Estimation of the parameters of a Markov-modulated loss process in insurance

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Estimation of the parameters of a Markov-modulated loss process in insurance

Armelle Guillou, Stéphane Loisel, Gilles Stupfler

To cite this version:

Armelle Guillou, Stéphane Loisel, Gilles Stupfler. Estimation of the parameters of a Markov-

modulated loss process in insurance. Insurance: Mathematics and Economics, Elsevier, 2013, 53,

pp.388-404. �10.1016/j.insmatheco.2013.07.003�. �hal-00589696�

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loss process in insurance

Armelle Guillou

(1)

, Stéphane Loisel

(2)

& GillesStuper

(1)

UniversitédeStrasbourg&CNRS,IRMA,UMR7501,7rueRenéDescartes,

67084Strasbourgcedex,France

(2)

UniversitédeLyon,UniversitéLyon1,InstitutdeScienceFinancièreetd'Assurances,50

avenueTonyGarnier,69007Lyon,France

Abstract. We present a new model of loss processes in insurance. The process is a

couple

(N, L)

^where

N

îsâûnivariateMarkov-modulatedPoissonprocess(MMPP)and

L

^is

amultivariatelossprocesswhosebehaviourisdrivenby

N

^. ^We^prove^the^strongconsistency of the maximumlikelihood estimator of theparameters of this model, and presentan EM

algorithmtocomputeitinpractice. Themethodisillustratedwithsimulationsandrealsets

ofinsurancedata.

Keywords: Markov-modulatedPoisson process, maximumlikelihood estimator,strong

consistency,EMalgorithm.

1 Introduction

AMarkov-modulatedPoissonprocess(MMPP)isadoublystochasticPoissonprocesswhose

intensityisdrivenbyanon-observablecontinuous-timeMarkovchainwithnitestatespace.

A comprehensive surveyof the properties of MMPPs is given in [15]. Such processes are

used tomodelcommunicationnetworks (see[18, 21]), environmentalphenomena asin[13],

and the surplus of an insurance company as in [1]. It has then been crucial to develop

methodstoestimatetheparametersofsuchprocesses. Fromatheoreticalpointofview,the

strongconsistencyof themaximumlikelihoodestimator(MLE) foranMMPPis shownby

Rydénin [32];his proofisstronglyinuencedby[23], inwhichconsistencyfortheMLEfor

general hiddenMarkov models (HMMs) is established. Theproperties of theMLE in this

contexthavebeenextensivelystudiedsinceBaumandPetrie [3]: in additiontoconsistency

in [23], asymptotic normality was proved in [5]. Now, from a practicalpoint of view, the

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the MLE. For other referenceson EM algorithms, we refer thereader to Baum et al. [4],

whorstpresentedsuchalgorithmforHMMs;recentsurveysonEMalgorithmsincludethe

monograph byMcLachlan and Krishnan[27]. Other possible approachesinclude matching

moments and covariance functions, see [17, 31], or maximizing a split-time likelihood, as

introduced by Rydén in [33, 34], further studied by Vandekerkhove[36] in the context of

hiddenmixturesofMarkovprocesses. In[25],Loiselsuggestedthatcorrelationbetweenlines

of business of an insurance company could be caused by common shocks and modulation

by a common Markovian environment process. Our goal is to extend the MLE approach

to estimatethe parametersofaprocess

(N, L)

^where

N

îs âûnivariate^MMPPând

L

^is^a

(possiblymultivariate)lossprocesswhosebehaviorisdrivenby

N

^, ⁱⁿ^order ^to^estimate^the

parametersofsuchaprocessintworealsetsofinsurancedata. Wealsocarryoutasimulation

study of loss processesfor 2 and 3lines of business modulated by acommon environment

process. Our results conrm that themethod works quitewell aslong as theobservation

periodcontainsenoughchangesoftheMarkovianenvironmentprocess.

2 Model, assumptions and notation

We consider an MMPP

(J, N )

^, ^where

J

^is ^an irreducible continuous-time Markov process with generator

L

^on ^the^state ^space

{1, . . . , r}

^, ^where

r ∈ N \ {0}

^, ^and

N

îs â ûnivâriate

countingprocess such that, when

J

^is ⁱⁿ ^state

i

^,

N

îs â^Poisson ^process ^with întensity

λ i

^.

Wefurther consideralossprocess

S = (S 1 , . . . , S n )

^(namely^, ^the

S k

^are^piecewise^constant

processeswithnonnegativeincrements)whosebehaviorisdrivenby

N

ⁱⁿ^the^following^sense:

assume that the

S k

^can^only ^jump ^when

N

^does, ^and ^that ^if

N

^jumps ^at ^time

t

^and ^if

J

is in state

i

^, ^then ^asimultaneous jump ofthe processes

S k 1 , . . . , S k p

^at ^time

t

^occurs ^with

probability

p i (e)

^where

e = {k 1 , . . . , k p }

îs â ^subset ôf

{1, . . . , n}

^. ^We ^then ^assume ^that

the random variables

E s

^, ^such ^that ^the

S k

^with

k ∈ E s

^jumped ^(and ^only ^these) ^at ^the

timeofthe

s−

^th^jump ^of

N

^,^areindependentgiventheprocess

(J, N )

^. ^Finally^,^assume^that

thevalue ofthejump

X s

^hasdistribution

P _{θ(i, e)}

, where

( P _θ ) θ∈Θ

^is^a^parametric statistical model,thatis

P (X s = x | J (τ s ) = i, E s = e) = P _{θ(i, e)} (∀ m, m ∈ e ⇒ X m = x m )

where

τ s

^is^the^time ^of ^the

s−

^th^jump ^of

N

^, ^with^clearly

x m = 0

^if

m / ∈ e

^. ^Note ^that ^this

modelcanbeseenasacommonshockmodelasin[24]: itisassumedthatgiventheprocess

(J, N )

^and^the^sequence

(E s )

^,^the

X s

^areindependentrandomvariables.

Thecontextofourworkisthefollowing: letusassumethattheprocess

S

^has^been^observed

untiltime

T

^,^so^that^the^available^data^is:

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1. Thenumber

r

^of^states^of

J

^;

2. Thefullknowledgeoftheprocesses

N

^and

S

^between^time

0

^and^time

T

^,^both^assumed

to betimeswhen

N

^jumps.

Thegoalisto estimatetheunknownparametersofthemodel,namely:

1. Theelements

` ij

^of^the^transition^intensity^matrix

L

^of

J

^;

2. Thejump intensities

λ i

^of

N

^;

3. Theprobabilities

p i (e)

^,^where

e

îsâ^subsetôf

{1, . . . , n}

^;

4. Theparameters

θ(i, e)

^.

Remarkthattheprocess

J

îs^notôbserved,^whichînduces^technical^diculties. ^For^the^sake

ofshortness,welet

Φ

^be^the^global^parameter^of^the^model. ^Thedistributionoftheprocess withparameter

Φ

^is^then ^denoted^by

P _Φ

.

3 Asymptotic properties of the maximum likelihood esti-

mator

Our aim is to estimate the parameters with a maximum likelihood estimator (MLE). Let

then

Y i = τ i − τ i−1

^be^the^amount^of^time ^between^the

(i − 1)−

^th^and^the

i−

^th^shock,^and

Λ = diag(λ 1 , . . . , λ r )

^.

Theavailabledatais:

1. Thevalues

0 < t 1 < . . . < t k = T

^of^the

τ i

^,^i.e. ^the^times^when

N

^jumps(equivalently, theinter-eventtimes

y 1 , . . . , y k

^,^where

y j = t j − t j−1

^,

t 0 = 0

^);

2.

e 1 , . . . , e k

^the^successive^values^of^the

E k

^;

3.

x 1 , . . . , x k

^the^successive^values^of^the^jumps^of

S

^.

Letnow

f ij (t, Φ) dt := P _Φ (T 1 ∈ dt, J(t) = j | J (0) = i) F ij (t, Φ) := P _Φ (T 1 > t, J(t) = j | J (0) = i).

Therefore(see[28]),wehave

f (t, Φ) = exp(t(L(Φ) − Λ(Φ)))Λ(Φ), F(t, Φ) = exp(t(L(Φ) − Λ(Φ))).

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p(e, Φ) = diag((p i (e, Φ)) 1≤i≤r ),

P _{θ(·, e,} _Φ) (X = x) = diag(( P _{θ(i, e,} _Φ) (X = x)) 1≤i≤r ),

andin matrixnotation

∀ e ⊂ {1, . . . , n}, e 6= ∅ , g(t, e, x, Φ) = f (t, Φ) · p(e, Φ) · P _{θ(·, e,} _Φ) (X = x) g(t, ∅ , x, Φ) = f (t, Φ) · p( ∅ , Φ) · 1l {x=0} .

With thesenotations,the

(i, j)−

^th^element^of^the^matrix

g(t, e, x, Φ)

^is

∀ e ⊂ {1, . . . , n}, e 6= ∅ , g ij (t, e, x, Φ) = f ij (t, Φ) p j (e, Φ) P _{θ(j, e,} _Φ) (X = x) g ij (t, ∅ , x, Φ) = f ij (t, Φ) p j ( ∅ , Φ) 1l {x=0} .

It is now sucient to specify the starting distribution of

J

^to ^compute ^the ^likelihood ^of

theobservations. Denote by

P (Φ)

^the^transition ^matrix ^of^the discrete-timeMarkovchain

(J i = J (τ i ))

^: integrating

f

^,^one^gets

P (Φ) = (Λ(Φ) − L(Φ)) ⁻¹ Λ(Φ).

According to [32],

P (Φ)

^has^a ^unique^stationary distribution

π(Φ)

^and ^we ^have, ^if

a(Φ)

^is

theonlystationarydistributionofthecontinuous-timeprocess

(J(t)) t≥0

^and

η

^is^the^column

vectorofsize

r

^withâllêntriesêqual^to

1

^,

π(Φ) = 1

a(Φ)Λ(Φ)η a(Φ)Λ(Φ).

Weassumethatthestartingdistributionof

J

^is

π(Φ)

^; ^the^process

((J i , Y i , E i , X i )) i

^is^then

P _Φ −

stationary,becausethebivariateprocess

((J i , Y i )) i

îsâ^Markov^renewal^process^(seeê.g.

[12,p. 313]). Thus, thelikelihoodoftheobserveddataunderthedistribution

P _Φ

is

L((y i , e i , x i ) 1≤i≤k , Φ) = π(Φ) Y k i=1

g(y i , e i , x i , Φ)

! η.

Assuming nowthat weknowthestates

j 0 , j 1 , . . . , j k

^of^the ^(hidden) ^Markov^process

J

^at

thetimeswhen

N

^jumps,^the^complete^likelihood^of^the^data^is

L((j i ) 0≤i≤k , (y i , e i , x i ) 1≤i≤k , Φ) = π j 0 (Φ)

Y k i=1

g j i−1 , j i (y i , e i , x i , Φ)

! .

TogivearesultonthestrongconsistencyoftheMLE, werstneedsomenotations: foran

arbitraryparameter

Φ

^,^denote^by

F Φ

^the^set^of^all^parameters

Φ ⁰

^such^that^for^all

e (∀ j λ j (Φ) p j (e, Φ) = 0) ⇔ (∀ j λ j (Φ ⁰ ) p j (e, Φ ⁰ ) = 0).

F Φ

^can ^be ^thought ôf âs^the ^set ôf ^the êlements

Φ ⁰

^such ^that ^a simultaneous jump of the processes

S k 1 , . . . , S k q

îsâ.s. împossibleûnder^the^law

P _Φ

ifandonlyifitisa.s. impossible

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under the law

P _Φ ₀

. Writefurther

Φ ∼ Φ ⁰

^whenever

((Y i , E i , X i )) i

^has^the ^same^law^under

P _Φ

and under

P _Φ ₀

.

Wenallywritedownthehypothesesweneedto stateourmainresult:

(A 1 )

^F^or^all

e 6= ∅

,thedistributions

P _{θ(·, e)}

havethesamesupport,withnoatomat

0

^.

(A 2 )

^F^or^all

e 6= ∅

and all

Φ, Φ ⁰

^, ^there ^exists ^aneighborhood

G

^of

Φ ⁰

^such ^that ^for^every

subset

G Φ ⁰

^of

G

^and^all

i, j ∈ {1, . . . , r}

^,

Z ln sup

ϕ∈G _Φ0

P _{θ(i, e, ϕ)} (m ∈ e ⇒ X m = x m )

P _{θ(j, e,} _Φ) (m ∈ e ⇒ X m = x m ) dx < ∞.

(A 3 )

^F^or ^all

e 6= ∅

, all

i ∈ {1, . . . , r}

^and ^all

x

^,

ϕ 7→ P _{θ(i, e, ϕ)} (m ∈ e ⇒ X m = x m )

^is ^a

continuousfunction.

Thisallowsustostateourmain result:

Theorem 1. Assumethat

(A 1 − A 3 )

^hold. ^Let

Φ 0

^be ^the ^true ^value ^of ^the ^parameter, ^and

let

C

^be^a^compact ^set^of

F Φ 0

^such^that

Φ 0 ∈ C

^. ^Let

Φ b p

^be^the ^MLE^for

Φ 0

^on

C

^,^computed

with

p

observations. Then if

O ⊂ C

îsân ôpen ^set ⁱⁿ

F Φ 0

^containing ^the equivalence class of

Φ 0

^modulo

∼

^,^one^has

Φ b p ∈ C

^a.s. ^for

p

^large^enough.

Proof of Theorem 1. We closely follow the proof of Theorem 1 in [32]: pick

Φ

^and

Φ ⁰ ∈ F Φ 0

^such ^that

Φ ⁰ Φ

^. ^Lemma ⁸^implies^that ^there^exists

ε > 0

^such ^that

H (Φ, Φ ⁰ ) <

H (Φ, Φ) − 2ε

^. ^Now, ^with ^the ^notations ôf ^Lemma ^3, ^Lemma ⁵ êntails ^that ^there êxists

N ∈ N \ {0}

^with

1 N E _Φ (q 0N (Φ ⁰ )) − H (Φ, Φ ⁰ ) < ε

sothat

1 N E _Φ (q 0N (Φ ⁰ )) < H (Φ, Φ) − ε.

We then pick aneighborhood

G

^of

Φ ⁰

ⁱⁿ

F Φ 0

^given ^by ^Lemma ^3; ⁱⁿ particular, for every subset

G Φ ⁰

^of

G

^containing

Φ ⁰

^,

E _Φ ln sup

ϕ∈G Φ0

q 0N (ϕ) < ∞.

Letting

B 1/t

^be^the^open^ball^centered^at

Φ ⁰

^with^radius

1/t

^,^the^continuity^of

q 0N

^gives:

ln sup

ϕ∈G ∩ B _1/t

q 0N (ϕ) −−−→

t→∞ ln q 0N (Φ ⁰ ).

Set now

A t = (

sup

ϕ∈G ∩ B 1/t

q 0N (ϕ) ≤ 1 )

, and let

A ^c _t

^denote ^the ^complement ^of

A t

^. ^Notice

that

ln sup q 0N (ϕ) = − ln

"

sup q 0N (ϕ)

#

1l A + ln

"

sup q 0N (ϕ)

#

1l A ^c

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whichentails

ln sup

ϕ∈G ∩ B 1/t

q 0N (ϕ)

≤ | ln q 0N (Φ ⁰ )| + ln sup

ϕ∈G

q 0N (ϕ) .

Wecanthenusethedominatedconvergencetheoremto getaneighborhood

G Φ ⁰ ⊂ G

^of

Φ ⁰

in

F Φ 0

^such^that

1 N E _Φ

ln sup

ϕ∈G Φ0

q 0N (ϕ) ≤ 1

N E _Φ (ln q 0N (Φ ⁰ )) + ε

2 < H(Φ, Φ) − ε 2 .

Now,because

(Z st = ln sup _ϕ∈G _Φ0 q st (ϕ))

^is

P _Φ −

subadditiveandergodic,Kingman'stheorem (see[22])impliesthatthereexists aniteconstant

H(Φ, Φ ⁰ , G Φ ⁰ )

^such ^that

n→∞ lim 1 n E _Φ

"

ln sup

ϕ∈G Φ0

q 0n (ϕ)

#

= H (Φ, Φ ⁰ , G Φ ⁰ )

and

n→∞ lim 1

n ln sup

ϕ∈G Φ0

q 0n (ϕ) = H (Φ, Φ ⁰ , G Φ ⁰ ) P _Φ −

^a.s.

Theorem1.1in [22]entails

H (Φ, Φ ⁰ , G Φ ⁰ ) ≤ 1 N E _Φ

"

ln sup

ϕ∈G Φ0

q 0N (ϕ)

#

< H (Φ, Φ) − ε 2 ;

putting

p st (ϕ | J (0) = j) = L((Y i , E i , X i ) s+1≤i≤t , ϕ | J (0) = j)

andremarkingthat forall

ϕ ∈ G Φ ⁰

q 0n (ϕ) =



 X

i∈C(ϕ)

π i (ϕ)



 max

i∈C(ϕ) p 0n (ϕ | J (0) = i)

≥ X

i∈C(ϕ)

π i (ϕ)p 0n (ϕ | J(0) = i)

= p 0n (ϕ),

onegets

ln sup

ϕ∈G Φ0

p 0n (ϕ) − ln sup

ϕ∈G Φ0

q 0n (ϕ) ≤ 0

^and^thus

lim sup

n→∞

( 1 n ln sup

ϕ∈G _Φ0

p 0n (ϕ) )

≤ H (Φ, Φ ⁰ , G Φ ⁰ ) < H (Φ, Φ) − ε 2 .

Covernowthecompactset

O ^c ∩ C

^by^the

G Φ ⁰ _i

^,

1 ≤ i ≤ d

^. ^We^have

sup

ϕ∈O ^c

{ln p 0n (ϕ) − ln p 0n (Φ 0 )} ≤ max

1≤i≤d

( ln sup

ϕ∈G Φ0 i

p 0n (ϕ) − ln p 0n (Φ 0 ) )

−−−−→

n→∞ −∞

with

P _Φ

0 −

probability

1

^. ^This ^shows^that necessarily

Φ b p ∈ C

^a.s. ^for

p

^large ^enough, ^and

completestheproof.

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∼

^. În^that ^sense,^this^resultîs^the^best ^possibleône.

Undersomeadditionalassumptions,onecanapply theasymptoticnormalitytheorem in[5]

in order to obtain the one of our estimator. This result is rather technical: we refer the

readerto[16]fordetails.

4 An EM algorithm to compute the MLE

WenowgiveanEMalgorithm,adapted from[35],allowingus to computethe MLEin our

context. Recalltheavailabledata:

1. Thevalues

0 < t 1 < . . . < t k = T

^of^the

τ i

^,^i.e. ^the^times^when

N

^jumps(equivalently, theinter-eventtimes

y 1 , . . . , y k

^,^where

y j = t j − t j−1

^,

t 0 = 0

^);

2.

e 1 , . . . , e k

^the^successive^values^of^the

E k

^;

3.

x 1 , . . . , x k

^the^successive^values^of^the^jumps^of

S

^.

Wewantto estimate

1. Theelements

` ij

^of^the^transition^intensity^matrix

L

^of

J

^;

2. Thejump intensities

λ i

^of

N

^;

3. Theprobabilities

p i (e)

^,^where

e

îsâ^subsetôf

{1, . . . , n}

^;

4. Theparameters

θ(i, e)

^.

Welet

0 < u 1 < . . . < u m < T

^be^the^jump^times^of

J

ⁱⁿ^the^time^interval

[0, T ]

^,

u 0 = 0

^and

u m+1 = T

^;^let^further

s i

^be^the^state^of

J

^on^the^interval

[u i−1 , u i [

^,

∆u i = u i − u i−1

^and

z i

bethenumberofjumpsof

N

ⁱⁿ^the^interval

[u i−1 , u i [

^.

Recall that, if

N ⁰

^is ^an homogeneous Poisson process, then given

{N ⁰ (t) = n}

^, ^the ^event

times of

N ⁰

ⁱⁿ ^the ^interval

[0, t]

^are ^uniformly distributed. Consequently, Bayes' formula impliesthatthecompletelikelihoodofthedatais

L ^c = π s 1

" _m Y

i=1

` s i , s i+1

−` s i , s i

· (−` s i , s i exp(` s i , s i ∆u i ))

#

exp(` s m+1 , s m+1 ∆u m+1 )

×

" _m+1 Y

i=1

(λ s i ∆u i ) ^z ⁱ

z i ! exp(−λ s i ∆u i ) · z i ! (∆u i ) ^z ⁱ

#

× Y r i=1



 

 Y

e⊂{1, ..., n}

e6= ∅

p i (e) ^card(A ⁱ ^(e)) Y

j∈A i (e)

P _{θ(i, e)} (∀ m ∈ e, X m = x m, j )



 

 · p i ( ∅ ) ^card(A ⁱ ^(∅))

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where

A i (e) = {j ∈ {1, . . . , k} | J (t j ) = i, e j = e}

^stands^for^the^set^of ^the^jump ^times^of

N

whenthe

S k

^with

k ∈ e

^(and^only^these) ^jump ^and

J

^isⁱⁿ ^state

i

^;

A i ( ∅ )

^stands^for^the^set

ofthejumptimesof

N

^when^none^of^the

S k

^jumps^and

J

^isⁱⁿ ^state

i

^.

Fromthat identity,wededucethatthecompletelog-likelihoodis

ln L ^c = X r

i=1

1l {X(0)=i} ln(π i ) + X r

i=1

T i ` ii + X r

i=1

X r

j=1 j6=i

m ij (T ) ln(` ij ) + X r i=1

(n i ln(λ i ) − λ i T i )

+ X r i=1

X

e⊂{1, ..., n}

card(A i (e)) ln(p i (e))

+ X r i=1

X

e⊂{1, ..., n}

e6=∅

X k j=1

ln P _{θ(i, e)} (∀ m ∈ e, X m = x m, j )1l {j∈A i (e)}

where

1.

T i = Z T

0 1l {J(u)=i} du

^is^the^time^spent^by^the^process

J

ⁱⁿ ^state

i

^until^time

T

^;

2.

m ij (T ) = card({s : 0 < s ≤ T | J (s − ) = i, J (s) = j})

^is ^the ^number^of ^jumps ^from

state

i

^to^state

j

^of^the^process

J

^;

3.

n i = X k j=1

1l {J(t j )=i}

îs^the^numberôfêvents^thatôccurred^when

J

^isⁱⁿ^state

i

^.

The Mstep. Wenowcomputetheconditionalexpectationof

ln L ^c (Φ)

^under^a^parameter

ϕ

^,^given^the^event

{N(u), S(u), 0 ≤ u ≤ T }

^: ^one^has

E _ϕ (ln L ^c (Φ) | N (u), S(u), 0 ≤ u ≤ T )

= X r i=1

1l {X(0)=i} \ ln(π i ) + X r i=1

T b i ` ii + X r i=1

X r

j=1 j6=i

m \ ij (T ) ln(` ij ) + X r i=1

( n b i ln(λ i ) − λ i T b i )

+ X r i=1

X

e⊂{1, ..., n}

card(A \ i (e)) ln(p i (e))+

X r i=1

X

e⊂{1, ..., n}

e6= ∅

X k j=1

ln P _{θ(i, e)} (∀ m ∈ e, X m = x m, j ) 1l {j∈A \ i (e)}

where

A b = E _ϕ (A | N (u), S(u), 0 ≤ u ≤ T )

^.

For

T

^large^enough,^the^rst^term^may^be^neglected;^recalling^that

` ii = − X r

j=1 j6=i

` ij , p i ( ∅ ) = 1 − X

e⊂{1, ..., n}

e6=∅

p i (e), card(A \ i ( ∅ )) = n b i − X

e⊂{1, ..., n}

e6=∅

card(A \ i (e)),

onegets,forall

i, j ∈ {1, . . . , r}

^and

i 6= j

^,^the^identities

b

p i (e) =

card(A \ i (e)) b n i

, ` b ij =

m \ ij (T ) T b i

, λ b i = n b i

T b i

, X k

j=1

∂

∂θ(i, e) ln P _{θ(i, e)} (∀ m ∈ e, X m = x m, j )

_{θ(i, e)=b} _{θ(i, e)}

1l {j∈A \ i (e)} = 0

^,

(10)

where

b p i (e)

^,

` b ij

^and

λ b i

^are^the^desiredestimators,andthelastsetofequationsistobesolved takingthepropertiesofthestatisticalmodel

( P _θ )

^into^account.

The E step. Accordingto Lemma9,if

A(e) = [ r i=1

A i (e) = {j ∈ {1, . . . , k} | e j = e}

^,^then

T b i = Z T

0 P _ϕ (J (v) = i, N (u), S (u), 0 ≤ u < v) P _ϕ (N (u), S(u), 0 ≤ u ≤ T)

× P _ϕ (N(u), S (u), v ≤ u ≤ T | J (v) = i) dv, b

n i = X k q=1

P _ϕ (J(t q ) = i, N (u), S(u), 0 ≤ u ≤ T ) P _ϕ (N (u), S(u), 0 ≤ u ≤ T ) , 1l _{j∈A \ _i _(e)} = 1l _{j∈A(e)} P _ϕ (J (t j ) = i | N (u), S(u), 0 ≤ u ≤ T ), card(A \ i (e)) =

X k j=1

1l {j∈A \ i (e)} = X k j=1

1l {j∈A(e)} P _ϕ (J (t j ) = i | N(u), S(u), 0 ≤ u ≤ T ), m \ ij (T ) = ` ij (ϕ)

Z T 0

P _ϕ (J (v) = i, N (u), S(u), 0 ≤ u < v) P _ϕ (N (u), S(u), 0 ≤ u ≤ T )

× P _ϕ (N (u), S(u), v ≤ u ≤ T | J (v) = j) dv.

Let

w i

^be^the^column^vector^of^size

r

^withâllêntriesêxcept^the

i−

^th^equal^to

0

^,^and^its

i−

^th

entryequalto

1

^. ^Firstly^,

P _ϕ (N(u), S (u), 0 ≤ u < v, J (v) = i) = π(ϕ)





N(v) Y

q=1

g(y q , e q , x q , ϕ)



 F (v − t N (v) , ϕ)w i .

Secondly,if

w ^t _i

^is^the^transpose^of

w i

^,

P (N (u), S(u), v ≤ u ≤ T, ϕ | J(v) = i)

= w ^t _i g(t N (v)+1 − v, e N (v)+1 , x N (v)+1 , ϕ)



 Y k q=N(v)+2

g(y q , e q , x q , ϕ)



 η,

andnally

P _ϕ (J(t q ) = i, N (u), S(u), 0 ≤ u ≤ T )

= π(ϕ) Y q p=1

g(y p , e p , x p , ϕ)

! w i w ^t _i

Y k p=q+1

g(y p , e p , x p , ϕ)

! η.

θ

îs ^generallyêstimated^withâ^numerical^(e.g. quasi-Newton)method.

Procedure. Here,wedescribeawayto implementouralgorithm,byinduction on

` ∈ N

. Dene, if

Φ `

îs^the^parameterêstimateât^step

`

^,

1.

G ` (0) = π(Φ ` )

^and

∀ 0 ≤ q ≤ k − 1, G ` (q + 1) = G ` (q) · g(y q+1 , e q+1 , x q+1 , Φ ` );

2.

D ` (k) = η

^and

∀ 0 ≤ q ≤ k − 1, D ` (k − q − 1) = g(y k−q , e k−q , x k−q , Φ ` ) · D ` (k − q)

^.

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Setthen

A ij (Φ ` ) = B i (·, Φ ` ) = C i (Φ ` ) = 0

^and^do,^for^all

q ∈ N

suchthat

1 ≤ q ≤ k

^,

A ij (Φ ` ) ← A ij (Φ ` ) +

Z t q

t q − 1

G ` (q − 1) F (t − t q−1 , Φ ` )w i w ^t _j g(t q − t, e q , x q , Φ ` ) D ` (q) dt, B i (q, Φ ` ) ← G ` (q)w i w ^t _i D ` (q),

C i (Φ ` ) ← C i (Φ ` ) + B i (q, Φ ` ).

Theestimatesatstep

` + 1

^are^then

b

p i (e) = P k

j=1 1l {j∈A(e)} B i (j, Φ ` )

C i (Φ ` ) , ` b ij = ` ij (Φ ` ) · A ij (Φ ` )

A ii (Φ ` ) , λ b i = C i (Φ ` ) A ii (Φ ` ) ,

andthe

θ(i, e) b

^that ^maximize^thefunctionals

θ 7→

X k j=1

ln P _θ (∀ m ∈ e, X m = x m, j )B i (j, Φ ` )1l {j∈A(e)} .

5 A posteriori reconstruction of the states, with a maxi-

mum likelihood method

Once theparametersof themodel are estimated, itcanbeinterestingto estimate thesuc-

cessivestatesoftheMarkovchain

(J i )

^. ^To^this^end, ^we^can^adapt ^the^procedure ^described

in[28]: considerthelog-likelihoodofboththeobservedandmissing data

(j 0 , . . . , j k ) 7→ ln(π j 0 (b Φ)) + X k i=1

ln g j i−1 , j i (y i , e i , x i , Φ). b

An estimatorof

(j 0 , . . . , j k )

^is^then^a

(k + 1)−

^tuple

( j b 0 , . . . , j b k )

^which^maximizes^this^func-

tional. Such an estimatorhasexcellentproperties, see [8]. Froma practicalpointof view,

onemayreconstructthestatesusingtheViterbialgorithm(see[37]), namely:

1. Set

V j = 0

^and

C j = [j]

^for^all

j ∈ {1, . . . , r}

^,^and

q = 1

^.

2. If

q ≥ k + 1

^,^go^to^step^6. ^Otherwise,^set

α ^(q) _{i, j} = ln g ij (y k−q+1 , e k−q+1 , x k−q+1 , Φ). b

3. Forall

i

^,

j ∈ {1, . . . , r}

^,^compute

β _{i, j} ^(q) = α ^(q) _{i, j} + V j

ândânîndex

j _i ^(q)

^such^that

β ^(q)

i, j ^(q) _i =

j∈{1,..., r} max β _{i, j} ^(q)

^.

4. Forall

i ∈ {1, . . . , r}

^,^replace

V i

^by

β ^(q)

i, j _i ^(q)

and

C i

^by

[j _i ^(q) , C i ]

^.

5. Replace

q

^by

q + 1

^and^go^back^to^step^2.

6. Find anindex

i

^such^that

V i = max

j∈{1,..., r} V j

^.

An estimateof thestatesisthenthesequence

( j b 0 , . . . , j b k ) = C i

^.

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6.1 Computing a rst estimate

Providing a rst estimate for an iterative algorithm is usually a daunting task. Here, we

describeaprocedure,adapted fromtheonedescribedin [28], thatworkedquitewellin our

examples:

1. Compute the average of the inter-eventtimes

λ c ^∗ = k/T

^, ând ^mobile âverages ôf ^the

inter-eventtimes

y i

^,^denoted^by

z i

^(for^the^rstând^last^timesôf^theôbserved^sample,

put

z i = y i

^).

2. Set

J b (·) = 0

^;^pick

q 1 ≤ 1 < q 2 < · · · < q r−1

^. ^For^all

i ∈ {1, . . . , k}

^:

(a) if

z i > 1/(q 1 c λ ^∗ )

^, ^set

J b (t i ) = 1

^;

(b) forall

j ∈ {1, . . . , r − 2}

^,^if

1/(q j+1 λ c ^∗ ) < z i ≤ 1/(q j λ c ^∗ )

^,^set

J b (t i ) = j + 1

^;

(c) if

z i ≤ 1/(q r−1 λ c ^∗ )

^,^set

J b (t i ) = r

^.

3. Compute

n b j =

k−1 X

i=1

1l _{ _J(t _b _i _)=j}

^for

j ∈ {1, . . . , r}

^.

4. Compute,forall

i, j ∈ {1, . . . , r}

P b ij = X k

`=2

1l _{ _J(t _b _`

− 1 )=i, J(t b ` )=j}

b

n i ,

which is the rst estimate of

P ij

^, ^the probability that the Markov chain

(J (t k )) k≥0

jumps fromstate

i

^to^state

j

^.

5. Calculate,forall

j ∈ {1, . . . , r}

^,

b π j = n b j + 1l _{ _J(t _b _k _)=j}

k

^,^the^rst^estimate^of

π j

^.

6. Thankstotheidentities

∀ j ∈ {1, . . . , r} λ j = λ ^∗ π j a ⁻¹ _j

^and

L = Λ(Id −P ⁻¹ ),

(where

λ ^∗ = P r

j=1 λ j a j

îs^theâverage^jump ^rateôf

N

^),^consider

L

^and

Λ

^as^functions

of

a 1 , . . . , a r−1

^,^and^maximize^the^complete^likelihood^with^respect^to ^the^parameters

a 1 , . . . , a r−1

^given

c λ ^∗

^,

π b 1 , . . . , π b r

^,

P b

^,

y 1 , . . . , y k

^and

J b

^: ^let

b a 1 , . . . , b a r−1

^be^the^estimate

obtainedthisway.

7. For all

j ∈ {1, . . . , r}

^, ^compute

c λ j = λ c ^∗ b π j b a ⁻¹ _j

^, ^let

Λ b

^be ^the ^diagonal ^matrix ^with

coecients

c λ 1 , . . . , λ c r

ⁱⁿ ^that ^order ^and^compute

L b = Λ(Id b − P b ⁻¹ )

^. ^These ^are^rough

estimatesfor

Λ

^and

L

^.

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8. Use

L b

^and

Λ b

âsînitial^values^forânÊMâlgorithm^to^provideêstimates^for

L

^and

Λ

^(see

[35]),whichwedenoteby

L

^and

Λ

^. ^Compute^thecorrespondingstationarydistributions

a

^and

π

^.

9. Performastatereconstructionof

J

^with^the^Viterbi^algorithm ^using

L

^and

Λ

^,^and^let

J

^be^the^process^obtained^this^way^.

10. Forall

j ∈ {1, . . . , r}

^,^calculate

n j =

k−1 X

i=1

1l _{J(t _i _)=j}

^.

11. Forall

i 1 , . . . , i n ∈ {0, 1}

^and

j ∈ {1, . . . , r}

^,^if

e

^is^the^subset^of

{1, . . . , n}

^such^that

k ∈ e ⇔ i k = 1

^,^compute

p j (e) = 1 n j

k−1 X

`=1

1l _{J(t _` _)=j} 1l {∀p∈{1, ..., n}, S p (t ` )−S p (t ` − 1 )>0 ⇔ i s =1}

whichistheinitialestimateof

p j (e)

^.

12. Forall

j = 1, . . . , r

^and

e 6= ∅

, consider the

X i

^such ^that

J(t i ) = j

^and

E i = e

^as

independent andidentically distributed randomvariableswith parameter

θ(j, e)

^, ^and

estimate

θ(j, e)

^with^a^standard^method^(maximum^likelihood^method ^for^instance).

This procedure isadapted in the particularcasewhen

λ 1 < · · · < λ r

^strongly ^dier,^which

shallbethecaseinournumericalstudybelow.

6.2 A non-life insurance example

We now use our algorithm on a real set of non-life insurance data. From January

2004

to November

2009

^,

594

^accidents corresponding to blazes causing industrial damages or losses were observed. The days of these events were recorded, and so were, if necessary,

the compensationsfor thevictims; the processes

N

^and

S

ôbtained^this^wayâre ^shown ôn

Figure 12. This situation corresponds to the case

α = n = 1

^of ^our ^model. ^We ^nally

choose

r = 2

^, ^which ^is^justied^by^the^fact ^that^the^MLE, ^computed^only^for

L

^and

Λ

^with

r = 3

^sets^all^parameterscorresponding tothethird stateto

0

^. ^Before^modeling^the^claims

themselves,theparametersofthismodelare

1.

` 12

^and

` 21

^,^the^jump ^rates^of^the^hidden^Markov^process

J

^;

2.

λ 1

^and

λ 2

^,^the^jump intensitiesoftheshockcountingprocess

N

^;

3.

p 1 (1)

^and

p 2 (1)

^, ^theprobabilitiesthat, when anaccidenthappens, theinsurancerm hastocompensate.

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0 500 1000 1500 2000 0

100 200 300 400 500

Figure1: Thecountingprocess

N

0 500 1000 1500 2000

0e+000 1e+006 2e+006 3e+006 4e+006 5e+006 6e+006 7e+006 8e+006

Figure2: Thelossprocess

S

Asfortheclaimsizes, aquickanalysisofthedatashowsthatsomeclaimshaveasmallsize

and afew others are very large, which prevents us from modeling the situation by a log-

Normal,GammaorGeneralizedParetodistribution (GPD).Inactuarialstatistics,onemay

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inmanySolvencyIIpartialinternalmodels,ordealdirectlywithamixtureofdistributions,

or with a distribution that looks like Lognormal or Gamma distributions for small values

and gets moreand more Pareto-typefor largevalues, likethe Champernownedistribution

(see[9, 10] and[20]). Anotherpossibility isto useaclassicalkerneldensity estimatorafter

transformingthedata (see[6]). Here,we useamixtureof alight-tailed and aheavy-tailed

distribution,namelyaGammadistributionandaGPD.

P _θ

thenhasdensity

x 7→ q (bx) ^a−1

Γ(a) be ^−bx 1l {x>0} + (1 − q) 1 σ

1 + ξ(x − µ) σ

^−1−1/ξ 1l {x>µ}

where

a, b, σ, ξ > 0

^,

0 < q < 1

^and

µ = 49.33

^is^the^minimal^(observed)^claim^size ^(the^unit

istheeuro).

Consequently,theparameterstobeestimatedare

` 12

^,

` 21

^,

λ 1

^,

λ 2

^,

p 1 (1)

^,

p 2 (1)

^,

a 1

^,

a 2

^,

b 1

^,

b 2

^,

σ 1

^,

σ 2

^,

ξ 1

^,

ξ 2

^,

q 1

^and

q 2

^.

EstimatingtheparametersviatheEMalgorithm,withaquasi-Newtonalgorithmtoestimate

theparameters

a i

^,

b i

^,

σ i

^,

ξ i

^and

q i

^during^the^M^step^gives^the^following^results:

L b =



 −0.0065 0.0065 0.0018 −0.0018



 , Λ = b



 0.462 0 0 0.214



 ,

b p(1) =



 0.963 0 0 0.947



 , p(0) = b



 0.037 0 0 0.053



 ,

b a =



 4.52 4.14



 , b b =



 0.011 0.0073



 , b σ =



 1145 1216



 , ξ b =



 1.45 1.31



 , q b =



 0.230 0.335



 .

Theclaim sizesthus haveinnitemeansin bothstatesin theory. This meansthat thetail

of the claim size distribution is very heavy. However, reinsurance mechanisms and other

guaranteesmayenable theinsurer to provide insurancecoverage of those risksupto some

highthresholdlevel. Afurtheranalysis thenshowsthat

1. Sojourntimesin state

1

âreônâverage

3.5

^times^shorter^thanⁱⁿ^state

2

^;

2. There aremoreaccidentswhen

J

^isⁱⁿ ^state

1

^thanⁱⁿ^state

2

^;

3. Because

p b 1 (1)

^is ^slightly ^greater ^than

b p 2 (1)

^, ^these ^accidents ^cause ^more^losses ^to ^the

insurancerm;

4. Losses instate

1

^are^more^likely^to^beheavy-tailedthaninstate

2

^.

An aposteriorireconstruction ofthestatesof

J

^is^givenⁱⁿ^Figure^3.

6.3 A life insurance data set

Let us now present an application in the life insurance eld. From January

2006

^to ^July

2010

^,

1507

^closures^of^savings^accounts^(also^calledsurrenders)were observed. Themonths

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0 500 1000 1500 2000 1.0

1.2 1.4 1.6 1.8 2.0

Figure3: Aposteriorireconstructionofthestatesof

J

oftheseeventswererecorded,alongwiththeamountofmoneywithdrawn. Earlysurrenders

canberegardedasclaimsfor theinsurancecompanyin somecases,becauseit corresponds

toadropinfuturebusiness,andbecausesometimestheinsurerhasbeenunabletochargeall

thefees(thatareoftenpartlypaidbythepolicyholderateachtimeperiodandnotupfront)

beforethe surrender. Surrender riskiscomplex: taxand penalty relief,interest ratelevels,

competitionbetweeninsurancecompanies,aswellasotherfactorsareatstake. Forareview

on surrendertriggers, theinterested readermightconsult [29] or[26]. Inthe present data

study, we are interested in the big picture in aquite stable regime (and notin prediction

offuture surrenderrates): intheconsideredperiod, theportfolioseemstohavebeenpretty

stable,mainlysensitivetoexternalcompetition(whichisdiculttoobserveinpractice). We

assumethat conditionally with respect to thestateof theenvironment,the probabilityfor

onepolicyholdertosurrenderhercontractdoesnotdependontheamountofsavings. Toset

aprecisedateforthe

k−

^th^surrender,^we^drawâûniform^random^variableândâddît^to^the

month ofthis eventto obtainan exactdate. Here, theclaims arethe amountsof money

withdrawn;theprocesses

N

^and

S

^are representedonFigure 45. Again,thissituation ts the case

α = n = 1

ôf ôur^model; ^weûse â^two-state ^model ^for ^this ^situation,^so ^that ^the

parametersare

1.

` 12

^and

` 21

^,^the^jump ^rates^of^the^hidden^Markov^process

J

^;

λ 1 λ 2 N

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Notethatinthisexample,thereisnoneedtoestimate

p 1 (1)

^and

p 2 (1)

^. ^On^the^graphs^below,

theunitoftimeis themonth:

0 10 20 30 40 50

0 200 400 600 800 1000 1200 1400

Figure4: Thecountingprocess

N

0 10 20 30 40 50

0.0e+000 5.0e+006 1.0e+007 1.5e+007 2.0e+007 2.5e+007 3.0e+007 3.5e+007

Figure5: Theprocess

S

representingthecumulativeamountofmoneywithdrawn

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Instate

1

^,^weûseâ^mixtureôfâlight-tailedandaheavy-taileddistribution,namelyaWeibull distributionandaGPD,thedensityof

P _θ

thenbeing

x 7→ q a b

x − µ b

a−1

e ^{−((x−µ)/b)} ^a 1l {x>µ} + (1 − q) 1 σ

1 + ξ(x − µ) σ

−1−1/ξ

1l {x>µ}

where

a, b, σ, ξ > 0

^,

0 < q < 1

^and

µ = 1.1

îs ^the^minimal ^(observed)âmount^(the ûnit îs

theeuro). Instate

2

^,^we^t^a^GPD, ^whose^density^is

x 7→ 1

σ

1 + ξ(x − µ) σ

−1−1/ξ

1l {x>µ}

⁽¹⁾

where

µ, σ, ξ > 0

^. Ôf ^course, ^surrender âmounts âre ^not ^completely independent at the microscopic level aseach policyholder has acertain balance onhis savingsaccountthat is

known at a precise date. We are aware that in theory, the

X i

^are ^not independent and identically distributed in each state, but in practice there are enough policyholders and

enough randomness in the surrendered amounts for this assumption to be acceptable in

practice at the macroscopic level in each state of the environment (this is supported by

statisticaltests).

Consequently,theparameterstobeestimated are

` 12

^,

` 21

^,

λ 1

^,

λ 2

^,

a

^,

b

^,

σ 1

^,

σ 2

^,

ξ 1

^,

ξ 2

^and

q

^.

EstimatingtheparametersviatheEMalgorithm,withaquasi-Newtonalgorithmtoestimate

theparameters

a

^,

b

^,

σ i

^,

ξ i

^and

q

^during ^the^M^step^gives^the^following^results:

L b =



 −0.254 0.254 0.373 −0.373



 , Λ = b



 34.2 0 0 17.4



 ,

b a = 1.65, b b = 9141, σ b =



 22350 14591



 , ξ b =



 0.17 0.40



 , q b = 0.306.

An a posteriori reconstruction of the states of

J

^is ^shown ⁱⁿ ^Figure ^6. ^Note ^that ^results

showthat during someerce competition periods, surrenderrates become moreimportant

(they double from one stateto the other). In the statewhere surrender rates are higher,

the surrenderedamount tteddistribution is composed of a light-tailed part and aheavy-

tailed part,whereas forsmaller surrenderrates, this distribution does notincorporateany

light-tailed part. This suggeststhat policieswith smallerfacialamountsaremoresensitive

tochangesin theenvironment. Onceagain,here,theheavy-tailed partmustberegardedas

astatisticalt,andthetailwouldhaveto becutatanappropriatelevelaposteriori.

6.4 Simulations in the multivariate setting

6.4.1 Motivation

Oneof themain purposes of insuranceisrisk diversicationand mutualization: thelaw of

largenumbersandthecentral limittheorem oftenapply inpracticewhenindependencebe-

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0 10 20 30 40 50 1.0

1.2 1.4 1.6 1.8 2.0

Figure6: Aposteriorireconstructionofthestatesof

J

insuranceportfolios(withoutmotorliabilityinsurance)atthenationallevel. However,when

itcomes to hurricane risksorearthquakerisks,individualrisksare onlyconditionallyinde-

pendentwithrespecttotheoccurrenceornotofsucheventsinthecountry. Thiscorrelation

makesitdiculttodiversifythoserisksatthenationallevel,andoneoftenusesreinsurance:

risksarethendiversiedatthegloballevel(oodsinAustralia,tsunamisinAsia,hurricanes

intheEastCoastofNorthAmerica,earthquakesinJapan,MonteCarloandSanFrancisco,

stormsinEuropeforinstance). Nevertheless,thoserisksarenotreallyindependent,assome

(often ignored) correlation factors are present. Even if they are geographically scattered,

meteorological phenomena like theEl Nino-La Nina Southern Oscillation (ENSO) may si-

multaneouslyinuenceclaimoccurrenceandseverityin thosedierentzones. Forexample,

itis nowacceptedthat theprobabilitiesofsevereoods in Australia,strongsnowstormsin

NorthAmericaandhurricanesontheUSEastCoastincreaseduringLaNinaepisodes,while

other kindsofeventsare morelikelyduring El Nino episodes. Tobuild amodel forENSO

and tounderstand allits impactsondierentareasof theworldis farbeyond thescopeof

this paper. Ofcourse, ENSO isobservedand canbe(partly) measured,itsbehaviorisnot

really Markovianand claim arrivalprocesses feature seasonality. There are certainly other

kindsofunobservedenvironmentprocessesthatjointlymodulateclaimprocessesindierent

regions of the world. In our illustrative example, we just imagine that some unobserved

Markov process inuences claim frequencies in three regions A (

k = 1

^), ^B⁽

k = 2

⁾ ^and ^C

(

k = 3

^). ^RegionsÂ ând ^Bâre âssumed ^to ^be ^close ^to êach ôther, ^so ^that ^common ^shocks

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changesaremorefrequentthanfortheENSO cycle. Wesimulate thecorrespondingmulti-

variateriskprocess,andwecheckwhetheritwouldbepossibleornotforustoestimatethe

parametersof themodel and to re-buildthestates ofthe environment modulating process

(without observingitofcourse).

6.4.2 Amodel with 2states ofthe environment

Werst assumethat

r = 2

^: ⁱⁿ ^state^1, ^claims^are^less^frequent^and ^less^severeⁱⁿ ^the ^three

zones, and common shocks are not present (

p 1 (e) = 0

^if

Card(e) ≥ 2

^). ^In ^state ^2, ^claims

are more likely and more severe in average, and common shocks are possible for zones A

and B(

p 2 ({1, 2}) > 0

^). ^T^ake

λ 1 = 20

^,

λ 2 = 200

^,

p 1 ({1}) = p 1 ({2}) = 0.3

^,

p 1 ({3}) = 0.4

^,

p 2 ({1}) = p 2 ({2}) = 0.2

^,

p 2 ({3}) = 0.4

^and

p 2 ({1, 2}) = 0.2

^. ^The^univariate^claim^severity

distributions arechosentobeGPdistributedasin(1),withtheparametersbeing

µ({1}) = µ({2}) = µ({3}) = 1, σ(1, {1}) = σ(1, {2}) = σ(1, {3}) = 1, σ(2, {1}) = σ(2, {2}) = σ(2, {3}) = 20, ξ(1, {1}) = ξ(1, {2}) = ξ(1, {3}) = 1/2, ξ(2, {1}) = ξ(2, {2}) = ξ(2, {3}) = 2

^.

Univariateclaimsarethereforemoresevereinaverageandinthetailforstate2forallthree

lines. Asfarasthebivariateclaimsinstate2areconcerned,wemodelthembyabivariate

GPDasin[7,11];namely,theirdensityhastheform

(x, y) 7→ α(α + 1) σ 1 σ 2

1 + x − µ 1

σ 1

+ x − µ 2

σ 2

−α−2

1l {x>µ 1 } 1l {y>µ 2 }

where

α, µ 1 , µ 2 , σ 1 , σ 2 > 0

^,^and^we^choose

µ({1, 2}) =



 3 3



 , σ(2, {1, 2}) =



 30 20



 , α(2, {1, 2}) = 2.

Assumethatweobservethemultivariateclaimprocessduring30years,andthattheaverage

timespentin state1(before switchingtostate2)is1year,whiletheaveragetimespentin

state2(beforeswitchingtostate1)is3months. Namely,

` 12 = 1

^and

` 21 = 4

^.

The estimate of

µ({e})

^,

e 6= ∅

ischosenas thevectorof theminima of the claims arising whenashockaectssimultaneouslythelines

L k 1 , . . . , L k p

^, ^with

e = {k 1 , . . . , k p }

^. ^Results

aregivenbelow:

b ` 12 = 1.064, ` b 21 = 3.891,

b λ 1 = 21.21, b λ 2 = 195.7,

(21)

b

p 2 ({1}) = 0.227, p b 2 ({2}) = 0.182, p b 2 ({3}) = 0.394, b

p 2 ({1, 2}) = 0.197, b

µ({1}) = 1.002, µ({2}) = 1.000, b µ({3}) = 1.004, b b

σ(1, {1}) = 0.950, b σ(1, {2}) = 1.393, b σ(1, {3}) = 0.999, b

σ(2, {1}) = 18.22, b σ(2, {2}) = 19.18, b σ(2, {3}) = 24.83, ξ(1, b {1}) = 0.552, ξ(1, b {2}) = 0.507, ξ(1, b {3}) = 0.493, ξ(2, b {1}) = 2.206, ξ(2, b {2}) = 2.220, ξ(2, b {3}) = 1.888, b

µ({1, 2}) =



 3.142 3.040



 , b σ(2, {1, 2}) =



 25.98 18.06



 , α(2, b {1, 2}) = 1.79

^.

Theestimationprocedureworksquitewellandthestatesarecorrectlyretrieved,seeFigure9.

Ofcourse,iftheobservationperiodwasshorter,orifthephasechangeintensitiesweresmaller,

thenitwouldbeimpossibletoestimatetransitionratesaccurately.

0 5 10 15 20 25 30

1.0 1.2 1.4 1.6 1.8 2.0

0 5 10 15 20 25 30

0 100 200 300 400 500 600 700 800

0 5 10 15 20 25 30

0 100 200 300 400 500 600 700

0 5 10 15 20 25 30

0 100 200 300 400 500 600 700

Figure7: Thecountingprocesses: topleft: thetrueprocess

J

^,^top^right:^the^counting^process

related to

S 1

^, ^bottom ^left: ^the ^counting ^process ^related ^to

S 2

^, ^bottom^right: ^the ^counting

processrelatedto

S 3

6.4.3 Amodel with 3states ofthe environment

Wenowassumethat

r = 3

^and^that^common^shocks^are^not^present^(for

i = 1, 2, 3

^,

p i (e) = 0

if

Card(e) ≥ 2

^). În^state ^1, ^claims âre^not ^very^frequent ând ^not^very^severe ⁱⁿ ^the ^three

zones. In state 2, claims are more likely and moresevere in average for the three zones.

State3correspondstoexceptionalconditionsthatfavorextremelysevereclaimsforzonesA

(22)

0 5 10 15 20 25 30 1.0

1.2 1.4 1.6 1.8 2.0

0 5 10 15 20 25 30

0 2.0e+5 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 1.4e+6

0 5 10 15 20 25 30

0 2.0e+5 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 1.4e+6 1.6e+6

0 5 10 15 20 25 30

0 2.0e+5 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 1.4e+6 1.6e+6 1.8e+6

Figure 8: Thelossprocesses

S k

^,^top^left: ^the^true^process

J

^, ^top^right: ^the^loss^process

S 1

^,

bottomleft: thelossprocess

S 2

^,^bottom^right: ^the^loss^process

S 3

0 5 10 15 20 25 30

1.0 1.2 1.4 1.6 1.8 2.0

0 5 10 15 20 25 30

1.0 1.2 1.4 1.6 1.8 2.0

Figure9: ReconstructionofthehiddenMarkovprocess

J

^: ^top: ^the^true^process

J

^,^bottom:

thereconstructedprocess

J b

(23)

and B but protectzone C.Take

λ 1 = 20

^,

λ 2 = 200

^,

λ 3 = 1000

^,

p 1 ({1}) = p 1 ({2}) = 0.3

^,

p 1 ({3}) = 0.4

^,

p 2 ({1}) = p 2 ({2}) = 0.3

^,

p 2 ({3}) = 0.4

^,

p 3 ({1}) = p 1 ({2}) = 0.45

^and

p 1 ({3}) = 0.1

^. ^The^claim^severitydistributionsareonceagainmodeledbyGPdistributions, with

µ({1}) = µ({2}) = µ({3}) = 1, σ(1, {1}) = σ(1, {2}) = σ(1, {3}) = 1, σ(2, {1}) = σ(2, {2}) = σ(2, {3}) = 20, σ(3, {1}) = σ(3, {2}) = 200, σ(3, {3}) = 0.5,

ξ(1, {1}) = ξ(1, {2}) = ξ(1, {3}) = 1/4, ξ(2, {1}) = ξ(2, {2}) = ξ(2, {3}) = 1/2, ξ(2, {1}) = ξ(2, {2}) = 1, ξ(2, {3}) = 1/3,

These parametersare chosensothat claimsfor zoneC in state3 areverysmall compared

to those forzones Aand B.Assume that weobservethe multivariateclaim processduring

30years,that theaveragetimespentinstate1(before switchingtoanotherstate)is1year

(resp. 3 months for state2, 1 month for state3), and that jumps from state 1to state3

orfrom state3to state1area.s. impossible. Assume nally that whenoneleavesstate2,

theprobabilitytogotostate1is

2/3

^. ^The^intensity^transition^parameters^are^then

` 12 = 1

^,

` 13 = 0

^,

` 21 = 8/3

^,

` 23 = 4/3

^,

` 31 = 0

^,

` 32 = 12

^.

Again, theestimate of

µ({i})

^,

i = 1, 2, 3

îs ^chosenâs^the^minimum ôf ^the^claims âecting

line

i

^. ^The^results^are^the^following:

` b 12 = 1.691, b ` 13 = 0, b ` 21 = 2.513, ` b 23 = 1.288, b ` 31 = 0, b ` 32 = 10.76, b λ 1 = 27.44, b λ 2 = 198.3, b λ 3 = 976.3,

b

p 1 ({1}) = 0.289, p b 1 ({2}) = 0.332, p b 1 ({3}) = 0.379, b

p 2 ({1}) = 0.306, p b 2 ({2}) = 0.298, p b 2 ({3}) = 0.396, b

p 3 ({1}) = 0.448, p b 3 ({2}) = 0.444, p b 3 ({3}) = 0.109, b

µ({1}) = 1.003, µ({2}) = 1.001, b µ({3}) = 1.000, b b

σ(1, {1}) = 1.013, b σ(1, {2}) = 1.065, b σ(1, {3}) = 1.016, b

σ(2, {1}) = 19.17, b σ(2, {2}) = 19.85, b σ(2, {3}) = 20.83, b

σ(3, {1}) = 191.9, b σ(3, {2}) = 191.2, b σ(3, {3}) = 0.472, ξ(1, b {1}) = 0.356, ξ(1, b {2}) = 0.298, ξ(1, b {3}) = 0.251, ξ(2, b {1}) = 0.504, ξ(2, b {2}) = 0.437, ξ(2, b {3}) = 0.433, ξ(2, b {1}) = 0.957, ξ(2, b {2}) = 0.948, ξ(2, b {3}) = 0.443

^.

Onceagain,resultsarecorrectbecausewehaveenoughenvironmentprocesschangesduring

our observation period, see Figure 12. Results are slightly less accurate than in the 2-

dimensional case, forexample regarding

λ 1

^. ^Note ^that ^even ^if^results^would ^be ^completely

(24)

reconstructionresultsareacceptablefor

3

^lines^and

3

^states^of^theenvironment.

0 5 10 15 20 25 30

1.0 1.5 2.0 2.5 3.0

0 5 10 15 20 25 30

0 500 1000 1500

0 5 10 15 20 25 30

0 500 1000 1500

0 5 10 15 20 25 30

0 200 400 600 800 1000 1200 1400

Figure 10: The counting processes: top left: the true process

J

^, ^top ^right: ^the ^counting

process related to

S 1

^, ^bottom ^left: ^the ^counting ^process ^related ^to

S 2

^, ^bottom ^right: ^the

countingprocessrelatedto

S 3

Acknowledgments

TheauthorsthankverymuchAlexandreYouforhisvaluablehelponthisproject,andJean-

Baptiste Gouere for a useful comment. The second author acknowledges partial support

from the research chair ActuariatDurable sponsoredby Milliman, from the research chair

ActuariatResponsable sponsored byGenerali, and from French Research National Agency

(ANR)under thereferenceANR-08-BLAN-0314-01.

References

[1] Asmussen, S. (1989). Risk theory in a Markovian environment. Scand. Actuar. J. 2,

69100.

[2] Asmussen,S. (2000).Ruinprobabilities,WorldScientic.

[3] Baum,L.E., Petrie,T. (1966).Statisticalinferenceforprobabilisticfunctions of nite-

(25)

0 5 10 15 20 25 30 1.0

1.5 2.0 2.5 3.0

0 5 10 15 20 25 30

0 1.0e+5 2.0e+5 3.0e+5 4.0e+5 5.0e+5 6.0e+5 7.0e+5

0 5 10 15 20 25 30

0 1.0e+5 2.0e+5 3.0e+5 4.0e+5 5.0e+5 6.0e+5 7.0e+5 8.0e+5

0 5 10 15 20 25 30

0 5000 10000 15000 20000 25000 30000 35000 40000

Figure11: Thelossprocesses

S k

^,^top^left: ^the^true^process

J

^,^top^right: ^the^loss^process

S 1

^,

bottomleft: thelossprocess

S 2

^,^bottom^right: ^the^loss^process

S 3

0 5 10 15 20 25 30

1.0 1.5 2.0 2.5 3.0

0 5 10 15 20 25 30

1.0 1.5 2.0 2.5 3.0

Figure12: ReconstructionofthehiddenMarkovprocess

J

^: ^top: ^the^true^process

J

^,^bottom:

thereconstructedprocess

J b

(26)

inthestatisticalanalysisofprobabilisticfunctionsofMarkovchains.Ann.Math.Statist.

41,164171.

[5] Bickel, P.J., Ritov, Y., Rydén, T. (1998). Asymptotic normality of the maximum-

likelihoodestimatorforgeneralhiddenMarkovmodels.Ann.Statist.26(4),16141635.

[6] Buch-Larsen,T.,Nielsen,J.P.,Guillen,M.,Bolanc,C.(2005).Kerneldensityestimation

forheavy-taileddistributionsusingtheChampernownetransformation.Statistics39(6),

503516.

[7] Cai,J.,Tan,K.S. (2007).Optimalretentionforastop-lossreinsurance undertheVaR

andCTEriskmeasure.ASTINBull. 37(1),93112.

[8] Caliebe, A. (2006). Properties of the maximum aposterioripath estimator in hidden

Markovmodels.IEEETrans.Inform. Theory 52(1),4151.

[9] Champernowne,D.G.(1936).TheOxfordMeeting,September2529.Econometrica5,

October1937.

[10] Champernowne,D.G. (1937).Thetheoryofincomedistribution,Econometrica5,379

381.

[11] Chiragiev,A.,Landsman,Z.(2007).MultivariateParetoportfolios: TCE-based capital

allocationanddivideddierences.Scand.Actuar. J.2007(4),261280.

[12] Çinlar,E.(1975).Introduction tostochastic processes,Prentice-Hall.

[13] Davison, A.C., Ramesh, N.I. (1993). A stochastic model for times of exposures to air

pollutionfromapointsource,inStatisticsfor theenvironment, editors: V.Barnettand

K.F. Turkman,Wiley,NewYork.

[14] Deng, L., Mark, J.W. (1993). Parameter estimation for Markov modulated Poisson

processesviatheEMalgorithm withtimediscretization.Telecomm. Syst.1,321338.

[15] Fischer, W., Meier-Hellstern, K.S. (1993). The Markov-modulated Poisson process

(MMPP)cookbook.Perf. Eval.18,149171.

[16] Guillou, A., Loisel, S., Stuper, G. (2011). Asymptotic normality of the maximum

likelihood estimator of a loss process, available on the webpage http://www-irma.u-

strasbg.fr/guillou/supplement.pdf.

[17] Gusella, R. (1991). Characterizing the variability of arrival processes with indexes of

Estimation of the parameters of a Markov-modulated loss process in insurance

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Estimation of the parameters of a Markov-modulated loss process in insurance

Armelle Guillou, Stéphane Loisel, Gilles Stupfler

To cite this version:

Armelle Guillou, Stéphane Loisel, Gilles Stupfler. Estimation of the parameters of a Markov-

modulated loss process in insurance. Insurance: Mathematics and Economics, Elsevier, 2013, 53,

pp.388-404. �10.1016/j.insmatheco.2013.07.003�. �hal-00589696�

(1)

(2)

(1)

(1)

(2)

(N, L)

N

L

N

(N, L)

N

L

N

(J, N )

J

L

{1, . . . , r}

r ∈ N \ {0}

N

J

i

N

λ i

S = (S 1 , . . . , S n )

S k

N

S k

N

N

t

J

i

S k 1 , . . . , S k p

t

p i (e)

e = {k 1 , . . . , k p }

{1, . . . , n}

E s

S k

k ∈ E s

s−

N

(J, N )

X s

P θ(i, e)

( P θ ) θ∈Θ

P (X s = x | J (τ s ) = i, E s = e) = P θ(i, e) (∀ m, m ∈ e ⇒ X m = x m )

τ s

s−

N

x m = 0

m / ∈ e

(J, N )

(E s )

X s

S

T

r

J

N

S

0

T

N

` ij

L

J

λ i

P _{θ(i, e)}

( P _θ ) θ∈Θ

P (X s = x | J (τ s ) = i, E s = e) = P _{θ(i, e)} (∀ m, m ∈ e ⇒ X m = x m )

P _Φ

f ij (t, Φ) dt := P _Φ (T 1 ∈ dt, J(t) = j | J (0) = i) F ij (t, Φ) := P _Φ (T 1 > t, J(t) = j | J (0) = i).

P _{θ(·, e,} _Φ) (X = x) = diag(( P _{θ(i, e,} _Φ) (X = x)) 1≤i≤r ),

∀ e ⊂ {1, . . . , n}, e 6= ∅ , g(t, e, x, Φ) = f (t, Φ) · p(e, Φ) · P _{θ(·, e,} _Φ) (X = x) g(t, ∅ , x, Φ) = f (t, Φ) · p( ∅ , Φ) · 1l {x=0} .

∀ e ⊂ {1, . . . , n}, e 6= ∅ , g ij (t, e, x, Φ) = f ij (t, Φ) p j (e, Φ) P _{θ(j, e,} _Φ) (X = x) g ij (t, ∅ , x, Φ) = f ij (t, Φ) p j ( ∅ , Φ) 1l {x=0} .

P (Φ) = (Λ(Φ) − L(Φ)) ⁻¹ Λ(Φ).

P _Φ −

P _Φ

Φ ⁰

e (∀ j λ j (Φ) p j (e, Φ) = 0) ⇔ (∀ j λ j (Φ ⁰ ) p j (e, Φ ⁰ ) = 0).

Φ ⁰

P _Φ

P _Φ ₀