Multivariate Risk Processes with Interacting Intensities

Transcription

1 Multivariate Risk Processes with Interacting Intensities Nicole Bäuerle (joint work with Rudolf Grübel) Luminy, April 2010

2 Outline Multivariate pure birth processes Multivariate Risk Processes Fluid Limits Directionally Mixed Poisson Models Attracting Intensities

3 A Multivariate Pure Birth Process Let N = (N(t)) t 0 with N(t) = (N 1 (t),..., N d (t)) be a vector process consisting of d one-dim. counting processes s.t. IP [ N(t + h) = k + e j N(t) = k ] = hβ j (k) + o(h) as h 0, for all k = (k 1,..., k d ) Z d +. We assume that β(k) > 0 for all k Z d +, η := sup k E β(k) <. Interpretation: N j (t) = number of claims in business line j up to time t.

4 Examples (d = 2) β 1 = β 2 1/2. β 1 (k 1, k 2 ) = 1+k 1 2+k 1 +k 2, β 2 (k 1, k 2 ) = 1+k 2 2+k 1 +k 2 Repelling. β 1 (k 1, k 2 ) = 1+k 2 2+k 1 +k 2, β 2 (k 1, k 2 ) = 1+k 1 2+k 1 +k 2 Attracting.

5 Examples (d = 2)

6 Examples (d = 2)

7 Dependence Structure A Markov process X is said to be stochastically monotone if x IE [ f (X(t)) X(s) = x ] is increasing for all increasing f : R d R and for all 0 s < t. A Markov process X is said to be associated if X(t 1 ),..., X(t n ) is associated for all 0 t 1 <... < t n, n N. Theorem If β is increasing, then N is stochastically monotone and associated.

8 Risk Models d business lines labeled j = 1,..., d. Claim sizes (U j,k ) k N are independent with distribution Q j, mean µ j and finite second moment. Premium income rate c j > 0 for the j-th business line. The multivariate risk reserve process R = (R t ) t 0 is then defined by R(t) = (R 1 (t),..., R d (t)), with R j (t) := R j (0) + c j t N j (t) k=1 U j,k for all t 0.

9 Fluid Limits Introduce scaling parameter γ > 0, N γ j (t) = 1 γ N j(γt), and R γ j (t) = R j (0) + c j t 1 γ In R d we have the probability simplex N j (γt) W d := { (x 1,..., x d ) R d + : x x d = 1 }. We assume that for a.a. x W d and all (x n ) n N R d +, lim x n =, n lim n k=1 U j,k x n = x = lim x n β( x n ) = β(x). n and extend β to R d + \ {0} by β(x) := β( x 1 x).

10 Examples of Fluid Scaling

11 Fluid Limits Theorem Every sequence (γ(k)) k N (0, ) with γ(k) has a subsequence (γ(k(l))) l N such that N γ(k(l)) φ, R γ(k(l)) r as l. Further, the paths of the limit processes φ and r are almost surely absolutely continuous, and satisfy φ(t) = t where x y = (x 1 y 1,..., x d y d ). 0 β ( φ(s) ) ds. r(t) = r(0) + ct µ φ(t).

12 Examples β 1 = β 2 1/2, then φ(t) = (t/2, t/2). 1, if k 2 = k 1 + 1, 1, if k 1 = k 2 + 1, β 1 (k 1, k 2 ) = 1 2, if k 2 = k 1, β 2 (k 1, k 2 ) = 1 2, if k 2 = k 1, 0, otherwise, 0, otherwise. Again φ(t) = (t/2, t/2).

13 Models with Repelling Intensities Assume that β 1 (k 1, k 2 ) = 1 + k k 1 + k 2, β 2 (k 1, k 2 ) = 1 + k k 1 + k 2. β exists and is given by β 1 (x 1, x 2 ) = x 1 x 1 + x 2, β 2 (x 1, x 2 ) = x 2 x 1 + x 2 for all x 1, x 2 R 2 +\{0}. The associated fluid limit equation has a whole family {φ α (t) = ( αt, (1 α)t ) : 0 α 1} of solutions. Relation to the Polya Eggenberger urn model provides an U U(0, 1) s.t. 1 γ N(γt) ( Ut (1 U)t ), γ.

14 Models with Repelling Intensities In particular the fluid limit of the first risk process is r 1 (t) = r 1 (0) + c 1 t µ 1 Ut, t 0. 1 If c 1 = µ δ for δ > 0 we obtain ψ 1 (u) = IP(R 1 1 (t) < 0 for some t 0 R1 1 (0) = u) ψ 1 (uγ) IP(R γ 1 (t) < 0 Rγ 1 (0) = u), γ 1, t > 0 ( IP U > δ + 1 ) = 1 µ δ. µ 1 In particular ψ 1 (u) 0 for u.

15 Directionally Mixed Poisson Models Let Z be a r.v. with values in W d and distribution ν. Let L be a Poisson process with intensity λ. Given Z = (z 1,..., z d ), construct N = (N 1,..., N d ) by independently assigning the events of L to the j-th component with probability z j. Theorem N is a time-homogeneous Markov chain with intensities β j (k) = λ IE Z k+e j IE Z k for all k E, where x k = x k x k d d. The fluid limit for N is of the form φ(t) = tλz for all t 0.

16 Example The model with repelling intensities fits into the framework of directionally mixed Poisson models by setting d = 2, λ = 1, Z := (U, 1 U) where U U(0, 1). This gives β 1 (k 1, k 2 ) = IE Uk 1+1 (1 U) k 2 IE U k 1 (1 U) k 2 = 1 + k k 1 + k 2.

17 Ruin Asymptotics for Directionally Mixed Models Consider the d-dimensional risk reserve process R = (R 1,..., R d ) with H j (t) := e tx Q j (dx) < for some t > 0, lim H j (t) = with t 0 := sup{t > 0 : H j (t) < }. t t0 Let R 0 (t) := R 1 (t) R d (t) and ψ 0 (u) := IP(R 0 (t) < 0 for some t 0 R 0 (0) = u).

18 Ruin Asymptotics for Directionally Mixed Models Theorem Suppose that c > λ sup{ z, µ : z supp(ν)}. Then ψ 0 (u) e κ0u 1 for all u > 0, lim u u log ψ 0(u) = κ 0, where the Lundberg exponent κ 0 is given by κ 0 = inf{κ(z) : z supp(ν)} and κ(z) is the Lundberg exponent for the one-dimensional model with premium income rate c, claim arrival intensity λ and claim size distribution d j=1 z jq j.

19 Attracting Intensities Suppose that β 1 (k 1, k 2 ) = 1+k 2 2+k 1 +k 2, β 2 (k 1, k 2 ) = 1+k 1 2+k 1 +k 2 and define Q := 1 2 Q Q 2 and H (r) = e rx Q (dx). Theorem Suppose that c > (µ 1 + µ 2 )/2 and that Q 1 st Q 2 or Q 2 st Q 1. Then it holds that 1 lim u u log ψ 0(u) = κ, where κ is the Lundberg exponent for the one-dimensional model with premium income rate c, claim arrival intensity 1 and claim size distribution Q.

20 References Bäuerle, N., Grübel, R. (2008) : Multivariate risk processes with interacting intensities. Adv. Appl. Prob., 40: Glynn P.W., Whitt W. (1994) : Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Prob. 31A: Liggett, T.M. (1985) : Interacting particle systems. Springer, New York. Zocher, M. (2005) : Multivariate mixed Poisson processes. Doctoral Thesis, TU Dresden Thank you very much for your attention!