A Claims Persistence Process and Insurance

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Pierre Vallois, Charles Tapiero. A Claims Persistence Process and Insurance. Insurance: Mathematics and Economics, Elsevier, 2009, 44 (3), pp.367-373. �10.1016/j.insmatheco.2008.11.009�. �hal-00591746�

DOI: 10.1016/j.insmatheco.2008.11.009 Reference: INSUMA 1346

To appear in: Insurance: Mathematics and Economics Received date: May 2008

Revised date: September 2008 Accepted date: 13 November 2008

Please cite this article as: Vallois, P., Tapiero, C.S., A claims persistence process and insurance.

Insurance: Mathematics and Economics(2008), doi:10.1016/j.insmatheco.2008.11.009

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1 R e v i s ed

S e p t e mb e r 2 7 , 2 0 0 8

A Claims Persistence Process and Insurance

Pierre Vallois

Département de Mathématiques, Université de Nancy I B.P. 239 - 54506 Vandoeuvre les Nancy Cedex - France

Charles S. Tapiero,

Department of Finance and Risk Engineering New York University Polytechnic Institute, New York

Keywords : Random walk, Persistence, Insurance Claims, Value at Risk

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22 A B S T R A C T

The purpose of this paper is to introduce and construct a state dependent counting and persistent random walk. Persistence is imbedded in a Markov chain for predicting insured claims based on their current and past period claim. We calculate for such a process the probability generating function of the number of claims over time and as a result are able to calculate their moments.

Further, given the claims severity probability distribution, we provide both the claim’s process generating function as well as the mean and the claim variance that an insurance firm confronts over a given period of time and in such circumstances. A number of results and applictions are then outlined (such as a Compount Claim Persistence Process).

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2 1 . In t r o d u c t i o n

Insured claims may be time dependent. For example, an insured who has claimed one year might be more careful the following year and thereby alter the probability that a claim occurs. By the same token, an insured whose health has been impaired one year may be more prone to claim subsequently. Such persistence recurs in many instances motivated and implied in insured attitudes and expectations as well as recurring due to events inherent to the claim process (see for example Denuit, et al., 2006). Questions such as, does car failure induce additional and subsequent car failures? Are terrorist attacks correlated (Telesca and Lovallo, 2006), one attack defining the propensity for a subsequent attack or a respite ? Does the severity of hurricanes one year imply the severity of the following year? etc., are obvously questions of importance for insurers. Similarly, does a patient relapses following a treatment or not? Do stock prices have an inertia, tending to increase (or decrease) following increases or decreases with the same probabilities? Are fear regarding financial markets persistent? (The Financial Times, February 7, 2008, p.28). Most actuarial counting approaches assume a Poisson distribution, implying that events are independent. In other words, prior events do not alter the basic probabilities laws that determine the occurrence of subsequent events. To circumvent this lacuna, the credibility theory approach in actuarial science, evaluates the objectivity and the subjectivity of a source—

the insured that may potentially claim, and devises a statistical “learning” meachanism that allows the updating of the underlying claim probability. Using Bayesian statistics for example, credibility theory divides insured into classes that have various propensities to claim, which are updated using subjective prior estimates of risk classes and an accrued experience—the claim history of insured which is observed. The goal of credibility theory is then to set up an experience rating system to determine next year's premium, accounting for the individual and the collective group experience. Unlike credibility theory, this paper presumes that there may be an inherent persistence in an underlying process that will dictate the probability laws with which subsequent events occur. The probability of a subsequent claim for example, will then be determined by the past memory (in our case, the single past event of a claim or no claim) rather than be determined by a statistical estimator based on the accrued evidence of past claims.

Explicitly, while credibility theory seeks to integrate “experience” in estimating the propensity to claim, a persistence to claim is an inherent property of the underlying

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3

claiming (stochastic process) that determines, conditionally on the “claim memory”, the actual probabilities with which an insured will claim or not. For example, if an insured claiming (or not) one year will be more careful (or less careful) the following year, then the underlying claiming process will account for such a behavior in defining the effects of the “process memory” on the claiming stochastic process. The credibility approach however, will use the fact that a claim has been made to revise the probability that he will claim again in the following year with a credibly (larger) probability. In this sense these approaches differ fundamentally.

The purpose of this paper is to consider such a counting persistent process based on the Markov memory of the immediate and past event (and not only its state) and claculate the generating function of such a process (for example, see Patlak [1953], Weiss and Rubin [1983], Balinth [1986], Claes and Van den Brock [1987], Weiss [1994, 2002], Pottier [1996], Vallois and Tapiero [2007] for prior research on such processes). The resulting persistent counting process will be shown on the one hand to depart from the Poisson counting process and recognize the effects of the past memory on the claim process. Given such a process, we assess the effects of persistent claims on an insurance contract and use such observation to better assess the risk premium needed to compensate its risk exposure. The result obtain in this paper extend the results obtained in Vallois and Tapiero [2007]. Explicitly, we extend our previous result by providing the probability generating function of the persistent claim process, its moments as well as an explicit expresison for the probabilities of such a process. In particular, the process kurtosis due to persistence is expressed explicitly. Finally, a general expression for Persistent Compound processes is provided and an explicit recursive equations for its probability moments is given explicitly, generalizing thereby the often used Compount Poisson process. Numerical analyses are then used to highlight the effects of persistence counting compared to traditional counting and claiming processes.

2. The Markov Memory Based Persistent Counting Process

A s s u me t h a t a r e p r e se n t a t i v e i n s u r ed c a n i n a n y o n e y e a r c l a i m o r no t, d eno ting t h e s e ev e n t s b y ( 0 , 1 ) . T h e e v en t « 0 » s t a t e s t h a t n o c l a i m i s m a d e w i t h i n t h e y e a r w h i l e a « 1 » s t a t e s t h at t h e i n s u r e d h a s f i l e d a c l a i m d u r i n g t h e y e a r . C l a i m r e c o r d s i n d i c a t e t h a t w h en a c l a i m i s ma d e i n a g i v e n y e a r , t h en t h e f o ll o wi n g y e a r , t h e p r o b a b i l ity o f n o cl a im i s β >0. W h e n a c l a i m i s n o t ma d e

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4

i n a g i v en y e ar , th en t h e p r o b ab i li ty o f a c l a i m b e i n g ma d e in th e f o l lo w i n g y e a r i sα. T h i s d e f i n e s a s i mp l e t w o - s t a t e s M a r k o v c h a i n , g iv e n b y :

( 1 ) 1

, 0 , 1 1

α α β β α β

⎡ − ⎤

=⎢⎣ − ⎥⎦ < <

P

I f w e d en o te b y y_t t h e v a l u e s t h e r a n d o m e v en t c a n a s s u me a t t i me t t h en :

( 2 )

( ) ( )

(

¹1

) (

¹ 1

)

1 0 , 0 0 1

1 1 1 , 0 1

t t t t

t t t t

P y y P y y

P y y P y y

α α

β β

− −

− −

⎧ = = = = = = −

⎪⎨ = = = − = = =

⎪⎩

O r

( 3 ) ( ) ( ) ( 1 0) ( 1 1) 1

0 1

1

t t

t t

P y P y

P y P y α α

β β

− −

⎡ = = ⎤ −⎡ ⎤

⎣ ⎦

⎡ = = ⎤ = ⎢ ⎥

⎣ ⎦ ⎣ − ⎦

O v e r a p er i o d o f t ime t , t h e t o t al n u mb e r o f ev ent s ( c l ai ms ma d e , e t c . ) i s t h e r e f o r e g i v en b y :

( 4 )

0 t

t j

j

x y

=

=

∑

T h e n u mb e r o f c l ai ms a r e c o n d iti o n a l o n the i n i ti a l eve n t ( th e cur r e n t me m o ry o f th e e v en t ), d en o t ed b y : y₀=0 o r y₀=1. I n t h i s me mo r y -b as e d p e r si s t e n t co un ti ng pro cess, we cal cul a t e fi rst th e pro b ab il ity gen e rati ng fu n ct io n (PGF ), s u m ma r i z e d i n P ro p o s i t io n 1 and su b s equ en t ly u s e thi s P GF to o b t a in so me o f t h e c h a r a ct e r i s t i c s o f a n u n d e rly in g c l ai m p ro c e s s .

P r o p o si ti o n 1

L e t {x tt^, ≥⁰} be a counting random variable of the number of claim events in a time interval (0, )t . And let (α^,1−β) be the probabilities that an event occurs at time t, conditional on its current (or not) occurrence in the previous period t-1. Define ^{ρ= − −1 α β ρ, ∈ −}] [^1,1 , as a “persistence index. Let ^{G( , )λ t =E}

{ }

^λxt be the probability generating function for the persistent counting process, for counting the number of claims, given by equation (4), in a time interval ( )^{0,t and let} P x( 0=0) be the probability that initially no claim is made while P x( ₀=1) denotes the probability that a claim was made in itially and set for notational convenience. Then:

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(5) ( ,1) ( , 0) ₁

( , ) G _t G _t , 1

Gλ t λ λρ λ t

δ δ ⁻

= Ψ − Ψ ≥

with

(6) ( ) ⁰ 0⁰

(

( )( )

)

0

( ,0) ( 0) ( 1)

,1 (1 ) ( 0) 1 1 ( 1)

G P x P x

G P x P x

λ λ

λ α λα λ λ α ρ

= = + =

= − + = + + − + =

where

( )

( )

2

0 1

0

1

1 ,

( ) 4

1 ,

2 1 2

t t

t

a a

a a α λα

δ λρ λρ

μ μ

μ λρ δ

μ λρ δ

= − +

= + −

Ψ = −

= + +

= + −

P r o o f : S e e Ap p en d i x 1

S u ch a g en e r a ti ng fu n ct io n pro vi d e s t h e me a n s t o c a l c u l a t e h i g h e r o r d e r mo me n t s of th e un d er ly i ng p er si s t e nt c o u n t i n g c l a i m p r o c e s s a s w e l l a s t h e p ro b ab il it ies o f th e n u mb er of cl ai ms. F u rt h er, it cl early po in t s o ut to th e e f f e c t s o f t h e p e r s i st e n c e i n d ex o n t h e c o u n t i n g p r o c e s s . W h e n ρ=0 then β= −1 α and whatever the previous outcome (whether a claim or no claim), the subsequent probability to claim is α while that of no claim is 1−α. When the persistence index is positive, ρ>0 then β = − −1 α ρ. That is, if in a given year a claim is made, then the probability that in a subsequent year a claim is made has a smaller probability. And vice versa, when the persistence index is negative ρ<0 and the underlying stochastic proces s would point out to a “contagious” claim process (for example, with Hurricanes of a high category following Hurricanes of High category).

Inversely, for a positive persistence index ρ>0 it will indicate that the underlying claim process has a built-in “incentive effect”, reducing a claim probability in a given year following a claim made in the previous one.

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6

O u r r e s u l t s i n p ro p o s it i o n 1 a ll o w a n e x p lici t cal cul a ti on o f th e mo men t s o f t he p e rsistent co un ting pro cess, d efin e d recu r s iv e ly a s s how n b e lo w . N ot e th a t:

(

¹

)

( , ) _t ^xt

G λt E xλ λ

∂ = −

∂ , ²G2( , )

(

_t( _t 1) ^{xt 2}

)

t E x x

λ λ

λ

∂ = − −

∂ , ³G3( , )

(

_t( _t 1)( _t 2) ^{xt 3}

)

t E x x x

λ λ

λ

∂ = − − −

∂

a n d s o o n f o r h ig h e r o r d e r t e r ms . U s i n g t h e s e t e r ms a n d s e t t i n g λ=1, w e o b t a in t h e n e c e s sa r y e q u a t i o n s w h i ch al l o w t h e c al c u l at i o n o f t h e me a n , t h e v a r i an c e , t h e k ur to s is a n d o th e r mo me n t s o f th e p e r si s t ing c ount i ng p ro c e ss d i s tr ibu t io n.

I t i s al s o u s e f u l to d er i v e a n u mb e r o f sp e ci a l an d w ell k n o w n c as e s t o con fi r m t h e v a li d i ty o f o u r r es u l t s. Fi r st n o t e t h at w h e n λ=1 t h en G(1, ) 1t = a s e x p e cte d . F u r t h er , wh e n t h e r e i s n o p e r si s t e n c e ( i . e ρ=0) , w e h a v e ( ,1)

( , ) G _t

G t

a

λ = λ Ψ ,

0 a, 1 0 and _t a^t

μ = μ = Ψ = a n d t h e r e f o r e : ( 7 ) G( , )λt ={P x( 0= +0) λP x( 0=1)}a^t

I n p a rt i cu la r , i f in i tia l ly , P x( ₀=0)= 1 , t h e n G( , )λt =a^t w h i c h c o r r e sp o n d s a s e x p e ct e d t o t h e P r o b a b i l i t y G en e r a t i n g F u n ct i o n o f a b i n o mi a l d i s t r i b u t i o n . H o w e v e r , i f P x( ₀= =1) 1, th e n G( , )λt =λa^t .

A co nv en ien t recu rsive ex pression fo r th e gen e rating fu n ctio n can b e foun d by n ot in g th a t Ψ =_t μ₀^t− μ₁^t w h er e μ₀ a n d μ₁ solve ^{μ2−μ λ ρ α}( ^{( + ) 1+ −α})^+λρ=0,

( p rov e d in A p p end ix 1 ) and v e rify th e s e con d o r d er equ a t ion : ( 8 ) Ψ_t+2−(a+λρ)Ψ +_t+1 λρΨ =_t 0

As a resu lt, (5 ) i mp lies t h at th e p ro b ab il ity g e n erat ing fun c t ion G( , )λt s a t i s f i e s a s w e l l t h e s e c o n d o r d e r re c ur s iv e eq u at ion g iv en by :

( 9 ) ^{G( ,}λ^t+²⁾= − +(¹ α λ α ρ⁽ + ⁾)^{G( ,}λ^t+ −¹⁾ λρ λ^{G( , )t}

De r i v i ng ( 9 ) w ith r es p e ct to λ w i t h λ=1 w e o b t ai n a r e c u r si v e e x p r e s si o n f o r t h e mo me n t s o f t h e co u n t i n g p r o c e s s . C o n c e n t r a ti n g o u r at t en ti o n o n th e f i r s t mo me n t s on ly , d er iv at i v e s o f ( 9) y i e ld s t h e f o ll o wi ng re c u r s iv e equ a t ion :

( 1 0 )

( )

( )

1 1

1 1

( , 2) 1 ( ) ( , 1) ( , )

( , 2) ( , 1) ( , )

1 ( )

( , 1) ( , )

( ) , 1, 2,3,....

k k k

k k k

k k

k k

G t G t G t

G t G t G t

G t G t

k k k

λ α λ α ρ λ λρ λ

λ α λ α ρ λ λρ λ

λ λ λ

λ λ

α ρ ρ

λ λ

− −

− −

+ = − + + + −

∂ + = − + + ∂ + − ∂

∂ ∂ ∂

∂ + ∂

+ + − =

∂ ∂

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7 W i th i n i ti al co n d it io n s :

( 1 1 )

( )

(

( )( )

)

( )( )

( )

( )

0 0

0 0

0 0 0

2 2

2 2 0

( ,0) ( 0) ( 1)

,1 (1 ) ( 0) 1 1 ( 1)

( , 0) ( ,1)

( 1), ( 0) 1 2 1 ( 1)

( ,0) ( ,1)

0, 2 ( 1)

( , 0) ( ,1)

0, 0, for 3

j j

j j

G P x P x

G P x P x

G G

P x P x P x

G G

P x

G G

j

λ λ

λ α λα λ λ α ρ

λ λ α λ α ρ

λ λ

λ λ α ρ

λ λ

λ λ

λ λ

= = + =

= − + = + + − + =

∂ = = ∂ = = + + − + =

∂ ∂

∂ = ∂ = + =

∂ ∂

∂ = ∂ = ≥

∂ ∂

T h e s e in i tia l c o n d it ion s a r e s p ec i f i ed u s ing e q u a t io n ( 6 ) i n Pr o p o si ti o n 1 . A t λ= 1 , w e c an w r i t e t h e s e e x p r e ss i o n s i n t h e f o l l o w i n g ma n n e r ( t o g e t h e r w i t h e q u at io n (9) ) wh i ch simp l i f i e s t h e i r n u me r i ca l so lu t io n :

(12 )

( )

( ) ¹ 1 ¹ 1

(1, 2) 1 (1, 1) (1, )

(1, 2) (1, 1) (1, ) (1, 1) (1, )

1 ( ) ,

k k k k k

k k k k k

G t G t G t

G t G t G t G t G t

k k

ρ ρ

ρ ρ α ρ ρ

λ λ λ λ λ

− −

− −

+ = + + −

∂ + = + ∂ + − ∂ + + ∂ + − ∂

∂ ∂ ∂ ∂ ∂

W h il e t h e in i ti a l c o n d i t io n s s t at ed ab o v e in e q u at io n (11 ) and l e a d i n g t o :

( 1 3 )

( )

( )

( )

0 0

2 2

2 2 0

(1, 0) 1, 1,1 1

(1, 0) (1,1)

( 1), 1 ( 1)

(1,0) (1,1)

0, 2 ( 1)

(1, 0) (1,1)

0, 0, for 3.

j j

j j

G G

G G

P x P x

G G

P x

G G

j

α ρ

λ λ

λ λ α ρ

λ λ

= =

∂ = = ∂ = + + =

∂ ∂

∂ = ∂ = + =

∂ ∂

∂ = ∂ = ≥

∂ ∂

S i mi l a r ly , w e c a n c al c u l at e th e p ro b ab il iti e s o f p e r si s t e nt co unt i ng p ro c e ss by s e t t in g λ= 0 i n t h e d er iv a t iv e s o f t h e g en e r at i ng f un c tio n s. I n thi s c a s e , the p ro b ab il it ie s a r e g iv en by :

1 ( , )

( ) , 0,1, 2,3,...,

!

i

i i

G t

p t i t

i λ λ

= ∂ =

∂ wi th :

( 1 4 )

( )

( )

1 1

1 1

(0, 2) 1 (0, 1)

(0, 2) (0, 1)

1

(0, 1) (0, )

( ) ,

k k

k k

k k

k k

G t G t

G t G t

G t G t

k k

α

λ α λ

α ρ ρ

λ λ

− −

− −

+ = − +

∂ + = − ∂ +

∂ ∂

∂ + ∂

+ + −

∂ ∂

W i th t h e ini t i al co n d iti o n s ( s p e cif i e d b y e q u a t i o n (6 ) ):

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8 ( 1 5 )

( )

( )

( )

0 0

0 0 0

2 2

2 2 0

(0,0) ( 0), 0,1 (1 ) ( 0)

(0, 0) (0,1)

( 1), ( 0) 1 ( 1)

(0,0) (0,1)

0, 2 ( 1)

( , 0) ( ,1)

0, 0, for 3.

j j

j j

G P x G P x

G G

P x P x P x

G G

P x

G G

j α

α α ρ

λ λ

λ λ α ρ

λ λ

λ λ

= = = − =

∂ = = ∂ = = + − − =

∂ ∂

∂ = ∂ = + =

∂ ∂

∂ = ∂ = ≥

∂ ∂

T h e s e e q u at i o n s d ef i n e a n u me r i c a l a p p ro ac h t o c a l cu l a tin g b oth t h e mo m e n t s a n d th e pro b ab il it i e s o f a p e r s is t e n t p ro c es s . A mo r e d i r e c t app ro a c h w ill b e o ut li n ed sub s e qu en t ly h o w ev e r.

E x p l i ci t r es u l t s fo r the f i r st t w o mo me n t s ar e p r o v id ed b e lo w wit h p r o o f s fo u n d d i r e ct ly f rom e q u a t ions ( 12 ) an d ( 13 ) .

P r op o si ti on 2 L e t x₀=0, th en:

( 1 6 ) ^{E x t}( ^{( )}) (= +¹ ρ) (^{E x t(} −¹⁾)−ρ^{E x t}( ⁽ −²⁾)+α^{, t}≥² ( 1 7 )

(

²

)

^{( )}

(

²

) (

²

)

1

( ) 1 ( 1) ( 2)

(1 )

2 1 2 , 2

1 1

t t

E x t E x t E x t

t t

ρ ρ

α α ρ α α ρ ρ

ρ ρ

+

= + − − − +

⎡ + + − ⎛ + − − ⎞⎤ ≥

⎢ ⎜ ⎟⎥

− ⎣ ⎝ − ⎠⎦

In th ese equ a tion s, note th at we h a v e as exp ected (Vallois and Tap i ero [2007 ]) :

( 1 8 ) ( ^{( )}) ^{1 1 1}

1 1

t

E x t α t ρ

ρ ρ

⎡ − + ⎤

= − ⎢⎣ + − − ⎥⎦

T h i s cl e a r l y i n d i c a t e s t h e n o n l i n e a r t i me e f f e c t s o f p er s i s t en c e i n s u ch c o u n t i n g p ro c e s s e s . A v e r i fi ca t i on o f (17 ) c a n al so b e r e a ched . Fi r s t n ot e th a t w h e n t h e r e i s n o p er s i s t e n c e , t h e n ρ= 0 and ^{E x t}

(

^{( )2}

) (

^{=E x t( −1)2}

)

^+α[^{1 2 (+ α t−1)}]^.

S u mmi n g f o r 1≤ ≤t n, w e o b t a in ^{E x n}

(

^{( )2}

)

^=nα

(

^1+α(ⁿ⁻¹)

)

. S i n c e ( )x n h a s a b in o mi a l d i s tr ib ut ion B n( , )α , we ha v e E x n( ( ))=nα, ^Var(^{x n( )})^{=nα(1−α) a n d}

t h e r eby : ^{E x n}

(

^{( )2}

)

^{=nα(1−α)+n2α2=nα}

(

¹⁺(ⁿ⁻¹)^α

)

a s e x p e c t ed .

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9

E xp l i ci t exp r e s s ion s fo r th e p e r si s t e nt c ount i ng p ro b ab i li ti e s c a n b e d et e r mi n e d a s w e l l u s in g t h e r e cu r s iv e p r o b a b i li ty g en e r a ti n g fun c t io n s . In th i s c a s e, w e c a l c u l at e (p tk^{( );0}≤ ≤k t) b y r e cu r r en c e . I n i t i a l l y t h e s e a r e s p e c i f i ed b y :

( 1 9 ) p₀(0) 1, (1) 1= p₀ = −α, (1)p₁ =α

F u rt h er , (p tk⁽ +^2);0≤ ≤ +k t ²) i s d e fi n ed a s a fu n ct io n o f (p tk⁽ +^1);0≤ ≤ +k t ¹) and

(p tk^{( );0}≤ ≤k t) b y u s i n g t h e r e cu r si v e e q u a t i o n

( 2 0 ) p t_k( +2)= −(1 α)p t_k( + +1) (α ρ+ )p_k−1(t+ −1) ρp_k−1( ),t

f o r a l l 0≤ ≤ +k t 2, and by co nv en t ion , we se t p t_k( )=0 i f k<0 o r k>t.

S ub s e qu en t c a l c u l at ion s w il l indi c a t e th e un d er ly in g p ro c e s s pro b ab il it i e s . I n p a r ti c ul a r, w e h av e for th e f i r st 3 p ro b abi l it i e s :

( 2 1 ) ( )

( )

0

2 1

2

2 4 2

( ) (1 )

( ) (1 ) (1 ) , 1

( )(1 )

( ) 1 , 2

(1 ) (1 )( 2) (1 ) 2

2

t t

t

t

p t

p t t t

p t t

t t

α

α α α ρ ρ

α ρ α

α α α ρ α α ρ α ρ α αρ

−

−

−

= −

= − − − + ≥

⎧ + − ⎫

⎪ ⎪

= ⎨ ⎬ ≥

+ − − − − − − + + − +

⎪ ⎪

⎩ ⎭

O f c o u r s e, w h e n t h e r e i s n o p e r si s t e n c e , t h i s i s r e d u c ed a s ex p e ct ed t o :

(22)

0

1 1

2 2

2

( ) (1 )

( ) (1 ) , 1

( ) 1 (1 ) ( 1), 2

2

t t

t

p t

p t t t

p t t t t

α

α α

α α

−

−

= −

= − ≥

= − − ≥

3 . A p p l i ca ti o n : The Compound Counting Persistent Process I n ma ny a pp l i c at ion s , c o un ti ng i s u s ed fo r su m mi n g eve n t s th a t are i nd ep e nde n t ( f o r e x a mp l e , t h e C o mp o u n d P o i s s o n p r o ce s s e s ) . E x p e c ted l y , w h e n c o u n t i n g i s p e r s i st e nt ( a nd th e r e fo r e d e p ende n t ), su c h a s t a t i st i c al c h a r a ct e ri s t i c h a s to b e a c c o u n t ed f o r . T h e an a l y t i c a l r e s u l t s r eg ar d i n g t h e co u n t i n g p er s i s t en t p r o c e s s c a n b e u se d a n d a p p l i ed t o n u me r o u s p r o b l e ms . F o r ex a mp l e , s ay t h at a n i n su r an c e f i r m s e e k s t o c a l c u l a t e t h e s u m o f c l a i ms ( i n d ep en d en tly di s tr ib ute d b ut d e p ende n t on t h e p e r si s t en t c ou nt ing p ro c e s s — an d th e r e for e th e Pois s o n d i s tr ib ut ion c a nno t be u s e d) . Le t th e s u m o f cl a i ms b e

1 xt

t j

j

ξ Z

=

=

∑

^% w h e r e c l a i ms

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Z%i a r e s t a t i s t i c a l l y i n d e p en d e n t w i t h me a n μ=E Z

( )

%i a n d k no wn v a ri a n c e

( )

var Z%_i . T h en , for in d ep en de n t ( Po i s son ) cou nt ing p ro c e s s es , th e me a n c l ai m s u p to t i me t a r e:

( )

( )

1 xt

i t i

i

E Z E x E Z

=

⎛ ⎞=

⎜ ⎟

⎝

∑

^%⎠ ^% w h i l e c l aims v a r i a n ce i s g iv en by :

( )

^{2 ( )}

( )

1

var ( )

xt

i i t i t

i

Z E Z Var x Var Z E x

=

⎛ ⎞ ⎡= ⎤ +

⎜ ⎟ ⎣ ⎦

⎝

∑

^%⎠ ^{% % .}

H o w e v e r, w h e n cl a i ms a r e p er s i st e n c e -d ep en d en t a mo r e g en e r a l e x pr e s s io n c a n b e fou n d . E x p l i ci tly , c o n si d er t h e f o l lo w in g r a n d o m c l a i ms :

(23) ⁰ _{{ 0}} ¹_{{ 1}}

0 0

1 1

i i

t t

t i y i y

i i

S Z ₌ Z ₌

= =

=

∑

^% +

∑

^{% where}

0 t

t i

i

x y

=

=

∑

whereZ%_i⁰ is a “normal claim” occurring in any regular period (defined by the fact that no specific event has occurred) while Z%¹_i is a “large claim”

(of course, if Z%_i⁰=Z%_i¹ then S_t=ξ_t as stated above). We assume that

{

Z% %i^0,Z¹i

}

are random variables independent of each other and independent of the Markov (persistent) claims. In this case, the compound claim mean and variance and the claim probability generating function are given by the following (with proofs provided in the appendix):

(24) ^{E S}( )^{t = +(t 1)E Z}

( )

^{%10 +}

(

^{E Z}

( ) ( )

^{%11 −E Z%10}

)

^{E x( )t}

(25 )

( ) ( ) ( ( ) ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

2 2

2 0 0 2 1 0

1 1 1 1

1 0 0 1 0

1 1 1 1 1

( 1) var ( 1)

var v ar 2( 1)

t t

t

E S t Z t E Z E x E Z E Z

E x Z Z t E Z E Z E Z

⎡ ⎤ ⎡ ⎤

= + ⎢⎣ + + ⎥⎦+ ⎣ − ⎦

⎡ ⎤

+ ⎣ − + + − ⎦

% % % %

% % % % %

W h i l e t h ei r L a p l a c e T r a n s f o r m i s:

( 2 6 )

( ) ( )

^{( )}

( )

( )

1 1 0

0 1 1

, ,

t i

Z

S Z t

Z

E e

E e E e G z t z

E e

λ

λ λ

λ + −

− −

−

⎡ ⎤

=⎢⎣ ⎥⎦ =

%

%

% , λ≥0

w h e r e ^{G z t}( )^, i s a p rob a bi li ty g e n er a t ing fun c ti on :

( )

^{1 ( )}

0

( , ) ^t

t

x i

t i

G z t E z z P x i

+

=

= =

∑

= ^.

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Note that when Z%₁⁰ i s nu l l, th e n, fr o m e qu a t io n (2 6) w e f i nd th a t : ( 2 7 ) E S( )t =E Z

( )

%1¹ E x( )t a n d ^{Var S}( )^{t =E x Var Z}( )^t

( )

^{%11 +Var x( )t}

(

^{E Z}

( )

^%11

)

²

T h e s e r e sul t s c a n t h en b e u s e d t o o b t ai n app r o x i ma t e p r i c e s f o r t h e p r e mi u m t o b e c h a r g ed ( b a s ed o n me a n – v a r i an c e r u l e s ) wh en t h e co u n t in g p ro ce s s i s p e r s i st e nt . N o t e h o we v e r t h at th e v a r i an ce o f th i s pr o c e s s h a s i n c r e a s ed o v er t i me d u e to th e p ro ce s s p e r s i sten c e . F urt h e r, b a s e d o n th e s e mo men t s , t h e V a l u e a t Ri s k ( V a R) r i s k exp o su r e c an b e d e t e r mi n ed w h i c h wo u l d u s e t h e s e t w o mo me n t s a s a f ir s t a pp rox ima t i o n ( a lth oug h h igher o rd e r mo men t s c a n b e c a l c u l at e d a s w e l l u s ing (26 ) , th e g e n e r a ti n g f u n c t io n o f t h e p ersi s t e n t Co mp o und Pro cess).

4 . S o m e N u me r i c a l R e su l t s

A nu mer i c a l a n aly si s o f ou r equ at i on s w il l r e v e a l so me o f t h e ch ar a c t e r i st i c s o f a p e r sist e n t p ro ce s s . A s e x p e ct e d, th e me a n e v ol ut ion of t h e p er s ist e n t p ro c e s s h as a n a l mo s t l in e a r g ro w th a s ind i c a t ed in o ur e qu a tio n. In th e l on g r u n , t h e v ar i a n c e t u r n s o u t t o b e a l s o a l mo s t l i n e ar , a s i t i s t h e c a s e f o r r a n d o m w a l k s . How e v e r , p e rsi s t e n c e (ρ> 0) has t he ef f ect of i n cr ea s i ng th e va r i an c e a s s h o wn in Fi gu r e -1 b elo w . I n th e s h o rt t e r m h o w ev e r , t h e v ar i an ce e v o lu ti on i s n on li n e ar as ou r e qu a ti on s h av e in di c a t ed .

I n t er e s t ingly , t h e r a te o f c h a nge i n v a ri a nc e i s n o t co n st a nt a nd g ro w ing o v er t i me w h i c h in d i c a t e s a “ p e r s i ste n t v o l a t ili ty ”. O f p a r ti c u l a r i n t er e s t i s t h e e v ol ut ion o f th e t hi rd mo men t o f th e p e r si s t e nt c l aim d i s t r ib uti on . I ni ti al ly , it w a s i n cr e a s i ng (ov e r 4 p e r iod s ) a n d s ub s equ e nt ly d e c li n ing ( al tho ugh r e mai n ing p o si ti v e f o r ρ p o si t iv e) . W h en ρ i s n eg a t i v e w e n o t e t h a t f o r t h e f i r st f e w p e r i o d s t h e e v o l u t i o n o f t h e me a n a n d t he v a r i an c e a r e i n d e e d n o n l i n e a r . T h i s is part icula r ly the c ase f or t he vari an ce as s ho w n in F igu r e 2 b e lo w . In t hi s F i g u r e, t h e v a r i an c e i n i t i a l l y d e c l i n e s, t h e n i n c r e as e s a n d ag a i n d e c r eas e s . F i n al ly , it c on v er g ed to a l in e ar gr o wt h. T hi s b eh av io r i s i ndi c a t iv e o f t h e s h o r t t e r m e f f e c t s o f me mo r y o n t h e s to c h as t i c p ro c e ss a s in d i c at e d e ar l i er . In F i g u r e 3 , w e n o t e t h e d i v er g en c e i n t h e g r o wt h o f v o lat i li ty whe n the p e r s i st e n c e p a r a met e r i s n e g at ive. F in al ly , t h e th ir d mo men t i s p o si ti v e w h i ch d e mo n s t r ate t h a t th e d i s t ri but io n i s sk ewe d r e f l e ct i ng th e pr o c e s s me mo r y -

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p e r s i st e n c e. T h i s l a t te r o b s er v ati o n i s p ar t icu l a rly i mp or t an t fo r i t may b e u s ed t o exp l a in p a r tly th e s k e w o f c er t a in ti me s e r i e s , p r esu mi n g th at th i s s k e w i s d u e to t h e sh or t t er m me mo r y eff ect s p r ev al en t in su ch seri es (fo r ex amp l e, in f i n an c i al t ime s e r i e s ) .

Figure 1: Persistence and Variance of the Counting process

Figure 2 : Rho=-0.3

Figure 3: Rho=-0.3

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Similarly, numerical analysis of some of the first persistent probabilities reveal a certain cyclicality, a function of the peristent indexr. As shown in Figures 4 and 5, the evolution of the claim probability is complex when there is persistence (memory) compared to that without persistence, oscillating initially and subsequently converging.

Figure 4: The Probability of a Claim, ρ=-0.3

.