Prediction for Risk Processes

Transcription

1 Predicion for Risk Processes Egber Deweiler Universiä übingen Absrac A risk process is defined as a marked poin process (( n, X n )) n 1 on a cerain probabiliy space (Ω, F, P), where he ime poins 1 < 2 < are he claim arrival imes of claims from a given porfolio of risks he marks X n are he claim amouns a ime n. If N () denoes he number of claims up o ime wih claim amoun in a orel se, hen (( n, X n )) n 1 can equivalenly be described by he family of processes (N ()) wih (R + ). Suppose ha (, ) is a measurable space, Θ a -valued rom variable, ha (F Θ ) is he filraion defined by F Θ σ(θ) σ({n s () : s, (R + ). Assume ha here is a family of (F Θ )-adaped processes (λ ()) ( (R + )) such ha all processes (N () λ s()ds) are local (F Θ )- maringales. hen (( n, X n )) n 1 is called a Θ-mixed risk process, for a number of reasons he rom variable Θ is called he porfolio srucure. Now suppose ha Z (Z ) is an (F Θ )-adaped process ha (F ) is a subfilraion of (F Θ ). he filering problem for Z given (F ) is jus he problem o deermine he process (E{Z F ), he predicion problem is he problem o deermine for a given h > he process (E{Z +h F ). For a number of relevan processes Z one can use a maringale propery inheried from he maringale propery of (( n, X n )) n 1 o solve he filering he predicion problem. A ypical example is he process (S ) ( (R + )) given by S n 1 X n1 {n 1 {Xn. In his case he process (S xλ s(dx)ds) is a local (F Θ )-maringale. 1 Mixed Risk Processes Le (E, E) be a measurable space le denoe an arificial elemen ouside of E. We se E : E { provide E wih he σ-algebra E : σ(e {{ ). Now suppose ha ( n ) n 1 is a claim arrival process on he probabiliy space (Ω, F, P), i.e. ha P-a.s. : 1 2 wih < n, if <, ha (X n ) n 1 is a sequence of E -valued rom variables such ha he following condiion holds: n X n. (1) 1

2 hen he double sequence (( n, X n )) n 1 is called a risk process wih claim space E. For n we make he convenion X : ɛ wih a fixed ɛ E. We will always assume in he following ha (E, E) is a polish space provided wih is orel field. In mos applicaions E R +, i.e. E R +. hen every X n is inerpreed as he claim size of he n-h claim (X n ) n 1 will be called he claim size process or he claim amoun process of he risk process. Le us denoe by M z +(X, X ) he space of all Z + -valued measures on a measurable space (X, X ).hen a risk process (( n, X n )) n 1 wih claim space E can be equivalenly described by he rom measure defined by N : (Ω, F, P) M z +(R + E, (R + ) E), N : n 1 δ (n,x n ). For E we se N () : N ([, ] ) n 1 1 {n 1 {Xn. hen every N () is a rom variable we can idenify he rom measure N wih he family ( (N ()) of sochasic processes. We will call ) E N ( (N ()) ) E he risk measure of he risk process (( n, X n )) n 1. Since (N (E)) is jus he claim number process of ( n ) n 1, we will also wrie N insead of N (E). Now le F (F ) be a given righ coninuous filraion on (Ω, F, P). A family Λ ( (λ ()) ) E of R +-valued sochasic processes is called an F-inensiy measure, if he following properies hold: (i) for every fixed E he process (λ ()) is an F-progressively measurable process wih values in R +. (ii) for every fixed, λ () is a finie measure on (E, E), (iii) for every, λ s (E) ds < P-a.s.. In he following we will jus wrie λ insead of λ (E). 2

3 Now suppose ha (( n, X n )) n 1 is a risk process wih associaed risk measure N ( (N ()) assume ha N is F-adaped. hen we say ha N ) E (or (( n, X n )) n 1 ) has he F-inensiy measure Λ ( (λ ()) ) E, if ( n Nn () λ s () ds ) (2) is an F-maringale for every n 1 every E. If he rom variables N : N (E) ( > ) are inegrable (in his case we will also say ha he risk process (( n, X n )) n 1 is inegrable), hen (2) jus means ha for all E he processes are F-maringales. ( N () λ s () ds ) (3) For E he measure λ () (d dp) is obviously absoluely coninuous relaive o λ (E) (d dp). Hence here exiss a Radon-Nikodym-densiy γ () relaive o λ (E) (d dp), i is no difficul o prove ha hese densiies can be chosen in such a way ha γ () is a probabiliy measure for all ω Ω. In case ha always λ (E) >, one can jus se In he following we assume always ha γ () : λ () λ (E). λ () γ ()λ, where γ is a probabiliy measure on (E, E). In his paper we will consider essenially wo filraions. he firs filraion is he canonical filraion F N (F N ) of N, defined by F N : σ ({ N s () s, E ). For he second filraion we ake a measurable space (, ), a measurable map Θ : (Ω, F, P) (, ), define he filraion F Θ (F Θ ) by F Θ : σ(θ) F N. Θ will shorly be called he porfolio srucure, if (( n, X n )) n 1 is a risk process wih a F Θ -inensiy measure Λ ( (λ ()) ) E, hen (( n, X n )) n 1 is called a Θ-mixed risk process. 3

4 Suppose ha G (G ) is a F Θ -adaped process. hen i is easily shown ha on he se { < n G only depends on ( k, X k ) k on Θ we express his dependence by he noaion G G (Θ, ( k, X k ) k ). We will also ofen make use of he following convenion: If a funcion f depends on ( j, x j ) j n, hen we will use freely he differen noaions f( 1,, n, x 1,, x n ), f(( j, x j ) j n ), or f(( i, x i ) 1 i k 1, ( j, x j ) k j n ) for 1 k n. For he purposes of his paper we need some furher regulariy properies of he F Θ -inensiy measure of a Θ-mixed risk process (( n, X n )) n 1. Λ ( (λ ()) ) E 1.1 Definiion. he F Θ -inensiy measure Λ is said o be regular, if he following condiion holds: here exiss a σ-finie measure γ on (E, E) such ha on { < n, such ha γ g ( ; Θ, 1,,, X 1,, X )γ (, x, θ, 1,,, x 1,, x ) g (x; θ, 1,,, x 1,, x ) is measurable. In case ha Λ is regular, he disribuion of (Θ, 1, 2,, X 1, X 2, ) can easily be compued. Denoe by β he disribuion of Θ suppose ha u 1,, u n R +, 1,, n E, C are given. hen P { 1 u 1,, n u n, X 1 1,, X n n, Θ C (4) u1 un G (n) (y, ( i, x i ) i n ) C 1 u n n γ(dx n )d n γ(dx 1 )d 1 β(dy), where he inegr G (n) (y, ( i, x i ) i n ) is given by G (n) (y, ( i, x i ) i n ) n { gi (x i ; y, ( j, x j ) j i 1 ) (5) i1 λ i (y, ( j, x j ) j i 1 )e R i i 1 λ s(y,( j,x j ) j i 1 ) ds. 4

5 here are also explici formulas for a number of imporan condiional disribuions. Define G (n,k) Θ,( i,x i ) i n (( j, x j ) 1 j k ) k { : gl (Θ, ( i, X i ) i n, ( j, x j ) 1 j l 1 ) (6) l1 λ l (Θ, ( i, X i ) i n, ( j, x j ) 1 j l 1 )e R l l 1 λ s(θ,( i,x i ) i n,( j,x j ) 1 j l 1 ) ds wih he convenion n. hen for n 1, k 1, u 1,, u k >, 1,, k E we have P { n+1 u 1,, n+k u k, X n+1 1,, X n+k k F Θ n u1 We remark ha n u 1 1 uk k 1 u k k G (n,k) Θ,( i,x i ) i n (( j, x j ) 1 j k ) (7) F Θ n σ(θ, ( j, X j ) j n ). γ(dx k )d k γ(dx 1 )d 1. here is also an explici formula for he condiional disribuion relaive o he σ- algebra F Θ for (cf. Deweiler [24]). For every n 1, k 1, u 1,, u k >, 1,, k E one has 1 { { < np n u 1,, n+k 1 u k, (8) X n 1,, X n+k 1 k F Θ u1 uk 1 { < n G,(,k) Θ,( i,x i ) i (( j, x j ) 1 j k ) 1 k u 1 where he densiy G,(,k) Θ,( i,x i ) i G,(,k) s k 1 u k is given by γ(dx k )d k γ(dx 1 )d 1, Θ,( i,x i ) i (( j, x j ) 1 j k ) (9) k { : gl (Θ, ( i, X i ) i, ( j, x j ) 1 j l 1 ) l1 λ l (Θ, ( i, X i ) i, ( j, x j ) j l 1 )e R l l 1 λ s(θ,( i,x i ) i,( j,x j ) j l 1 ) ds wih he convenion. In his paper we will consider - beside he general resuls - hree classes of special risk processes. (a) Mixed (homogeneous) Poisson Risk Processes: his is he case, if (λ ) is a consan process only depending on Θ if also he densiies g above only depend on Θ ( also no on ). his means ha we assume λ λ(θ), (1) 5

6 g ( ; Θ, 1,,, X 1,, X ) g( ; Θ). (11) (b) Mixed (inhomogeneous) Poisson Risk Processes: In his case we assume λ λ (Θ), (12) g ( ; Θ, 1,,, X 1,, X ) g ( ; Θ). (13) (c) Mixed Markovian Risk Processes: Here we suppose ha we have on he ses { < n (n 1) λ λ (n) (Θ), (14) g ( ; Θ, 1,,, X 1,, X ) g (n) ( ; Θ), (15) he mixed Markovian risk process is said o be homogeneous, if λ (n) no depend on. g (n) do 2 Predicion From now on we will always assume ha (( n, X n )) n 1 is a Θ-mixed risk process wih a regular F Θ -inensiy measure Λ as described in definiion 1.1. Suppose ha Z (Z s ) s is an inegrable, F Θ -adaped process, ha < u are wo given fixed ime poins ha G is a fixed sub-σ-algebra of F Θ. hen we will call E{Z u G he predicion of Z u on he basis of he informaions given by G (or more shorly: he predicion of Z u given G ). his predicion problem will be solved in wo seps: In his secion we will firs consider he predicion of Z u given F Θ. hen in he nex secion we will solve he predicion of Z u given G by filering E{Z u F Θ, which simply means ha we use he ieraion formula for condiional expecaions: E{Z u G E { E{Z u F Θ G. Since Z (Zs ) s is assumed o be F Θ -adaped, we have Z u 1 { < n 1 {+k 1 u< +k Z u (Θ, ( i, X i ) i +k 1 ). n 1 k 1 hus we obain from (8) he general formula E{Z u F Θ 1 { < n n 1 where k 1 I,u n,k (Θ, ( i, X i ) i ), (16) I,u n,k (Θ, ( i, X i ) i ) : E { 1 {+k 1 u< +k Z u (Θ, ( i, X i ) i +k 1 ) F Θ. For k 1 we have I,u n,1(θ, ( i, X i ) i ) Z u (Θ, ( i, X i ) i e R u λ r(θ,( i,x i ) i )dr, (17) 6

7 for k > 1 we have (wih he convenion ) I,u n,k (Θ, ( i, X i ) i ) (18) u u { Z u (Θ, ( i, X i ) i, ( j, x j ) 1 j k 1 ) E k 2 E e R u λr(θ,( i,x i ) i,( j,x j ) 1 j k 1 )dr G,(,k 1) Θ,( i,x i ) i (( j, x j ) 1 j k 1 ) γ(dx k 1 )d k 1 γ(dx 1 )d 1. he double series in formula (16) reduces o a finie sum, if Z u does no fully depend on (( n, X n )) n 1, i.e. if Z u Z u (Θ, ( i, X i ) i m ) for some fixed m. We omi he deails. We will resric now o more special predicion problems. For his we assume ha in he following always E R + (or a leas R d ), bu we will sill use he noaion E o disinguish he claim space from he ime axis. For a fixed orel subse of E we se S : n 1 X n 1 {Xn 1 {n. (19) hen S x N(ds, dx). We will assume ha for all E x g s (x)λ s γ(dx)ds <. (2) hen one knows (cf. Deweiler [24]) ha ( S x g s (x)λ s γ(dx)ds ) is an F Θ -maringale. his implies E{Su S F Θ E { u x g s (x)λ s γ(dx)ds F Θ. (21) hus he predicion of Su S given F Θ is he same as he predicion of u Z u : x g s (x)λ s γ(dx)ds (22) given F Θ. 7

8 If he densiy processes g λ fully depend on (( n, X n )) n 1, one has o use he general formula (16) for he Z u given by (22), which clearly is no an easy pracical ask. u here are wo imporan special cases, where he predicion is quie easy, since g λ only depend on Θ. 2.1 Proposiion. Suppose ha (( n, X n )) n 1 is a mixed inhomogeneous Poisson risk process, for which he inegrabiliy assumpion (2) holds. hen u E{Su S F Θ xg s (x; Θ)λ s (Θ)γ(dx)ds. (23) If especially (( n, X n )) n 1 is a mixed homogeneous Poisson risk process, hen E{Su S F Θ (u )λ(θ) xg(x; Θ)γ(dx). (24) he predicion problem for general mixed risk processes is geing more simple in he following siuaion. We inroduce he sopping ime : inf{v > N v N 1, consider he predicion problem for he incremen S u S. Since n on { < n, we have E{S u S F Θ n 1 1 { < n E{1 {n u1 {Xn X n F Θ, (25) one obains E{S u S F Θ n 1 1 { < n J,u n (Θ, ( i, X i ) i ), (26) where J,u n (Θ, ( i, X i ) i ) (27) u ( ) xg s (x; Θ, ( i, X i ) i )γ(dx) Especially, we have he following resul: λ s (Θ, ( i, X i ) i )e R s λ r(θ,( i,x i ) i ) dr ds. 2.2 Proposiion. Suppose ha he inegrabiliy condiion (2) holds. (a) If (( n, X n )) n 1 is a mixed Markovian risk process (see (14) (15), hen E{S u S F Θ (28) u ( ) 1 { < n xg s (n) (x; Θ)γ(dx) λ (n) s (Θ)e R s λ(n) r (Θ) dr ds. n 1 hus - in case ha (( n, X n )) n 1 is homogeneous - 8

9 E{S u S F Θ (29) ( )( ) 1 { < n 1 e (u )λ(n) (Θ) xg (n) (x; Θ)γ(dx). n 1 (b) If (( n, X n )) n 1 is a mixed inhomogeneous Poisson risk process, hen u ( E{S u S F Θ xg s (x; Θ)γ(dx) )λ s (Θ)e R s λ r(θ) dr ds, (3) in case of a homogeneous Poisson risk process ( )( E{S u S F Θ 1 e (u )λ( Θ) ) xg(x; Θ)γ(dx). (31) In he nex ( las) predicion problem we replace he deerminisic ime poins u by he sopping imes n, consider he predicion of X n 1 {Xn given F Θ. his means ha we deermine From (7) we obain E{X n 1 {Xn F Θ E{X n 1 {Xn Θ, ( i, X i ) i. E{X n 1 {Xn F Θ x G (,1) Θ,( i,x i ) i (, x)γ(dx) d ( ) x g (x; Θ, ( i, X i ) i )γ(dx) λ (Θ, ( i, X i ) i )e R λ r(θ,( i,x i ) i )dr d. his implies for our examples of mixed risk processes he following proposiion: 2.3 Proposiion. Suppose ha he inegrabiliy condiion (2) holds. (a) If (( n, X n )) n 1 is a mixed Markovian risk process, hen (32) E{X n 1 {Xn F Θ (33) ( ) xg (n) (x; Θ)γ(dx) λ (n) (Θ)e R λ (n) r (Θ) dr d. hus - in case ha (( n, X n )) n 1 is homogeneous - E{X n 1 {Xn F Θ xg (n) (x; Θ)γ(dx). (34) (b) If (( n, X n )) n 1 is a mixed inhomogeneous Poisson risk process, hen ( E{X n 1 {Xn F Θ xg (x; Θ)γ(dx) )λ (Θ)e R λ r (Θ) dr d,(35) in case of a mixed homogeneous Poisson risk process E{X n 1 {Xn F Θ xg(x; Θ)γ(dx). (36) 9

10 3 Filering In his secion we filer he predicion formulas of he foregoing secion relaive o sub-σ-algebras G of F N (resp. sub-σ-algebras G of F N ). Firs we consider filering relaive o F N. Le us make he following convenion: elow here will ofen occur - in connecion wih condiional expecaions - quoiens of he form F (x) G(x), where i may happen ha he denominaor G(x) is zero. In ha case he value of ha quoien is defined o be zero. he following lemma will be he basis of mos of he resuls in his secion. 3.1 Lemma. Suppose ha F F (Θ, ( i, X i ) i ) is inegrable. hen for every every n 1 he following filering formula holds on he se { < n : wih E{F (Θ, ( i, X i ) i ) F N Φn,F (( i, X i ) i ) Ψ n (( i, X i ) i ), (37) Φ n,f (( i, X i ) i ) (38) {F (y, ( i, X i ) i )e R λ r(y,( i,x i ) i ) dr G () (y, ( i, X i ) i ) β(dy) Ψ n (( i, X i ) i ) (39) {e R λ r(y,( i,x i ) i ) dr G () (y, ( i, X i ) i ) β(dy). Proof. Le H be an arbirary bounded, F N -measurable funcion. Since H H(( i, X i ) i ) on { < n (cf. Deweiler [24]), we obain from (4) 1 { < n H F (Θ, ( i, X i ) i ) dp Ω ( E n 2 E G () (y, ( i, x i ) i )λ n (y, ( i, x i ) i ) { H(( i, x i ) i )F (y, ( i, x i ) i ) e R n λ r (y,( i,x i ) i ) dr d n γ(dx )d γ(dx 1 )d 1 )β(dy) 1

11 ( E { H(( i, x i ) i )F (y, ( i, x i ) i ) n 2 E G () (y, ( i, x i ) i )e R λ r (y,( i,x i ) j )dr Since γ(dx )d γ(dx 1 )d 1 )β(dy) H(( i, x i ) i ) E n 2 E ( {F (y, ( i, x i ) i )e R λ r (y,(, x i ) i )dr ) G () (y, ( i, x i ) i ) β(dy) γ(dx )d γ(dx 1 )d 1 { H(( i, x i ) i ) E n 2 E Φ n,f (( i, x i ) i ) γ(dx )d γ(dx 1 )d 1 {H(( i, x i ) i ) Φn,F (( i, x i ) i ) E n 2 E Ψ n (( i, x i ) i ) Ψ n (( i, x i ) i ) γ(dx )d γ(dx 1 )d 1 ( { H(( i, x i ) i ) E n 2 E Φ n,f (( i, x i ) i ) Ψ n (( i, x i ) i ) G() (y, ( i, x i ) i )λ n (y, ( i, x i ) i ) e R n λ r(y,( i,x i ) i ) dr d n γ(dx )d γ(dx 1 )d 1 )β(dy) 1 { < n H Φn,F (( i, X i ) i ) Ψ n (( i, X i ) i ) dp. Ω 1 { < n Φ n,f (( i, X i ) i ) Ψ n (( i, X i ) i ) is F N -measurable, he asserion of he lemma is proved. he lemma could be applied quie general o he predicion formula (16). hus we would ge E{Z u F N n 1 1 { < n We will no pursue his general seing. k 1 Φ n,i,u n,k (( i, X i ) i ) Ψ n (( i, X i ) i ). (4) As a firs concree applicaion, we compue he predicion of S u S given F N for mixed Poisson processes(cf. proposiion 2.1). We remark ha for a mixed inhomogeneous Poisson process he densiy G () is given by G () (y, ( i, x i ) i ) e R λ r (y)dr 11 ( gi (x i ; y)λ i (y) ), (41) i1

12 for a mixed homogeneous Poisson process we have G () (y, ( i, x i ) i ) λ(y) e λ(y) g(x i ; y). (42) hus he following resuls follow immediaely from lemma Proposiion. Suppose ha (( n, X n )) n 1 is a mixed inhomogeneous Poisson risk process, for which he inegrabiliy assumpion (2) holds. hen E{Su S F N Φ n,,u (( i, X i ) i ) 1 { < n Ψ n n 1 (( i, X i ) i ), (43) i1 wih Φ n,,u (( i, X i ) i ) {( u xg s (x; y)λ s (y)γ(dx)ds )e R λr(y) dr (44) ( gi (X i ; y)λ i (y) ) β(dy), Ψ n (( i, X i ) i ) i1 e R λ r(y) dr ( gi (X i ; y)λ i (y) ) β(dy). (45) i1 If (( n, X n )) n 1 is a mixed homogeneous Poisson risk process, hen wih Φ n, E{Su S F N Φ n,,u ((X i ) i ) 1 { < n Ψ n n 1 ((X i ) i ), (46),u ((X i ) i ) (47) { ( (u ) xg(x; y) γ(dx) ) λ(y) n e λ(y) g(x i ; y) β(dy) i1 Ψ n ((X i ) i ) {λ(y) e λ(y) g(x i ; y) β(dy). (48) i1 Remark. Suppose ha he measure γ is a probabiliy measure. hen he inensiy measure Λ, defined by λ () : γ(), (49) is exremely simple, i is no difficul o see ha here is a probabiliy measure P on (Ω, F), such ha relaive o P he risk process ( n, X n ) n 1 has he F Θ -inensiy 12

13 measure Λ. I is easily proved (cf. also rémaud [1981]) ha he resricion of he original probabiliy measure P o F Θ is absoluely coninuous relaive o he resricion of P o F Θ has he Radon-Nikodym-densiy L, given by L e e R λ r(θ,( i,x i ) i )dr G () (Θ, ( i, X i ) i ) (5) on { < n. I follows ha (43) is jus he formula E P {S u S F N E P {(S u S )L F N E P {L F N, (51) which is well known in he lieraure (cf. rémaud [1981]). We will no pursue his idea (i.e. using a ype of Girsanov ransformaion for predicion), since our formulas are immediae consequences from he consrucion of marked poin processes. In connecion wih he above heorem we consider a relaed predicion problem, which occurs, if for he given ime poin here is only he informaion on { < n, (X i ) i (n 1). available. o model his siuaion we se G n : σ({1 { < n, X 1,, X ) (52) σ ({ { < n {X j j j E (1 j n 1) ) j1 G : n 1 G n. (53) For he filering relaive o G he following lemma is proved similarly as lemma Lemma. Suppose ha F F (Θ, ( i, X i ) i ) is inegrable. hen for every every n 1 he following filering formula holds on { < n : E{F (Θ, ( i, X i ) i ) G E{F (Θ, ( i, X i ) i ) G n (54) Φn,F (X i ) i ) Ψ n ((X i ) i ), wih Φ n,f ((X i ) i ) n 2 {F (y, ( i, X i ) i )e R λ r (y,( i,x i ) i ) dr G () (y, ( i, X i ) i ) d d 1 β(dy) 13

14 Ψ n ((X i ) i ) n 2 {e R λ r (y,( i,x i ) i ) dr G () (y, ( i, X i ) i ) d d 1 β(dy). If we apply his lemma o he filering of E{Su S Poisson risk processes we ge: F Θ relaive o G for mixed 3.5 Proposiion. Suppose ha (( n, X n )) n 1 is a mixed inhomogeneous Poisson risk process, for which he inegrabiliy assumpion (2) holds. hen E{Su S G Φ,u((X i ) i ) 1 { < n Ψ n 1 ((X i ) i ), (55) wih Φ n,,u ((X i ) i ) n 2 {( u xg s (x; y)λ s (y)γ(dx)ds )e R λ r(y) dr ( gi (X i ; y)λ i (y) ) d d 1 β(dy), i1 Ψ n ((X i ) i ) e R λr(y) dr n 2 i1 ( gi (X i ; y)λ i (y) ) d d 1 β(dy). If (( n, X n )) n 1 is a mixed homogeneous Poisson risk process, hen (cf. proposiion 3.2) E{S u S F N E{S u S G. (56) Now we consider he filering of E{S u S F Θ relaive o F N also o G. Using he formula (26) we ge from lemma 3.1 lemma 3.4 he following general resul: 3.6 Proposiion. Le (( n, X n )) n 1 be a mixed risk process wih regular F Θ - inensiy measure Λ suppose ha S u S is inegrable. If Jn,u is defined by (27), hen E{S u S F N n 1 1 [, n [() Φn,u(( i, X i ) i ) Ψ n (( i, X i ) i ), (57) 14

15 where Φ n,u Ψ n are given by Φ n,u(( i, X i ) i ) {J n,u(y, ( i, X i ) i )e R λ r(y,( i,x i ) i )dr G () (y, ( i, X i ) i ) β(dy) Ψ n (( i, X i ) i ) {e R λ r(y,( i,x i ) i )dr G () (y, ( i, X i ) i ) β(dy). For he filering relaive o G we have where Φ n,u Ψ n Φ n,u((x i ) i ) E{S u S G n 1 are given by 1 [, n [() Φn,u((X i ) i ) Ψ n ((X i ) i ), (58) {J n,u(y, ( i, X i ) i )e R λ r(y,( i,x i ) i )dr n 2 G () (y, ( i, X i ) i ) d d 1 β(dy) Ψ n ((X i ) i ) n 2 {e R λ r(y,( i,x i ) i )dr G () (y, ( i, X i ) i ) d d 1 β(dy). he above general resul can easily be applied o more special mixed risk processes. We jus give wo examples. 3.7 Proposiion. Le (( n, X n )) n 1 be a mixed risk process such ha S u S is inegrable. (a) If (( n, X n )) n 1 is a mixed homogeneous Markovian risk process, hen wih E{S u S F N n 1 Φ n,u(( i, X i ) i ) { ((1 e (u )λ (n)(y) ) e P i1 ( i i 1 )λ (i) (y) 1 [, n[() Φn,u(( i, X i ) i ) Ψ n (( i, X i ) i ), (59) 15 xg (n) (x, y)γ(dx) ) e ( )λ (n) (y) ( g (i) (x; y)λ (i) (y) ) β(dy) i1

16 Ψ n,u(( i, X i ) i ) {e ( )λ (n) (y) e P i1 ( i i 1 )λ (i) (y) ( g (i) (x; y)λ (i) (y) ) β(dy). i1 Similarly, E{S u S G n 1 1 [, n [() Φn,u((X i ) i ) Ψ n ((X i ) i ), (6) wih Φ n,u((x i ) i ) { ((1 e (u )λ (n)(y) ) ( xg (n) (x, y)γ(dx) ) ( g (i) (x; y)λ (i) (y) ) e ( P )λ(n) (y) i1 ( i i 1 )λ (i)(y) d d 1 )β(dy) n 2 i1 Ψ n,u((x i ) i ) { ( g (i) (x; y)λ (i) (y) ) i1 e ( P )λ(n) (y) n 2 ( i1 ( i i 1 )λ (i) (y) d d 1 )β(dy). (b) If (( n, X n )) n 1 is a mixed homogeneous Poisson risk process, hen E{S u S F N E{S u S G (61) Φ n,u((x i ) i ) 1 { < n Ψ n n 1 ((X i ) i ), wih Φ n,u((x i ) i ) { (1 e (u )λ(y) ) xg(x; y) γ(dx)λ(y) e λ(y) g(x i ; y) β(dy) i1 Ψ ((X i ) i ) {λ(y) e λ(y) g(x i ; y) β(dy). i1 16

17 Now we consider he predicion of X n 1 {Xn given G σ({x 1,, X ), i.e. he filering of E{X n 1 {Xn F Θ relaive o G. If we wrie J n (Θ, ( i, X i ) i ) for he righ h side of equaion (32), we have he following general resul: 3.8 heorem. Le (( n, X n )) n 1 be a mixed risk process wih regular F Θ -inensiy measure suppose ha X n 1 {Xn is inegrable. hen E{X n 1 {Xn (X i ) i Φ n((x i ) i ) Ψ n ((X i ) i ) (62) wih Φ n ((X i ) i ) { J n (y, ( i, X i ) i ) n 2 G () (y, ( i, X i ) i ) d d 1 β(dy), Ψ n ((X i ) i ) { G () (y, ( i, X i ) i ) d d 1 β(dy). n 2 his resul can easily be applied o special mixed risk processes. hus we ge e.g.: 3.9 Proposiion. (a) If ( n, X n ) n 1 is a mixed Markovian risk process, hen J n (y, ( i, x i ) i ) J n (y, ) (63) ( xg (n) n (x; y)γ(dx) ) λ (n) n (y)e R n λ (n) s (y)ds d n, where E{X n 1 {Xn (X i ) 1 i (64) n 2 J n (y, )G () (y, ( i, X i ) i )d d 1 β(dy), n 2 G () (y, ( i, X i ) i )d d 1 β(dy) G () (y, ( i, x i ) i ) g (i) i (x; y)λ (i) i (y)e R i λ (i) s (y)ds i 1. i1 If ( n, X n ) n 1 is homogeneous, hen J n (y, ) J n (y) xg (n) (x; y)γ(dx), 17

18 E{X n 1 {Xn (X i ) i (65) { xg(n) (x; y)γ(dx) i1 g(i) (X i ; y) β(dy) { i1 g(i) (X i ; y). β(dy) (b)if ( n, X n ) n 1 is a mixed homogeneous Poisson risk process, hen E{X n 1 {Xn (X i ) i (66) { xg( x; y)γ(dx) i1 g(x i; y) β(dy) { i1 g(x i; y). β(dy) Proof. Formula (64) follows immediaely from heorem 3.8. We jus prove (65), which implies (66). From (64) we have E{X n 1 {Xn (X i ) i Φ n((x i ) i ) Ψ n ((X i ) i ) wih Φ n ((X i ) i ) {( ) { λ (i) (y)e ( i i 1 )λ (i)(y) d d 1 n 2 i1 xg (n) (x; y)γ(dx) i1 g (i) (X i ; y) β(dy) a similar formula for Ψ n ((X i ) 1 i ). hus (65) follows, since for every y. { λ (i) (y)e ( i i 1 )λ (i)(y) d d 1 1 n 2 i1 Remark. A a firs glance he formulas (65) (66) may a lile bi surprise, since here is no dependence on he disribuions of he claim arrival imes 1,,. u he reason is simple: he predicion of X n 1 {Xn given X 1,, X replaces he naural ime by he ime poins 1,, he independence of ( n ) n 1 (X n ) n 1 reduces he predicion o a predicion problem for he discree ime process (X n ) n 1. he siuaion becomes quie differen, if we consider he predicion of X n 1 {Xn given ( i, X i ) i : Suppose ha (( n, X n )) n 1 is a mixed homogeneous Poisson risk process. hen E{X n 1 {Xn ( i, X i ) i E{X n 1 {Xn, X 1,, X (67) {( xg(x; y)γ(dx)) e λ(y) λ(y) i1 g(x i; y) β(dy) { e λ(y) λ(y) i1 g(x i; y). β(dy) 18

19 Since an increase of informaion gives surely more reliable predicion, formula (67) should be beer han (65) in case here is he informaion on. As a las filering problem we consider he problem of filering he inensiy measure Λ relaive o F N. I will urn ou ha he filered inensiy measure Λ is again regular. We have he following general resul, which follows easily from lemma 3.1: 3.11 heorem. Suppose ha he inensiies λ () ( E) are inegrable. hen he following formula holds: E{λ () F N n 1 1 [, n [() λ (; ( i, X i ) i ), (68) wih λ (; ( i, X i ) i ) φ (; ( i, X i ) i ) θ (( i, X i ) i ), (69) where φ θ are given by (cf. (5)) { φ (; ( i, x i ) i ) g (x; y, ( i, x i ) i )γ(dx)λ (y, ( i, x i ) i ) e R λ r(y,( i,x i ) i )dr G () (y, ( i, x i ) i ) β(dy), θ (( i, x i ) i ) {e R λ r(y,( i,x i ) i )dr G () (y, ( i, x i ) i ) β(dy), for < 1 < < x 1,, x E. he F N -inensiy measure Λ ( ( λ ()) ) E (7) given by (69) has a similar srucure as he F Θ -inensiy measure Λ. On he se { < n we have where λ (dx; ( i, X i ) i ) γ (dx; ( i, X i ) i ) λ (( i, X i ) i ), (71) λ (( i, X i ) i ) φ (E; ( i, X i ) i ) θ (( i, X i ) i ), (72) where he probabiliy measure γ (dx; ( i, X i ) i ) has a densiy g (x; ( i, X i ) i ) relaive o γ(dx), which is given as follows: 19

20 We se for x E φ (x; ( i, x i ) i ) { g (x; y, ( i, x i ) i )λ (y, ( i, x i ) i ) (73) e R λ r (y,( i,x i ) i )dr G () (y, ( i, x i ) i ) β(dy), choose a fixed, measurable g : E R + such ha g γ is a probabiliy measure. hen g (x; ( i, X i ) i ) φ (x; ( i, X i ) i ) φ (E; ( i, X i ) i ) 1 {φ (E;( i,x i ) i )> (74) + g (x)1 {φ (E;( i,x i ) i ) Corollary. Le (( n, X n )) n 1 be a mixed homogeneous Poisson risk process wih he regular F Θ -inensiy measure Λ given by (1) (11). hen λ (dx; ( n, x n ) n ) n 1 1 [, n [() λ (dx; ( i, x i ) i ) wih λ (dx; ( i, x i ) i ) g (x; ( i, x i ) i )γ(dx) λ (( i, x i ) i ), (75) where λ (( i, x i ) i ) i1 g(x i; y) λ(y) n e λ(y) β(dy) i1 g(x i; y) λ(y) e λ(y) β(dy) g (x; ( i, x i ) i ) g(x; y) i1 g(x i; y) λ(y) n e λ(y) β(dy) i1 g(x i; y) λ(y) n e λ(y) β(dy). Remark. I is well known (cf. e.g. Schmid [1996]) ha he filering of a mixed Poisson process gives a Markov process. Corollary 3.12 shows ha he filering of a mixed Poisson risk process gives no longer a Markovian risk process Corollary. Le (( n, X n )) n 1 be a mixed inhomogeneous Poisson risk process wih he regular F Θ -inensiy measure Λ given by (12) (13). hen λ (dx; ( n, x n ) n ) n 1 1 [, n[() λ (dx; ( i, x i ) i ) wih λ (dx; ( i, x i ) i ) g (x; ( i, x i ) i )γ(dx) λ (( i, x i ) i ), (76) 2

21 where λ (( i, x i ) i ) i1 i1 ( gi (x i ; y) λ i (y) ) e R λs(y) ds β(dy) ( gi (x i ; y) λ i (y) ) e R λs(y) ds β(dy) g (x; ( i, x i ) i ) g (x; y)λ (y) i1 g i (x i ; y) λ i (y)e R λ s(y) ds β(dy) ( gi (x i ; y) λ i (y) ) e R λ s(y) ds β(dy) i1. We omi he corresponding filering resul for he mixed Markovian risk process. heorem 3.11 has he following inerpreaion: 3.15 Proposiion. Le (( n, X n )) n 1 be an inegrable, mixed risk process wih regular F Θ -inensiy measure le Λ ( (λ ()) ) E, Λ ( ( λ ()) ) E, be he inensiy measure defined in heorem hen (( n, X n )) n 1 has he inensiy measure Λ for he filraion F N. Proof. We have o prove ha for every E he process ( N () λ s ds ) is an F N -maringale. his follows, since for s < E{N () N s () Fs N E { E{N () N s () Fs Θ F N s E { E{ E{ E{ s s s λ r dr F Θ s F N s λ r dr F N s λ r dr F N s. Remark. Proposiion 3.15 could be used o compue direcly he predicion formulas relaive o F N proved in his secion. u his looks much more complicaed han he mehod of firs predicing relaive o F Θ hen filering relaive o F N. 21

22 4 Predicion of he Claim Amoun Disribuion he predicion problems of he foregoing secions are only a firs sep o ge some insigh ino he fuure behaviour of mixed risk processes. One should have in mind ha e.g. he predicion of he firs claim amoun X,u : S u S (S S E ) in he planning period [, u] is jus a mean value on he basis of he informaions given by F N. For more reliable informaion one should no only consider he predicion of his mean value, bu also (a leas heoreically) he predicion of he disribuion of X,u. Suppose ha his disribuion is given by P,u (dz; F N ). (77) hen one can easily predic in addiion he disance d(x (), ϕ ) o some given F N - measurable funcion ϕ [one could imagine ha ϕ is (conneced wih) he individual premium prediced by he las secion]. his predicion would simply be given by he formula E{d(X,u, ϕ ) F N d(z, ϕ )P,u (dz; F N ). (78) Such a disance d(, ) could be he euclidian disance E d 2 (z, v) : (z v) 2. (79) hus for ϕ E{X,u F N he predicion of d 2 (X,u, ϕ ) is jus he predicion of he condiional variance of X,u given F N. u here are also oher disances (which are even more imporan): We se for a given ε > d ε (z, v) : 1 {z v+ε, (8) d + (z, v) : (z v) +. (81) hen he predicion of d ε (X,u, ϕ ) e.g. is jus he condiional probabiliy of {X,u ϕ + ε given F N. he formulas for he predicion of he claim amoun disribuion are derived similarly as he predicion formulas for X,u. Analogously o proposiion 3.6 we ge 4.1 heorem. Le (( n, X n )) n 1 be a inegrable, mixed risk process wih regular F Θ -inensiy measure Λ ( (λ ()) ) E, define for < u, n 1, y, < 1 < <, x 1,, x E A E 22

23 J,u n (A; y, ( i, x i ) i ) u ( A ) g n (x; y, ( i, x i ) i )γ(dx) λ n (y, ( i, x i ) i )e R n λ s (y,( i,x i ) i ) ds d n +1 A ()e R u λ s(y,( i,x i ) i )ds. (82) hen P{X,u A F N n 1 1 [, n[() Φ,u n (A; ( i, X i ) i ) Ψ n(( i, X i ) i ), (83) where Φ,u n (A) Ψ n are given by { Φ,u n (A; ( i, X i ) i ) J,u n (A; y, ( i, X i ) i )e R λ r(y,( i,x i ) i )dr G () (y, ( i, X i ) i ) β(dy), Ψ n(( i, X i ) i ) {e R λ r(y,( i,x i ) i )dr G () (y, ( i, X i ) i ) β(dy). hus we have he formula P,u (dz; F N ) n 1 1 [, n[() Φ,u n (dz; ( i, X i ) i ) Ψ n(( i, X i ) i ). (84) Proof. Since X,u n 1 1 { < n X n 1 {n u, we have {X,u A { < n {X n 1 {n u A { < n. If / A, hen {X n 1 {n u A {X n A { n u, if A, hen {X n 1 {n u A ({X n A { n u) {u < n where he union on he righ h side clearly is disjoin. Alogeher we have 1 {X,u A 1 { < n ( 1{Xn A { n u + 1 A ()1 {u<n ). n 1 23

24 hus he predicion of 1 {X,u A given F Θ is given by (cf. secion 2, esp. (27)) E{1 {X,u A F Θ n 1 1 { < n J,u n (A; Θ, ( i, x i ) i ), where J,u n is defined by (82). An applicaion of lemma 3.1 hen proves (83). hus we ge for example, if d denoes one of he disances inroduced above, he following predicion formula: 4.2 Corollary. Le ϕ be F N -measurable suppose ha d(x,u, ϕ ) is inegrable. hen where E{d(X,u, ϕ ) F N 1 { < n G,u Φ,u n (G; ( i, X i ) i ) Ψ n(( i, X i ) i ) n (y, ( i, x i ) i ) u ( ) d(x, ϕ (( i, x i ) i ))g n (x; y, ( i, x i ) i )γ(dx) E λ n (y, ( i, x i ) i )e R n λ s (y,( i,x i ) i ) ds d n +d(, ϕ (( i, x i ) i ))e R u λs(y,( i,x i ) i )ds, where Φ,u n (G) is defined analogously o Φ,u n (A) (wih Jn,u (A) replaced by G,u n ). Of course, here are corresponding resuls for P,u (dz; G ) : P{X,u dz G, (85) where G was defined in (53). Corresponding resuls also hold for or P n (dz; ( i, X i ) i ) : P{X n dz F (86) P n (dz; (X i ) i ) : P{X n dz X 1,, X. (cf. heorem 3.8 proposiion 3.9). For he mixed Poisson risk process we have he following corollary from heorem 4.1: 4.3 Corollary. Le (( n, X n )) n 1 be a mixed homogeneous Poisson risk process wih densiies g inensiies λ given by (1) (11), define for A E, y H(A, y) : g(x; y)γ(dx). A 24,

25 hen he following predicion formula holds: wih Φ,u P{X,u A F N n 1 1 { < n Φ,u n (A; (X i ) 1 i ) Ψ n((x i ) 1 i ) n (A; (x i ) 1 i ) {H(A, y)(1 e (u )λ(y) )e λ(y) λ(y) g(x i ; y) β(dy) +1 A ()e (u )λ(y), i1, Ψ n ((x i ) 1 i ) {e λ(y) λ(y) g(x i ; y) β(dy). i1 hus we have P,u (dz; F N ) n 1 1 { < n Φ,u n (dz; (X i ) i ) Ψ n((x i ) i ). Le us suppose ha ϕ only depends on X 1,, X on he se { < n, which is he case, if ϕ E{X,u F N (cf. (61). hen we obain from he corollary he following wo examples: (1) For he disance d + we have E{d + (X,u, ϕ ) F N n 1 1 { < n E d +(z, ϕ ((X i ) i ))Φ n(dz; (X i ) i ) Ψ n((x i ) i ), where d + (z, ϕ ((X i ) i ))Φ n(dz; (X i ) i ) E {( ) d + (x, ϕ ((X i ) i ))g(x; y)γ(dx) E (1 e (u )λ(y) )e λ(y) λ(y) g(x i ; y) β(dy). i1 (2) For he disance d ε we have 25

26 P{X,u ϕ + ε F N n 1 1 { < n E d ε(z, ϕ ((X i ) i ))Φ n(dz; (X i ) i ) Ψ n((x i ) i ), where d ε (z, ϕ ((X i ) i ))Φ n(dz; (X i ) i ) E {( ) 1 [ϕ ((X i ) i )+ε[(x)g(x; y)γ(dx) E (1 e (u )λ(y) )e λ(y) λ(y) g(x i ; y) β(dy). i1 References rémaud, P. [1981]: Poin Processes Queues - Maringale Dynamics. erlin Heidelberg New York: Springer. Deweiler, E. [24]: Risk Processes. Leipzig: Ediion am Guenbergplaz. Schmid, K.D. [1996]: Lecures on Risk heory. Sugar: eubner. Egber Deweiler Mahemaisches Insiu Universiä übingen Auf der Morgenselle 1 D 7276 übingen E mail: e.deweiler@web.de 3h Ocober 25 26