Séminaire de Statistiques de l'irma Modèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine Romain Biard LMB, Université de Franche-Comté en collaboration avec Søren Asmussen, Claude Lefèvre, Stéphane Loisel et Haikady Nagaraja 1 / 35
Outline 1 Introduction 2 Biard et al. (2010) 3 Model and regenerative structure 4 Heavy-tailed claim amounts 5 Light-tailed claim amounts Large deviation theory point of view Markov additive process point of view 2 / 35
Ruin theory Solvency of an insurance operation. Evolution of the reserve of an insurance company: starting with some initial capital premium inow claim payments. Main problem : probability that the reserve is always sucient to cover the claims that occur. If not : ruin. 3 / 35
Risk process : Insurance company's reserve evolution N(t) R(t) = u + ct X i. i=1 4 / 35
Classical assumptions where N(t) R(t) = u + ct X i, i=1 (N(t)) t 0 : Poisson process with parameter λ. Claim amounts (X i ) i 1 : sequence of independent and identically distributed positive random variables. Claim inter-occurrence times (V i ) i 1 : sequence of independent and identically distributed random variables. (X i ) i 1 is independent from (V i ) i 1. Remark : by convention, N(t) i =1 X i = 0 if N(t) = 0. 5 / 35
Classical problems Finite-time ruin probability: ψ(u, T ) = P( τ [0, T ], R(τ) < 0 R(0) = u), and innite-time ruin probability: ψ(u) = lim T ψ(u, T ). 6 / 35
Motivation In practice, (X i ) i 1 is generally not a i.i.d. sequence. The mutual independence of X 1,.., X n,..., V 1,..., V n is not realistic. The time elapsed since the last claim or since the last large claim may have a strong impact on the conditional distribution of the next claim amount. In some cases, the whole history of the risk process may play a role. Earthquake type phenomena. Flooding type phenomena. 7 / 35
1 Introduction 2 Biard et al. (2010) 3 Model and regenerative structure 4 Heavy-tailed claim amounts 5 Light-tailed claim amounts 8 / 35
Biard et al. (2010) In Biard et al. (2010), each claim amount X k is either Y k or Z k. The Y k, k 1 are dependent, and so are the Z j, j 1 (but independent from the Y k, k 1). = Derive asymptotics of nite-time ruin probabilities for large risks ψ(u, t) = P( τ [0, t], R(τ) < 0 R(0) = u), in more general models with inter-dependence between claim amounts and dependence between claim amounts and inter-occurrence times. 9 / 35
Framework : Regular variation distributions Univariate setting A random variable X is regularly varying if there exists some α > 0 such that, for all t > 0, P(X > tx) We denote X R α. t α P(X > x). x Heavy-tailed distribution. Multivariate setting A random vector X is regularly varying if there exists a non-null Radon measure µ such that, for all Borel sets A bounded away from 0 with µ( A) = 0, P(X xa) µ(a)p( X > x). x We have for all x > 0 and A dened as above, µ(xa) = x α µ(a). We denote X MR µ, α. Pareto-type distribution : P(X > x) x α, for some α > 0. The smaller the α, the heavier the tail. Extreme events : earthquake, ooding, terrorist attacks... 10 / 35
Finite-time ruin probability and regular variation distribution Ruin probability in the i.i.d. case Let, for t 0, S t = N(t) i=1 X i. Assume (X i ) i 1 is an i.i.d. regular variation sequence. Then, for T > 0, ψ(u, T ) P(S T > u) (λt ) P(X 1 > u). u u Useful result in the non-i.i.d. case For X = (X (1),..., X (d) ) MR µ, α, there exists q d,α such that P(X (1) +... + X (d) > x) q d,α P(X (1) > x). x Special copulas Independent Fréchet upper bound Gaussian q k,α k k α k Table: Values of q k,α for dierent copulas. 11 / 35
Earthquake 12 / 35
Let k > 0. If k successive inter-occurrence times are larger than τ, then the subsequent claim amount X j is equal to a r.v. Y j else it is equal to Z j. For all j 1, (Y 1,..., Y j ) is α-regularly varying (with cdf F ). For all j 1, (Z 1,..., Z j ) is β-regularly varying (with cdf G). M +(n, k, t, τ) = the random variable that counts the number of sequences, during (0, t), of k consecutive Poisson spacings which are larger than τ, given that N(t) = n ( 1). If α < β, for t > 0 and large u, we get the ruin probability min( n/k, t/kτ ) ψ + k (u, t) P[N(t) = n] P [M +(n, k, t, τ) = j] q j,α F (u+ct), where... n=1 j=1 13 / 35
Distribution of M + (n, k, t, τ)... for 0 j min ( n/k, t/kτ ), ( ) k 1 x 1 + + x k + j P [M +(n, k, t, τ) = j] = x 1,..., x k, j i=0 x 1,...,x k D k P[M(n, t, τ) = n x 1 x k ]/ where D k is the set of all nonnegative integers x 1,..., x k such that k rx r = n i kj and n r=1 k x r t/τ. ( n x 1 + + x k and M(n, t, τ) is the number of spacings of the process {N(s), 0 s t} larger than τ given that N(t) = n ( 1)... r=1 ), 14 / 35
Distribution of M(n, t, τ)... for 0 j n, P[M(n, t, τ) = j] = n i=j ( 1) i j ( n j )( n j i j ) P(V 1 > τ t,..., V i > τ t ), with P(V1 > v,..., V i > v) = 1, v 0, (1 iv) n, 0 < v < 1/i, 0, 1/i v < 1. Thanks to the Poisson hypothesis, claim instants U 1,..., U n are distributed as the n-th order statistics on [0, t]. Using inclusion-exclusion principle. 15 / 35
Impact of τ and k on ψ + (u, t) k τ α = 3 k 16 / 35
Flooding 17 / 35
Let k > 0. If k successive inter-occurrence times are smaller than τ and this sequence follows an inter-occurrence time larger than τ, or this sequence is constituted with the rst k inter-occurrence times, (such a sequence is called a good sequence). then X j is equal to a r.v. Z j else it is equal to Y j. For all j 1, (Y 1,..., Y j ) is α-regularly varying (with cdf F ). For all j 1, (Z 1,..., Z j ) is β-regularly varying (with cdf G). M (n, k, t, τ) = random variable that counts the number of good sequences during (0, t), given that N(t) = n ( 1). If α > β, for t > 0 and large u, we get the ruin probability min( (n+1)/(k+1), t/(τ 1) ) ψ k (u, t) P[N(t) = n] P [M (n, k, t, τ) = j] q j,β G(u+ct), where... n=1 j=1 18 / 35
Distribution of M (n, k, t, τ)... for 0 j min ( (n + 1)/(k + 1), t/(τ 1) ), ( ) n x 1 + + x n P[M (n, k, t, τ) = j] = x 1,..., x n i=0 x 1,...,x n E k P[M(n, t, τ) = x 1 + + x n]/ where E k is the set of all nonnegative integers x 1,..., x n such that ( n x 1 + + x n ), n rx r = n i, r=1 n r=k+1 x r = j 1 {i k} and n x r t/τ. r=1 19 / 35
1 Introduction 2 Biard et al. (2010) 3 Model and regenerative structure 4 Heavy-tailed claim amounts 5 Light-tailed claim amounts 20 / 35
Here If k = 2 successive inter-occurrence times are larger than τ then the cdf of the subsequent claim amount X j is G, else it is F. Independent claim amounts. Computation of asymptotics of innite-time ruin probability, ψ(u) = P ( t > 0, R(t) < 0 R(0) = u). Heavy-tailed case : Subexponential distribution Light-Tailed case Large deviation theory Markov additive process Key point : Regenerative structure. 21 / 35
Regenerative structure 22 / 35
Mean of the cycle length p + = P(V > τ) = e λτ, p = P(V τ) = 1 e λτ, m + = E[V V > τ] = 1 p + m = E[V V τ] = 1 p τ τ 0 xλe λx dx = τ + 1 λ, xλe λx dx = 1 λ τ e λτ 1 e λτ. E[ω] = µ = p p + m + m + + p + m + + p (m + µ). µ = 1 λ eλτ (1 + e λτ ). 23 / 35
1 Introduction 2 Biard et al. (2010) 3 Model and regenerative structure 4 Heavy-tailed claim amounts 5 Light-tailed claim amounts 24 / 35
Subexponential distribution A distribution K R + is said to be subexponential if, with K = 1 K, K K(x) lim = 2. x K(x) Subexponentiality of K implies that there exists a non-decreasing function δ with δ(x) as x, such that K ( x ± δ(x) ) K(x) as x. Assumption There exists a distribution K such that both K(x) and K I (x) = x K(y)dy are 0 subexponential and that F (x) c F K(x), G(x) c G K(x), x, where c F + c G > 0. 25 / 35
Result Let and. P(S > x) = P P(M > x) = P sup N(ω 2 ) i=1 ω 1 <t ω 2 X i cω 2 > x ω1 = 0 N(t) X i ct > x ω1 = 0 i=1 Let M be the number of claims in a cycle and A = M i=1 X i. We have, for large x, if e gδ(x) = o( K I (x) ) for all g > 0, P(S > x) P(A > x) d K(x), P(M > x) P(A > x) d K(x), where d = E(M 1)c F + c G = (λµ 1)c F + c G. 26 / 35
Sketch of Proof I Let A = M i=1 X i, with M the total number of claims in a cycle. We know that P (M 1 X i > ) x E(M 1)c F K(x). i=1 Since X M has tail c G K(x) and is independent of X 1,..., X M 1, we get that P(A > x) dk(x), where d = E(M 1)c F + c G. Let δ such that We have P ( A > x ± cδ(x) ) P ( A > x ) as x. P(A cω > x) = P ( A cω > x, ω δ(x) ) + P ( A cω > x, ω > δ(x) ). Since ω has nite exponential moments, we have, for some constant γ > 0, P ( A cω > x, ω > δ(x) ) P ( ω > δ(x) ) e γδ(x) = o( K(x) ). 27 / 35
Sketch of Proof II Moreover, P ( A cω > x, ω δ(x) ) P(A > x) = dk(x), P ( A cω > x, ω δ(x) ) P ( A cδ(x) > x, ω δ(x) ) Finally, as x, and from S M A. = P ( A cδ(x) > x ) P ( A cδ(x) > x, ω > δ(x) ) dk(x) o( K(x) ). P(A > x) dk(x), (1) P(S > x) dk(x), (2) P(M > x) dk(x) (3) 28 / 35
Result There exists random times ω 0 = 0, ω 1, ω 2,... such that the post-ω k process {S(t + ω k ) S(ω k )} t 0 is independent of the pre-ω k process {S(t)} 0 t ωk and its distribution does not depend on k. P(S > x) P(M > x), x. So, from Asmussen, Schmidli & Schmidt (1999) Under some other weak assumptions, ψ(u) d ηcµ K I (u), u, where η = 1 (λµ 1)m F +m G cµ and m F, m G denote the means of F, resp. G. 29 / 35
1 Introduction 2 Biard et al. (2010) 3 Model and regenerative structure 4 Heavy-tailed claim amounts 5 Light-tailed claim amounts Large deviation theory point of view Markov additive process point of view 30 / 35
Large deviation theory point of view Large deviation theory F, G light-tailed If, in a suitable α-range, 1 log EeαS(t) t κ(α) for some function κ and κ(γ) = 0 for some γ > 0, then 1 log ψ(u) γ. u 2 steps to nd γ. Computation of ϕ(α, β) = E [ e αs(ω2) e βω2 ω1 = ] 0. Choose κ = κ(α) as the solution of 1 = ϕ(α, κ). 31 / 35
Large deviation theory point of view Let F [ ] be the m.g.f. of F and Ĝ[ ] be the m.g.f. of G. Dene  + [α] = E[e αv V > τ] =  [α] = E[e αv V τ] = λ λ α eατ, λ ( 1 e (λ α)τ ) λ α 1 e λτ. ϕ(α, β) = p + 1 p F [α]  [β cα] F [α]â+ [β cα] ( ) p + Ĝ[α]Â+ [β cα] + p  [β cα] F [α]ϕ(α, β) 32 / 35
Markov additive process point of view Markov additive process F, G light-tailed Markov process J with state space E. Claim size distribution is F i when J(t) = i. a : time since the last arrival and b : the length of the previous interarrival interval. J(t) =, a if b, a τ, J(t) =, + if b τ, a > τ, J(t) = +, a if b > τ, a τ and J(t) = +, + if a, b > τ. E = { } [0, r] {, +} {+} [0, r] {+, +} F i = G when i = +, +, F i = F for all other i E. Note that after a G-claim, J is reset to, 0. ψ i (u) = P ( t > 0, R(t) < 0 R(0) = u, J(0) = i) e γu h γ (i)c. h θ (i) = E i e θs(ω 1) κ(θ)ω 1, computable for all i. C is a constant, hardly computable. 33 / 35
Markov additive process point of view Remarks k > 2 is also feasible. Flooding model is also feasible since the regenerative structure is preserved. Earthquake aftershocks : shot-noise process. Perspectives : Non-Poisson case. Model validation : Chaire Actuariat Responsable. 34 / 35
35 / 35 Markov additive process point of view Merci pour votre attention!