U_sT_−cx, (3.4)

i.e. thatX possesses the time stochastics-convexity property. Indeed, when (3.4) is satisfied,T1Ts−cxT2implies that for anyφ∈U_sS_−cx,

Eφ (XT1)=Eφ

∗_(T

1)≤Eφ∗(T2)=Eφ (XT2),

so that

XT1

S

s−cxXT2

holds. Of course, the above reasoning is still valid with thes-increasing convex order substituted for thes-convex one.

The problem (3.3) was examined by Shaked and Wong (1995) with different classes of order relations. Moreover, in Remark 2.3, these authors considered the Rolski orderings and obtained results similar to some of those discussed in Section 4. Nevertheless, as mentioned in Denuit et al. (1998), thes-increasing convex ordering and the Rolski ordering are close but mathematically distinct.

4. Claim processes

4.1. Binomial and compound binomial processes

In the compound binomial risk process, time is measured in discrete time unitst ∈ N≡ {0,1,2, . . .}and the number of insured claims is governed by a binomial process{Nt, t ∈ N}with parameterp ∈]0,1[ (i.e. in any time period, there occurs 1 or 0 claim with probabilitiespand 1−p, respectively, and occurrences of claims in different time intervals are independent events). Instead of establishing directly that the binomial process owns the time stochastics-convexity ands-increasing convexity properties, we prove the next general result stating that any random walk possesses this remarkable property.

Proposition 4.1. Let{Yk, k∈N0}be a sequence of independent and identically distributed (i.i.d., in short) random

variables valued inR+_{and consider the random walk processX} = {Xt, t ∈N}defined by

X₀=0 a.s., Xt = t X

k=1

Yk, t∈N0.

Then, (i)Xpossesses the time stochastic s-convexity property and (ii)Xpossesses the time stochastic s-increasing

Proof. Part (i) directly follows from Property 4.6 in Denuit et al. (1999d). To get (ii), it suffices to apply the reasoning provided in Denuit et al. (1999d), (Property 4.6) successively fork=1 tos. Corollary 4.2. The binomial process owns the time stochastic s-convexity and the time stochastic s-increasing convexity properties.

Proof. It suffices to notice that if{Nt, t∈N}is a binomial process, then

N₀=0 a.s., Nt = t X

k=1

Yk, t ∈N0,

where theYk’s are i.i.d. Bernoulli random variables. The announced result then follows from Proposition 4.1. As a consequence, we have that when the occurrence of the insured claims is described by a binomial process

{Nt, t∈N},

T1Ns−cxT2⇒NT1

N

s−cx NT2, T1

N

s−icxT2⇒NT1

N

s−icx NT2. (4.1)

The aggregate claim process{St, t ∈N}is modeled by a compound binomial process of the form

St =0 as long asNt =0, St = Nt

X

k=1

Zk whenNt ≥1, t∈N, (4.2)

where{Nt, t ∈N}is a binomial process with parameterp∈]0,1[ and{Zk, k∈N0}is a sequence of non-negative

i.i.d. random variables, independent of the occurrence process{Nt, t ∈ N};Zk represents the amount of claim during the period [k−1, k[,k∈N₀.

Proposition 4.3. LetT =N_orR+.Let{Zk, k∈N0}be a sequence of non-negative iid random variables and let

{Nt, t∈T}be an integer-valued stochastic process possessing the time stochastic s-increasing convexity property.

Define the compound processX = {St, t ∈T}as in(4.2).Then,Xstill possesses the time stochastic s-increasing

convexity property.

Proof. This result is immediate from Property 4.8 in Denuit et al. (1999d). We mention that the latter result only holds with the time stochastics-increasing convexity property, and not with thes-convex one. The reason is the proof uses the fact that a composition of twos-increasing functions is itself an

s-increasing function, but this is no more valid fors-convex functions.

From Proposition 4.3, a compound process of the form (4.2) built with a counting process{Nt, t ∈T}possessing the times-increasing convexity property will itself own this interesting property. As a consequence, we have from Corollary 4.2 that when the aggregate claim is described by a compound binomial process of the form (4.2),

T1 N

s−icxT2⇒ST1

R+

s−icx ST2. (4.3)

4.2. Poisson and the compound Poisson processes

Another classical model for the aggregate claim is the compound Poisson process. In this case, the number of insured claims is governed by a Poisson process{Nt, t ∈R+}with intensity rateλ >0. Let us prove the following result.

Proposition 4.4. The Poisson process{Nt, t ∈R+}owns the time stochastic s-convexity property, as well as the

time stochastic s-increasing convexity property.

Proof. In order to prove the result, we need to show that the functionφ∗given by

φ∗:R+→R, t7→φ∗(t )=X

i∈N

φ (i)e−λt(λt ) i

iss-convex wheneverφ:N→R. By Leibniz formula, we get ds

dtsφ

∗_{(t )=λ}sX i∈N

φ (i) min(i,s) X j=0 s j

(−1)s−je−λt(λt ) i−j

(i−j )! =λ s s X j=0 s j

(−1)s−j +∞ X

i=j

φ (i)e−λt(λt ) i−j

(i−j )!

=λs

s X j=0 s j

(−1)s−j +∞ X

i=0

φ (i+j )e−λt(λt ) i

i! .

The well known Newton Binomial formula (see, e.g., Agarwal, 1992) ensures that

1sφ (i)=

s X j=0 s j

(−1)s−jφ (i+j ),

so that it is easily checked that ds

dtsφ

∗_{(t )=λ}sX i∈N

1sφ (i)e−λt(λt ) i

i! ≥0,

whenceφ∗∈UsR−+cx(see (2.3) together with (2.5)) follows. This achieves the first part of the proof. The second part follows similarly by applying the same reasoning successively fork=1 tos.

As a consequence, we have that when the occurrence of the claims is described by a Poisson process{Nt, t∈R+},

T1R +

s−cxT2⇒NT1

N

s−cx NT2, T1

R+

s−icxT2⇒NT1

N

s−icx NT2. (4.4)

Now, the aggregate claim process{St, t∈R+}is described by a compound Poisson process of the form

St =0 as long asNt =0, St = Nt

X

k=1

Zk whenNt ≥1, t∈R+, (4.5)

where{Nt, t ∈ R+}is a Poisson process with parameterλ > 0 and{Zk, k ∈ N0}a sequence of non-negative

i.i.d. random variables, independent of the occurrence process{Nt, t ∈R+};Zkrepresents the amount of thekth claim affecting the insurance company,k∈ N₀. By Proposition 4.3, we have that the compound Poisson process also owns the time stochastics-increasing convexity property. As a consequence, we have that when the aggregate claim is described by a compound Poisson process{St, t∈R+},

T1R +

s−icxT2⇒ST1

R+

s−icx ST2. (4.6)

5. sss-Convex order lower bounds on aggregate claims

Assume that{St, t∈R+}is a compound Poisson process of the form (4.5) modeling the total amount of claims affecting an insurance company. LetT represent the first time when{St, t ∈ R+}hits the linear upper barrier

u+ct,u≥0,c >0, modeling the premium income of the company;T can be interpreted as the time of ruin and

ST as the aggregate claims affecting the company during [0, T], i.e. before ruin occurs.

Nevertheless, the distribution ofT is often complicated, so that the exact distribution ofST can be difficult to obtain. Our purpose here is to show how to get lower bounds onST in thes-convex sense when a few moments of theZk’s are known.

It is well known that ifc < λEZ1, thenT <+∞a.s., i.e. ruin occurs with probability 1. This corresponds for

whole company has a positive non-ruin probability. When theZk’s are valued inN0andu=0, Picard and Lefèvre

(1998) have shown that

µ1=ET=

1

c̺0

(5.1) and

µ2=ET2=

2µ2₁

1−exp(−̺0)g′{exp(−̺0)}/c

, (5.2)

where̺0is the positive solution of the equationλ−c̺0=λEexp(−̺0Z1)andg(s)=λP+∞j=1sjP[Z1=j].

When the moments ofT,µk =ETksay, are known fork=1,2, . . . , s−1, it is possible to construct a random variableT_min(s) with the same moments asT and such that

T_min(s) R_s−+_cxT . (5.3)

It is not possible to boundT from above since the support ofT is the whole half-positive real line (the maxima in thes-convex sense always put a positive probability mass on the upper bound of the support). Explicit expressions of the bounds involved in (5.3) are available in Denuit et al. (1998,1999a). As an illustration, fors=1 and 2, using (5.1) and (5.2), we have thatT_min(2) =µ1a.s. and

T_min(3) =

0 with probability(µ₂−µ2₁)/µ₂, µ1+(µ2−µ2₁)/µ1 with probabilityµ2₁/µ2.

Now, from (4.6), we have that

S

T_min(s)

S

s−icx ST, (5.4)

so that (5.4) provides a lower bound for the total amount of claim affecting an insurance company before ruin occurs. In particular, from the integral definition (2.1) of thes-increasing convex ordering, (5.4) provides lower bounds on

Eφ0(ST)for anyφ0∈UsS−icx.

Acknowledgements

This research was done while the author, supported by the Académie Royale des Sciences, des lettres et des beaux-arts de Belgique, visited the Institut de Sciences Actuarielles of the University of Lausanne, Switzerland. The warm hospitality of Professor Hans Gerber and Gérard Pafumi is gratefully acknowledged.

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with the convention that10φ≡φ. Moreover,

UDn

s−icx =

n

φ:Dn→R|1kφ≥0 onDn−kfork=1,2, . . . , s

o

. (2.6)

The orderingsDn

s−cxand

D_n

s−icxhave been studied in detail by Denuit and Lefèvre (1997) and Denuit et al. (1999b).

To end with, let us mention that thes-convex and thes-increasing convex orderings are closely related. More precisely, given two risksXandY valued inS, the following equivalence holds:

XS_s−cx Y ⇔

_EXk _=EYk _{fork=1,2, . . . , s−1,}

XS_s−icx Y. (2.7)

Thes-convex ordering can thus be seen as a strengthening of thes-increasing convex ordering obtained by requiring in addition that the firsts−1 moments of the random variables to be compared are equal. We mention that the aforementioned strengthening has been previously studied in the actuarial literature.

The reader interested in a deep study of thes-convex ands-increasing convex orderings is referred to Denuit (1997).

3. Time stochasticsss-convexity

The idea underlying the time stochastics-convexity is very similar to the philosophy of the stochastics-convexity. The latter concept has been recently introduced and investigated in details by Denuit and Lefèvre (1998) and Denuit et al. (1999d). We begin with a brief description of it.

Let us consider a family of random variables{Xθ, θ ∈ 2}valued in a subset S ⊆ Rindexed by a single

parameterθ ∈2⊆R. Now, given a functionφ:S→R, let us construct the new functionφ∗defined as

φ∗:2→R, θ7→φ∗(θ )≡Eφ (Xθ), (3.1)

provided that the expectation exists. A natural question is to which extent some properties of the functionφcan be transmitted to the functionφ∗. Such a question is rather general and has already been discussed in probability and statistics, for instance by Shaked and Shanthikumar (1988); see also Chapter VI of the book of Shaked and Shanthikumar (1994). Denuit et al. (1999d) defined the stochastic s-convexity and the stochastics-increasing convexity as follows.

Definition 3.1. The family{Xθ, θ ∈ 2} of random variables valued inS and indexed byθ ∈ 2is said to

be stochasticallys-convex when, given any functionφ ∈ US

s−cx, we have thatφ∗ ∈ Us2−cx, with φ∗ defined in

(3.1). Similarly, the family{Xθ, θ ∈2}is said to be stochasticallys-increasing convex when, given any function

φ∈U_sS_−icx, we have thatφ∗∈U2 s−icx.

Most parametric families of probability distributions possess the stochastics-convexity ands-increasing convexity properties, as Denuit and Lefèvre (1998) and Denuit et al. (1999d) showed, so that this notion possesses many applications, namely in relation with mixture models and compound sums.

Now, letX = {Xt, t ∈ T}be a stochastic process with time spaceT ⊆ R+and state space S ⊆ R, say.

Analogous to (3.1), let us define forφ:S→Rthe associated functionφ∗as

φ∗:T →R, t 7→φ∗(t )≡Eφ (Xt), (3.2)

provided that the expectation exists. The time stochastics-convexity ands-increasing convexity can then be defined as follows.

Definition 3.2. A stochastic processX = {Xt, ∈T}with time spaceT and state spaceS is said to possess the

time stochastics-convexity property when, given any functionφ∈US

s−cx, we have thatφ∗∈U

T

in (3.2). Similarly,Xis said to possess the time stochastics-increasing convexity property when, given any function φ∈U_sS_−icx, we have thatφ∗∈U_sT_−icx.

Let us now examplify the practical use of this notion. LetT₁andT₂be two random variables valued inT and independent ofX;XT1 andXT2 are the states occupied by X at the instantsT1 andT2, respectively. We study here conditions onX under which ans-convex ordering or ans-increasing convex ordering betweenT1andT2is transformed into an ordering of the same type betweenXT1 andXT2, i.e. under which

T1Ts−cxT2⇒XT1

S

s−cxXT2, T1Ts−icx T2⇒XT1

S

s−icx XT2. (3.3)

Let us examine thes-convex case, for instance. It is easily seen that Eφ (XTi)=Eφ

∗_(T

i) fori=1,2.

From the integral definition (2.1) of thes-convex ordering, we then deduce that a sufficient condition for (3.3) to hold is that

φ∈U_sS_−cx ⇒φ∗∈U_sT_−cx, (3.4)

i.e. thatX possesses the time stochastics-convexity property. Indeed, when (3.4) is satisfied,T1Ts−cxT2implies

that for anyφ∈U_sS_−cx,

Eφ (XT1)=Eφ∗(T1)≤Eφ∗(T2)=Eφ (XT2),

so that

XT1 Ss−cxXT2

holds. Of course, the above reasoning is still valid with thes-increasing convex order substituted for thes-convex one.

The problem (3.3) was examined by Shaked and Wong (1995) with different classes of order relations. Moreover, in Remark 2.3, these authors considered the Rolski orderings and obtained results similar to some of those discussed in Section 4. Nevertheless, as mentioned in Denuit et al. (1998), thes-increasing convex ordering and the Rolski ordering are close but mathematically distinct.

4. Claim processes

4.1. Binomial and compound binomial processes

In the compound binomial risk process, time is measured in discrete time unitst ∈ N≡ {0,1,2, . . .}and the number of insured claims is governed by a binomial process{Nt, t ∈ N}with parameterp ∈]0,1[ (i.e. in any

time period, there occurs 1 or 0 claim with probabilitiespand 1−p, respectively, and occurrences of claims in different time intervals are independent events). Instead of establishing directly that the binomial process owns the time stochastics-convexity ands-increasing convexity properties, we prove the next general result stating that any random walk possesses this remarkable property.

Proposition 4.1. Let{Yk, k∈N0}be a sequence of independent and identically distributed (i.i.d., in short) random variables valued inR+_{and consider the random walk processX} = {Xt, t ∈N}defined by

X₀=0 a.s., Xt = t

X

k=1

Yk, t∈N0.

Then, (i)Xpossesses the time stochastic s-convexity property and (ii)Xpossesses the time stochastic s-increasing convexity property.

Proof. Part (i) directly follows from Property 4.6 in Denuit et al. (1999d). To get (ii), it suffices to apply the reasoning provided in Denuit et al. (1999d), (Property 4.6) successively fork=1 tos.

Corollary 4.2. The binomial process owns the time stochastic s-convexity and the time stochastic s-increasing convexity properties.

Proof. It suffices to notice that if{Nt, t∈N}is a binomial process, then

N₀=0 a.s., Nt = t

X

k=1

Yk, t ∈N0,

where theYk’s are i.i.d. Bernoulli random variables. The announced result then follows from Proposition 4.1.

As a consequence, we have that when the occurrence of the insured claims is described by a binomial process

{Nt, t∈N},

T1Ns−cxT2⇒NT1

N

s−cx NT2, T1Ns−icxT2⇒NT1

N

s−icx NT2. (4.1)

The aggregate claim process{St, t ∈N}is modeled by a compound binomial process of the form

St =0 as long asNt =0, St = Nt X

k=1

Zk whenNt ≥1, t∈N, (4.2)

where{Nt, t ∈N}is a binomial process with parameterp∈]0,1[ and{Zk, k∈N0}is a sequence of non-negative i.i.d. random variables, independent of the occurrence process{Nt, t ∈ N};Zk represents the amount of claim

during the period [k−1, k[,k∈N₀.

Proposition 4.3. LetT =N_orR+.Let{Zk, k∈N0}be a sequence of non-negative iid random variables and let

{Nt, t∈T}be an integer-valued stochastic process possessing the time stochastic s-increasing convexity property.

Define the compound processX = {St, t ∈T}as in(4.2).Then,Xstill possesses the time stochastic s-increasing

convexity property.

Proof. This result is immediate from Property 4.8 in Denuit et al. (1999d). We mention that the latter result only holds with the time stochastics-increasing convexity property, and not with thes-convex one. The reason is the proof uses the fact that a composition of twos-increasing functions is itself an s-increasing function, but this is no more valid fors-convex functions.

From Proposition 4.3, a compound process of the form (4.2) built with a counting process{Nt, t ∈T}possessing

the times-increasing convexity property will itself own this interesting property. As a consequence, we have from Corollary 4.2 that when the aggregate claim is described by a compound binomial process of the form (4.2),

T1

N

s−icxT2⇒ST1

R+

s−icx ST2. (4.3)

4.2. Poisson and the compound Poisson processes

Another classical model for the aggregate claim is the compound Poisson process. In this case, the number of insured claims is governed by a Poisson process{Nt, t ∈R+}with intensity rateλ >0. Let us prove the following

result.

Proposition 4.4. The Poisson process{Nt, t ∈R+}owns the time stochastic s-convexity property, as well as the

time stochastic s-increasing convexity property.

Proof. In order to prove the result, we need to show that the functionφ∗given by

φ∗:R+→R, t7→φ∗(t )=X i∈N

φ (i)e−λt(λt )

i

iss-convex wheneverφ:N→R. By Leibniz formula, we get

ds

dtsφ

∗_{(t )=λ}sX

i∈N

φ (i) min(i,s) X j=0 s j

(−1)s−je−λt(λt )

i−j

(i−j )! =λ

s s X j=0 s j

(−1)s−j +∞ X

i=j

φ (i)e−λt(λt )

i−j

(i−j )!

=λs

s X j=0 s j

(−1)s−j +∞ X

i=0

φ (i+j )e−λt(λt )

i

i! .

The well known Newton Binomial formula (see, e.g., Agarwal, 1992) ensures that 1sφ (i)=

s X j=0 s j

(−1)s−jφ (i+j ),

so that it is easily checked that ds

dtsφ

∗_{(t )=λ}sX

i∈N

1sφ (i)e−λt(λt )

i

i! ≥0,

whenceφ∗∈UsR−+cx(see (2.3) together with (2.5)) follows. This achieves the first part of the proof. The second part

follows similarly by applying the same reasoning successively fork=1 tos.

As a consequence, we have that when the occurrence of the claims is described by a Poisson process{Nt, t∈R+},

T1R +

s−cxT2⇒NT1

N

s−cx NT2, T1R +

s−icxT2⇒NT1

N

s−icx NT2. (4.4)

Now, the aggregate claim process{St, t∈R+}is described by a compound Poisson process of the form

St =0 as long asNt =0, St = Nt X

k=1

Zk whenNt ≥1, t∈R+, (4.5)

where{Nt, t ∈ R+}is a Poisson process with parameterλ > 0 and{Zk, k ∈ N0}a sequence of non-negative i.i.d. random variables, independent of the occurrence process{Nt, t ∈R+};Zkrepresents the amount of thekth

claim affecting the insurance company,k∈ N₀. By Proposition 4.3, we have that the compound Poisson process also owns the time stochastics-increasing convexity property. As a consequence, we have that when the aggregate claim is described by a compound Poisson process{St, t∈R+},

T1R +

s−icxT2⇒ST1

R+

s−icx ST2. (4.6)

5. sss-Convex order lower bounds on aggregate claims

Assume that{St, t∈R+}is a compound Poisson process of the form (4.5) modeling the total amount of claims

affecting an insurance company. LetT represent the first time when{St, t ∈ R+}hits the linear upper barrier

u+ct,u≥0,c >0, modeling the premium income of the company;T can be interpreted as the time of ruin and ST as the aggregate claims affecting the company during [0, T], i.e. before ruin occurs.

Nevertheless, the distribution ofT is often complicated, so that the exact distribution ofST can be difficult to

obtain. Our purpose here is to show how to get lower bounds onST in thes-convex sense when a few moments of

theZk’s are known.

It is well known that ifc < λEZ1, thenT <+∞a.s., i.e. ruin occurs with probability 1. This corresponds for instance to a situation when one of the subsidiaries of an insurance company runs to ruin with probability 1, but the

whole company has a positive non-ruin probability. When theZk’s are valued inN0andu=0, Picard and Lefèvre (1998) have shown that

µ1=ET= 1 c̺0

(5.1) and

µ2=ET2=

2µ2₁

1−exp(−̺0)g′{exp(−̺0)}/c

, (5.2)

where̺0is the positive solution of the equationλ−c̺0=λEexp(−̺0Z1)andg(s)=λP+∞j=1sjP[Z1=j]. When the moments ofT,µk =ETksay, are known fork=1,2, . . . , s−1, it is possible to construct a random

variableT_min(s) with the same moments asT and such that

T_min(s) R_s−+_cxT . (5.3)

It is not possible to boundT from above since the support ofT is the whole half-positive real line (the maxima in thes-convex sense always put a positive probability mass on the upper bound of the support). Explicit expressions of the bounds involved in (5.3) are available in Denuit et al. (1998,1999a). As an illustration, fors=1 and 2, using (5.1) and (5.2), we have thatT_min(2) =µ1a.s. and

T_min(3) =

0 with probability(µ2−µ21)/µ2, µ1+(µ2−µ2₁)/µ1 with probabilityµ2₁/µ2. Now, from (4.6), we have that

S

T_min(s)

S

s−icx ST, (5.4)

so that (5.4) provides a lower bound for the total amount of claim affecting an insurance company before ruin occurs. In particular, from the integral definition (2.1) of thes-increasing convex ordering, (5.4) provides lower bounds on Eφ0(ST)for anyφ0∈UsS−icx.

Acknowledgements

This research was done while the author, supported by the Académie Royale des Sciences, des lettres et des beaux-arts de Belgique, visited the Institut de Sciences Actuarielles of the University of Lausanne, Switzerland. The warm hospitality of Professor Hans Gerber and Gérard Pafumi is gratefully acknowledged.

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