Insurance Claims Modulated by a Hidden Marked Point Process

Insurance Claims Modulated by a Hidden Marked Point Process

Robert J. Elliott, Tak Kuen Siu and Hailiang Yang

Abstract— Recently Markov-modulated compound Poisson models have gained its popularity in modelling insurance claims in the actuarial science literature. A Markov-modulated compound Poisson model can provide a realistic and flexibile way to model aggregate insurance claims by incorporating the impact of hidden states of an economy on claim frequencies and claim sizes. However, in practice, the Markov chain in the model is not observable. It is of practical interest to develop some methods to estimate the hidden state of the Markov chain and other unknown model parameters of the Markov- modulated compound Poisson model. This paper considers this important issue. We shall develop filters and smoothers for the hidden state of the economy underlying the Markov-modulated compound Poisson model. In general, we consider the case when both the stochastic intensity and the distribution of the claim sizes of the compound Poisson process depend on the hidden Markov chain. The filter and smoother provide an optimal way to estimate the insurance claims model in the “mean- squared-error” sense. We shall also develop estimators for the unknown model parameters of the Markov-modulated marked point process using the robust filter-based and smoother-based EM algorithms.

I. INTRODUCTION

The compound Poisson process is a standard, classical model in ruin theory which is used to describe aggregate insurance claims over a certain time horizon. The use of the compound Poisson process for modelling aggregate insurance claims has a natural interpretation. In particular, the number of claims over a time horizon can be modelled as a Poisson process while the amounts of individual claims can be modelled by a sequence of positive random variables, independent of the number of claims. One of the main reasons why the compound Poisson model is so popular is that analytically tractable results for ruin theory can be obtained.

In recent years there has been considerable interest in the applications of regime-switching models in risk theory, and the regime-switching model is becoming a popular model for aggregate insurance claims (see [1]). However, the regime-switching process is not directly observable. The unobservability of the states introduces model uncertainty.

In this paper we develop methods of filtering and smoothing the hidden states of the economy underlying a Markov- modulated compound Poisson model. In the general case,

R.J. Elliott is with Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada;[email protected]

T.K. Siu is with Department of Actuarial Mathematics and Statistics, School of Mathematical Sciences and the Maxwell Institute for Math- ematical Sciences, Heriot-Watt University, Edinburgh, United Kingdom;

[email protected]

H. Yang is with Department of Statistics and Actuarial Sci- ence, The University of Hong Kong, Pokfulam Road, Hong Kong;

[email protected]

robust filters and smoothers for a marked point process are derived in the form of O.D.E.s, which seem to be new in the literature. The filter and smoother of the hidden states provide an appropriate method to select, or estimate, a risk model in the “mean-squared-error” sense. They can also provide information about the underlying hidden economic states generating the claims data. We also discuss the estimation of the unknown parameters in the Markov-modulated market point process using the robust filter-based and smoother-based EM algorithms. This provides a convenient way to calibrate the models.

II. MODEL DYNAMICS AND CHANGE OF MEASURES

In this section we shall present a Markov-modulated compound Poisson model with a Markov-switching stochastic intensity only for aggregate insurance claims. We suppose that both the claim frequencies and the claim sizes are observable.

The claim frequencies are modelled as a Poisson process while the claim sizes are described as a sequence of i.i.d. random variables with finite common mean. We further assume that the claim frequencies and the claim sizes are independent.

The dynamics of the aggregate claim amounts are described by a Markov-modulated compound Poisson process with a Markov-switching compensator depending on the hidden states of an economy. We suppose that the hidden states of an economy are described by the states of a continuous-time hidden Markov chain process, which is the state process in our model. We shall describe the mathematical set up of our model in the sequel.

Consider a complete probability space (Ω, F , P), where P is a real-world probability measure. We suppose that (Ω, F , P) is rich enough to model the randomness of the observations process and the state process. Write T for the time index set [0, ∞) of the model. Let X := {Xt}t∈T denote a continuous-time hidden Markov chain process, defined on (Ω, F , P), with state space S := {s1, s₂, . . . , s_K} ⊆ <^K. Without loss of generality, we can take the state space of X to be the set of unit basis vectors L := {e1, e2, . . . , eK} ⊆ <^K, where ei := (0, 0, . . . , 1, . . . , 0, 0)^T ∈ <^K with “1” in the i^th position, and where y^T represents the transpose of the row vector y. This is called a canonical representation of the state space of X. Let A denote a constant rate matrix, or the Q-matrix of the Markov chain process. Then, using the canonical representation for the state space, [4] show that the dynamics of the Markov chain X have the following semi-martingale representation:

X_t= X₀+ Z t

0

AX_sds + M_t. (2.1) New York City, USA, July 11-13, 2007

Here {Mt}t∈T is an <^K-valued martingale increment process with respect to the filtration generated by X.

Consider a Poisson process N := {Nt}t∈T on (Ω, F , P), whose stochastic intensity is given by:

λt:= hλ, Xti =

K

X

i=1

hλ, eii I_{Xt=e_i} . (2.2)

Here λ := (λ1, λ2, . . . , λK) ∈ <^K and λk ≥ 0, for each k = 1, 2, . . . , K.

For each t ∈ T , Ntwill represent the number of claims over the time [0, t].

Consider right-continuous, complete versions of the filtra- tions

F^X := {F_t^X}t∈T , F_t^X := σ{Xu|u ∈ [0, t]} , F^N := {F_t^N}t∈T , F_t^N := σ{Nu|u ∈ [0, t]} ,

G := {Gt}t∈T , Gt:= F_t^X∨ F_t^N .

Then the Doob-Meyer decomposition for N is (see [6]):

N_t= Z t

0

hλ, X_ui du + V_t , (2.3) where V := {Vt}t∈T is a (P, G)-martingale.

Consider a probability distribution FY(·) on (R⁺, B(R⁺)).

The total amount of claims to time t is then supposed to be given by:

Z_t= Z t

0

Z ∞ 0

ydF_Y(y)dN_u . (2.4) Note that if we observe Z, we certainly observe N . The jump sizes here provide no extra information about X.

As in [6], we assume that under a reference probability measure P^†, the process N is a canonical Poisson process with unit intensity and is independent of the state process X.

Then,

Qt:= Nt− t , (2.5)

is a martingale under P^†.

Define a process Λ := {Λ0,t}t∈T as follows:

Λ_0,t := Y

0<u≤t

hXu, λi^∆Nuexp

 Z t 0

(1 − hX_u, λi)du



= 1 + Z t

0

Λ_0,u−(hX_u−, λi − 1)dQ_u . (2.6) Note that Λ is a (G, P^†)-martingale. To define the real-world probability measure P, we set

Λ_0,t:= dP dP^†

   Gt

. (2.7)

By the semi-group property of Λ,

Λ0,T = Λ0,tΛt,T , (2.8) where

Λ_t,T = 1 + Z T

t

Λ_t,u−(hX_u−, λi − 1)dQ_u . (2.9)

Suppose γ := {γt}t∈T is any G-adapted process. Given F_t^N, we can estimate γtby its least-square estimate E[γt|F_t^N].

By a form of Bayes’ rule (see [4]), E[γt|F_t^N] = E^†[Λ0,tγt|F_t^N]

E^†[Λ_0,t|F_t^N] =σt(γ)

σt(1) , (2.10) where E^† denote expectation with respect to P^†.

III. FILTERS AND SMOOTHERS

In this section, we state results without proofs which will be published later in [8]. Write qt:= E^†[Λ0,tXt|F_t^N], which is a vector in <^K giving the conditional distribution of X given the observations. Then, the following theorem provides the stochastic differential equation governing qt.

Theorem 3.1: q_tsatisfies the following stochastic differential equation:

qt= q0+ Z t

0

Aqudu + Z t

0

diag{hλ, eki − 1}qu−dQu .(3.1) By Bayes’ rule (2.10),

E[Xt|F_t^N] = E^†[Λ0,tXt|F_t^N]

E^†[Λ0,t|F_t^N] . (3.2) Let 1 := (1, 1, . . . , 1)^T ∈ <^K; then, hXt, 1i = 1. Hence,

E^†[Λ0,tXt|F_t^N], 1

= E^†[Λ0,thXt, 1i |F_t^N]

= E^†[Λ0,t|F_t^N] . (3.3) This implies that

p_t:= E[X_t|F_t^N] = q_t

hqt, 1i . (3.4) Now, define a matrix-valued stochastic process {Γt} such that Γt∈ R^K×K, for each t ∈ T , and

Γt:= diag{γ_t¹, γ_t², . . . , γ_t^K} , (3.5) where γ_t^k := exp[(1 − hλ, e_ki)t] hλ, eki^Nt, for each k = 1, 2, . . . , K.

Let ¯qt:= Γ⁻¹_t qt, which represents the transformed process.

Then, we have the following theorem.

Theorem 3.2: qt satisfies the following forward linear ordinary differential equation:

d¯qt

dt = Γ⁻¹_t AΓtq¯t, q¯0= q0∈ <^K . (3.6) Lemma 3.3: Define

π(Xt) := Γtq¯t

hΓ_tq¯_t, 1i . (3.7) Then, π(X_t) is a version of the expectation E[Xt|F_t^N], which is continuous in the observation process N in the Skorokhod topology.

For the proof of Lemma 3.3, refer to [10].

Following [6], the smoothed state estimates for X, namely, E[Xt|F_T^N], t ∈ [0, T ], can be derived by exploiting a duality.

We state the result in [6] here without proof.

Theorem 3.4: For t ∈ [0, T ], pt := E[Xt|F_T^N]

= 1

h¯qt, ¯νti

K

X

k=1

h¯q_t, e_ki h¯ν_t, e_ki e_k

= 1

hqt, νti

K

X

k=1

hqt, eki hνt, eki ek , (3.8)

where ¯q_t satisfies the following forward linear ordinary differential equation:

d¯q_t

dt = Γ⁻¹_t AΓ_tq¯_t , q¯₀= q₀ , (3.9) and ¯νt satisfies the following backward linear ordinary differential equation:

d¯νt

dt = −ΓtA^∗Γ⁻¹_t ν¯t , ν¯T = ΓT1 . (3.10) For the proof of Theorem 3.4, refer to the proof of Theorem 8 in [6].

IV. MARKOV-SWITCHING STOCHASTIC INTENSITY AND CLAIM SIZES

In this section, we consider a Markov-modulated marked point process for aggregate insurance claims with the stochastic intensity λ and the distribution of the claim sizes Y switching over time according to the state of the hidden Markov chain X. First, we shall write the Markov-modulated marked point process in terms of a random measure with a Markov- switching compensator as in [7]. Then, we adopt the general Girsanov theorem for jump processes in [3] and [7] for changing probability measures. By working under a reference probability measure, we derive robust filters and smoothers in the form of O.D.E.s. We present the results without proofs, which can be found in [8].

Let Z := {Z_t}_t∈T denote a generalized Markov-modulated marked point process under a real-world probability P. If

∆Zu:= Zu− Zu−, we can write Z_t= X

0<u≤t

∆Z_u , Z₀= 0 , P-a.s. (4.1)

Recall that the state space of random times of claim arrivals T is [0, T ] while the state space of random claim sizes Z is (0, ∞). Write X for the product space T × Z. Suppose γ(·, ·) is a random measure on X , which selects the random times of claim arrivals and random claim sizes y := Zu− Zu−. Then, we can write the process Z as follows:

Z_t= Z t

0

Z ∞ 0

yγ(dy, du) . (4.2)

Note that γ is a sum of random delta functions γ(dy, dt) =X

k

δ(Y_Tk(ω))δ(T_k(ω))

so that for suitable integrands f : (Ω × R × [0, ∞)) → <

Z t 0

Z ∞ 0

f (ω, y, u)γ(dy, du)

= X

T_k≤t

f (ω, YT_k(ω), Tk(ω)) . (4.3)

Write Nt=Rt 0

R∞

0 γ(dy, du).

Now, we shall specify a Markov-modulated compensator for Z under the real-world probability P.

First, for each k = 1, 2, . . . , K, write fk(y) for a probabil- ity density function of the random claim size y := Zu− Zu−

when Xu− = k. That is, fk(y) satisfies the following properties:

1) fk is defined on (<⁺, B(<⁺)) 2) fk(y) ≥ 0, for y ∈ Z

3) R∞

0 fk(y)dy = 1

Suppose the number of claim arrivals up to and including time t, N_t, is a random point process on (Ω, F , P) with stochastic intensity:

λ_t:= hλ, X_ti , (4.4)

where λ := (λ1, λ2, . . . , λK) ∈ <^K.

Here, we assume that the random times of claim arrivals and the random claim sizes are independent conditional on the hidden states X. Write ¯Gtfor the σ-algebra F_t^Z∨ F_t^X∨ F_t^N, for t ∈ T . Then, we suppose that the Markov-switching compensator of the random measure γ(dy, du) under P is:

ν(dy, du|Xu−) :=

K

X

k=1

hXu−, eki λkfk(y)dydu . (4.5) Now, we suppose that under a reference probability measure P^†, Z is a marked point process with unit intensity and a density function for the claim sizes f (y), which is independent of the hidden state X.

For each k = 1, 2, . . . , K, let hk(y) = ^λk_{f (y)}^fk(y). Then, we define a density process ¯Λ := { ¯Λ_0,t}_t∈T giving the change of measure as below:

Λ¯0,t := 1 + Z t

0

Λ¯0,u−

 ^K X

k=1

Z ∞ 0

hXu−, eki (hk(y) − 1)

(γ(dy, du) − f (y)dydu)



. (4.6)

Λ is a ( ¯¯ G, P^†)-local martingale.

Suppose ¯Λ is a ( ¯G, P^†)-martingale. To define the real-world probability measure P, we set

Λ¯0,t:= dP dP^†

   _G¯

t

. (4.7)

Then, by the general Girsanov theorem for jump processes in [3] and [7],

Mt:=

Z t 0

Z ∞ 0

y(γ(dy, du) − ν(dy, du|Xu−)) ,

(4.8)

is a ( ¯G, P)-local martingale.

We shall derive the recursive Zakai equation for the filter E(Xt|F_t^Z), where F_t^Z represents the σ-algebra generated by the observation process Z up to t. Again using a version of Bayes’ theorem,

E[Xt|F_t^Z] = E^†[ ¯Λ0,tXt|F_t^Z]

E^†[ ¯Λ_0,t|F_t^Z] = σt(Xt)

σt(1) , (4.9) where σt(X_t) := E^†[ ¯Λ_0,tX_t|F_t^Z], which is an unnormalized estimate of X.

For each time u ∈ (0, ∞), suppose Yu : Ω → (0, ∞) is a random variable with Yu(ω) > 0 and density function f under P^†. Write

Hk(u, ω) :=λkfk(Yu(ω))

f (Y_u(ω)) = hk(Yu(ω)) . (4.10) Then,

E^†(Hk) = Z ∞

0

hk(y)f (y)dy = λk . (4.11) We shall consider the diagonal matrices

diag(H − 1) := diag((H₁− 1), . . . , (H_K− 1)) , diag(λ − 1) := diag((λ1− 1), . . . , (λK− 1)) . Write qt = σt(Xt) = E^†( ¯Λ0,tXt|F_t^Z). We obtain the recursive Zakai equation governing qt in the following proposition:

Theorem 4.1:

q_t = q₀+ Z t

0

Aq_udu + Z t

0

diag(H − 1)q_u−dN_u

− Z t

0

diag(λ − 1)q_udu . (4.12) We shall derive a robust filter for the marked point process Z in the sequel. To our knowledge, this is a new result in the literature. First, consider the process

γ_t^k:= exp(X_t^k) ,

(4.13) where

X_t^k := (1 − λk)t + Z t

0

log Hk(u, ω)dNu . (4.14) Recall the differentiation rule from [3] (see Theorem 12.19 therein),

F (X_t) = F (X₀) + Z t

0

F⁰(X_u−)dX_u

+ X

0<u≤t

[F (Xu) − F (Xu−) − F⁰(Xu)∆Xu] . (4.15) We shall apply this to F (X) = e^−X so F⁰(X) = −e^−X, and with Xt= X_t^k,

∆Xu= log Hk(u, ω)∆Nu . (4.16)

Also,

F (Xu) − F (Xu−)

= e^−Xu−[exp(− log Hk(u, ω)∆Nu) − 1]

= e^−Xu−

 1

H_k(u, ω)− 1



∆Nu. (4.17) Therefore, from (4.17),

d(γ^k_t)⁻¹= −(γ^k_t−)⁻¹[(1 − λ_k)dt + log H_k(t, ω)∆N_t] + (γ_t−^k )⁻¹

 1

H_k(t, ω)− 1



∆Nt

+ log H_k(t, ω)∆N_t



= −(γ^k_t)⁻¹(1 − λ_k)dt + (γ_t−^k )⁻¹ 1 − Hk(t, ω)

H_k(t, ω)



∆Nt.

(4.18) Now we define Γ_t to be the diagonal matrix process diag(γ_t¹, γ_t², . . . , γ^K_t ). Then Γ⁻¹_t = diag((γ¹_t)⁻¹, (γ_t²)⁻¹, . . . , (γ_t^K)⁻¹), and from (4.18),

dΓ⁻¹_t = diag(λ − 1)Γ⁻¹_t dt + diag 1 − H

H



Γ⁻¹_t dNt,

where the diagonal matrix diag(^1−H_H ) :=

diag((^1−H_H ^1(u,ω)

1(u,ω) ), (^1−H_H ^2(u,ω)

2(u,ω) ), . . . , (^1−H_H ^K(u,ω)

K(u,ω) )).

Define ¯qt:= Γ⁻¹_t qt. Then, we have the following Theorem.

Theorem 4.2: ¯q satisfies the ordinary linear differential equation:

¯

q_t= q₀+ Z t

0

Γ⁻¹_u AΓ_uq¯_udu . (4.19) By noticing that Z has independent increments under P^†, we can apply Theorem 3.4 to evaluate pt:= E(X_t|F_T^Z). In this case, ptis given by Theorem 3.4 with F_t^N replaced by F_t^Z.

V. PARAMETER ESTIMATION BY THE EM ALGORITHM:

The estimation of the unknown parameters in the Markov- modulated compound Poisson model with stochastic intensity in Section 2 has been considered in [5] using the EM algorithm. In this section, we shall consider the estimation of the unknown parameters for the Markov-modulated marked point process with Markov-switching stochastic intensity and claim sizes described in Section 4 using the EM algorithm.

Here, we shall compute the estimates for the parameters in the rate matrix A := [aij]i,j=1,2,...,K and the vector of intensity parameters λ := (λ1, λ2, . . . , λK). We suppose that for each regime k = 1, 2, . . . K, the distribution of claim sizes Fk(y) is known. In practice, one can estimate the distributions F1(y), F2(y), . . . FK(y) by first dividing a given set of claims data into K groups. Since this is a set of historical claims data,

one can adopt some simple criteria to divide them into K groups. For example, one can first determine a set of threshold parameters, say R1< R₂ < · · · < R_K−1, and allocate the claim data Ytinto the k^th group if Yt∈ [Rk−1, R_k]. Then, one can estimate Fk(y) using the claims data in the k^thgroup using the methods outlined in [9].

First, we notice that since Fk(y) is supposed to be given, the distribution, the observation processes N and Z provide the same amount of information to estimate aijand λi. Hence, the estimators ˆaij and ˆλi here are given by:

ˆ

aij =E[N_T^ij|F_T^N]

E[O_Tⁱ|F_T^N] , (5.1) and

ˆλi=E[Gⁱ_T|F_T^N]

E[Oⁱ_T|F_T^N] . (5.2) See [8] for detail.

For any ¯G-adapted integrable process γ := {γt}t∈T, σ(γt) := E^†[ ¯Λ0,tγt|F_t^Z] , (5.3) where ¯Λ is governed by (4.6).

We shall present the dynamics for the measure-valued quantities σ(N_t^ijXt), σ(Oⁱ_tXt) and σ(Gⁱ_tXt) without proofs, where N_t^ij, Oⁱ_tand Gⁱ_tare given by:

O_tⁱ:=

Z t 0

hXu, eii du ∈ < , (5.4)

N_t^ij:=

Z t 0

hXu−, eii hdXu, eji ∈ < , (5.5) and

Gⁱ_t:=

Z t 0

hX_u, e_ii dN_u∈ < . (5.6) Write λ − 1 := (λ1− 1, λ2 − 1, . . . , λK − 1) and h :=

(h1(y), h2(y), . . . , hK(y)). Then, σ(Gⁱ_tX_t) =

Z t 0

Aσ(Gⁱ_uX_u)du +

Z t 0

diag(H − 1)σ(Gⁱ_u−Xu−)dNu

− Z t

0

diag(λ − 1)σ(Gⁱ_uXu)du +

Z t 0

hh, eii hqu−, eii dNuei

− Z t

0

hλ − 1, e_ii hq_u, e_ii due_i , (5.7)

σ(N_t^ijXt) = Z t

0

Aσ(N_u^ijXu)du +

Z t 0

hqu−, eii hAei, eji duej

+ Z t

0

diag(H − 1)σ(N_u−^ij Xu−)dNu

− Z t

0

diag(λ − 1)σ(N_u^ijX_u)du , (5.8) and

σ(Oⁱ_tXt) = Z t

0

Aσ(Oⁱ_uXu)du +

Z t 0

hqu, eii duei

+ Z t

0

diag(H − 1)σ(Oⁱ_u−Xu−)dNu

− Z t

0

diag(λ − 1)σ(Oⁱ_uXu)du . (5.9) The proofs for the above dynamics will be published in [8].

We shall then adopt the gauge transformation of [2] to derive robust filtering equations corresponding to σ(Gⁱ_tX_t), σ(N_t^ijX_t) and σ(Oⁱ_tX_t). Define Γt:= diag(γ_t¹, γ_t², . . . , γ^K_t ) as in Section 4. Write ¯σ(Gⁱ_tX_t) := Γ⁻¹_t σ(Gⁱ_tX_t). Then,

¯

σ(Gⁱ_tXt) = Z t

0

Γ⁻¹_u AΓuσ(G¯ ⁱ_uXu)du

− Z t

0

hλ − 1, eii h¯qu, eii duei

+ diag 1 H



hh, eii Ntei

− Z t

0

N_uhh, eii

 d



diag 1 H



¯ q_u

 , e_i

e_i , (5.10)

¯

σ(N_t^ijXt) = Z t

0

Γ⁻¹_u AΓu¯σ(N_u^ijXu)du +

Z t 0

h¯qu, eii hAei, eji duej , (5.11) and

¯ σ(O_tⁱXt)

= Z t

0

Γ⁻¹_u AΓ_uσ(O¯ ⁱ_uX_u)du + Z t

0

h¯q_u, e_ii duei . (5.12) Now, we present the time-domain discretizations of the robust filtering equations without proofs, which will be published in [8]. Let Φm,m−1:= Γt_mΓ⁻¹_tm−1. Then,

qt_m ≈ Φm,m−1[I + A∆]qt_m−1 , (5.13) σ(Gⁱ_t

mX_tm)

≈ Φm,m−1[I + A∆]σ(Gⁱ_t

m−1Xt_m−1) +Γt_mhh, eii

 diag

 1 Ht_m



h¯qt_m, eii Nt_mei

−diag

 1

H_tm−1



¯qt_m−1, ei Nt_m−1ei



− hλ − 1, eii ¯q_tm−1, e_i ∆ei

−Γt_mNt_mhh, eii

 diag

 1 Ht_m



¯ qt_m, ei

 ei

−Γt_mNt_m−1hh, eii

 diag

 1

Ht_m−1



Γ⁻¹_tm−1(A − I)qt_m−1, ei

∆ei , (5.14)

and

σ(N_t^ij_mXt_m) ≈ Φm,m−1[I + ∆A]σ(N_t^ij_m−1Xt_m−1) +Φm,m−1qt_m−1, ei hAei, eji ∆ej ,

(5.15) and

σ(Oⁱ_t

mX_tm) ≈ Φ_m,m−1[I + ∆A]σ(Oⁱ_t

m−1X_tm−1) +Φm,m−1qt_m−1, ei ∆ei , (5.16) Then, the estimators ˆaij and ˆλican be computed by following the three steps of the filter-based EM algorithm, which is described in [5].

In the sequel, we shall present the estimates from the robust smoother-based EM algorithm. By exploiting the duality,

σ(Gⁱ_TXT), νT

= diag 1 H



NThh, eii h¯qT, eii hei, ¯νTi

− Z T

0

N_thh, eii hei, ¯ν_ti

 d



diag 1 H



¯ q_t

 , e_i

 , (5.17) D

σ(N_T^ijXT), νT

E

= Z T

0

hAei, eji hqt, eii hνt, eji dt , (5.18) and

σ(Oⁱ_T, X_T), ν_T = Z T

0

hqt, e_ii hνt, e_ii dt . (5.19) Hence, the smoother-based update equations are:

ˆ

a_ij(k + 1) = ˆa_ij(k) RT

0 hq_t, e_ii hν_t, e_ji dt RT

0 hq_t, e_ii hν_t, e_ii dt , (5.20) and

λˆi(k + 1)

=



diag 1 H



NThh, eii h¯qT, eii hei, ¯qTi

− Z T

0

N_thh, e_ii he_i, ¯ν_ti

 d



diag 1 H



¯ q_t

 , e_i

 

 Z T 0

hqt, eii hνt, eii dt

−1

. (5.21)

The estimators ˆaij and ˆλi can then be computed using the above smoother-based update equations and following the three steps in the smoother-based EM algorithm, which resembles the filter-based EM algorithm.

VI. CONCLUSIONS

We developed a method to filter and smooth a Markov- modulated compound Poisson model which is used to model insurance claims. In general, we considered a Markov- modulated marked point process with both the stochastic intensity and the distribution of the jump amplitude switch

over time according to the state of a hidden Markov chain. The filters and the smoothers can be obtained from solving linear O.D.E.s. We also computed the estimates for the unknown parameters in the Markov-modulated marked point process based on the robust filter-based and smoother-based EM algorithms.

VII. ACKNOWLEDGMENTS

Robert Elliott would like to thank SSHRC for its contin- ued support. Hailiang Yang would like to acknowledge the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 7426/06H). The authors gratefully acknowledge the reviewers’ comments.

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Estimation and Control, Springer-Verlag, Berlin-Heidelberg-New York;

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[5] Elliott, R. J. and Malcolm, W. P. (2000). Robust EM alogarithms for Markov-modulated Poisson Processes. IEEE Conference on Decision and Control. Sydney, Australia, December, 2000.

[6] R.J. Elliott and W.P. Malcolm, General Smoothing Formulas for Markov-Modulated Poisson Observations, IEEE Transactions on Automatic Control, vol. 50(8), 2005, pp 1123-1134.

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10(2), 2006, pp 250-275.

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[9] S.A. Klugman, H.H. Panjer and G.E. Willmot, Loss Models: Data, Decisions, and Risks, John Wiley and Sons, New York; 2004.

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