Ruin probabilities in multivariate risk models with periodic common shock

Transcription

1 Ruin probabilities in multivariate risk models with periodic common shock January 31, 2014

2 3 rd Workshop on Insurance Mathematics Universite Laval, Quebec, January 31, 2014 Ruin probabilities in multivariate risk models with periodic common shock Ionica Groparu Cojocaru Postdoctoral Fellow, Concordia University, Montreal Supervisor Dr. Jose Garrido, Concordia University, Montreal Research funded by the Fonds de recherche du Quebec Nature et technologies (FRQNT )

3 Outline 1 Multivariate risk model description Ruin probabilities Review of the literature 2 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Numerical illustrations Multivariate risk model perturbed by a diffusion process 3 Asymptotic behavior of the finite-time ruin probabilities 4 References

4 Multivariate risk model description Great interest has been shown in developing multivariate risk models with dependence.

5 Multivariate risk model description The approach to modeling dependent classes of business by incorporating a common component into each of the associated claim-number processes has been studied by many authors, for example, Ambagaspitiya (1998), Cossette and Marceau (2000), Wang and Yuen (2005).

6 Multivariate risk model description Consider an insurance company with m 1 classes of business. The surplus process {U i (t), t 0} of the i-th class of business is given by where N p(t) U i (t) = u i + c i t X ik, t 0, i = 1,..., m, (1.1) k=1 u i = U i (0) 0 is the initial capital; c i > 0 is the constant premium income per unit time; N p (t) is the number of claims up to time t; X ik is the size of the k-th claim.

7 Multivariate risk model description Dependence structure For fixed i = 1,.., m, {X ik } k 1 are independent and identically distributed (i.i.d.) nonnegative random variables with distribution function (d.f.) F i (x i ) such that F i (0) = 0 and finite mean µ i = E[X i ]. {(X 1k,..., X mk )} k 1 is a sequence of i.i.d. m-dimensional random vectors with joint d.f. F (x 1,..., x m ). Assume that {N p (t), t 0}, {(X 1k,..., X mk )} k 1 are mutually independent.

8 Multivariate risk model description Ruin probabilities The time of ruin for the i-th class (1 i m) is defined by τ i = inf{t 0 : U i (t) < 0}, and the corresponding probability of ruin is ψ i (u i ) = P(τ i < U i (0) = u i ). If for each i, the process U i (t) 0 for all t 0 (no ruin occurs), we indicate this by writing τ i =. The net profit condition (see Rolski et al. (1999)): E[U i (t)] lim > 0, i = 1, 2,..., m. t t

9 Multivariate risk model description Ruin probabilities Different ruin concepts for multivariate risk processes are introduced by Chan et al. (2003). For instance, The first time when ruin occurs in all classes simultaneously or at the same instant in time is defined by τ sim = inf{t 0 : max{u 1 (t),..., U m (t)} < 0}. Then, the associated infinite time ruin probability is given by ψ sim (u 1,..., u m ) = P{τ sim < (U 1 (0),..., U m (0)) = (u 1,..., u m )}, and the finite-time ruin probability, for a fixed time t, is ψ sim (u 1,..., u m, t) = P{τ sim t (U 1 (0),..., U m (0)) = (u 1,..., u m )}.

10 Multivariate risk model description Ruin probabilities The first time when ruin occurs in all classes, but not necessarily simultaneously is defined by τ and = max(τ 1,..., τ m ). The first time when ruin occurs in at least one class of business is τ or = min(τ 1,..., τ m ). Based on these times, the ruin probabilities are denoted by ψ and (u 1,..., u m ) and ψ or (u 1,..., u m ), respectively.

11 Multivariate risk model description Review of the literature {N p (t), t 0}=Homogeneous Poisson process: Chan, Yang and Zhang (2003): partial integro-differential equation for φ or (u 1, u 2 ) and simple lower and upper bounds for ψ or (u 1, u 2 ) and ψ sim (u 1, u 2 ); Gong, Badescu and Cheung (2012): recursive integral formulas for ψ or (u 1, u 2,..., u m ) and results for ψ and (u 1, u 2 ); Cai and Li (2005, 2007): stochastic bounds for ψ and (u 1, u 2,..., u m ) and ψ or (u 1, u 2,..., u m ) ; Yuen, Guo and Wu (2006) : approximation to the finite-time φ or (u 1, u 2, t) and an upper bound for ψ or (u 1, u 2 );

12 Multivariate risk model description Review of the literature Li, Liu and Tang (2007) : Lundberg-type upper bound for ψ sim (u 1, u 2 ) and asymptotic estimate of ψ sim (u 1, u 2, t) (independent claims) in a bivariate perturbed risk model; Asmussen and Albrecher (2010) : Lundberg-type upper bound for ψ sim (u 1,..., u m ) and asymptotic estimate of ψ sim (u 1,..., u m, t) (independent claims).

13 Multivariate risk model description Review of the literature {N p (t), t 0}=Renewal process: Chen, Yuen and Ng (2011) :asymptotic estimates for ψ or (u 1, u 2, t) and ψ and (u 1, u 2, t) (independent claims); I. Groparu Cojocaru (2012) : Lundberg-type upper bound for ψ sim (u 1,..., u m ) in a multivariate Poisson model with renewal common shock.

14 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim {N p (t), t 0} is a non-homogeneous Poisson process with time-dependent periodic intensity function λ p (t), where λ p (t) 0 for t 0. Univariate periodic risk model was considered by, for example, Beard et al. (1984); Dassios and Embrechts (1989); Asmussen and Rolski (1994); Garrido et al. (1996); Morales (2005).

15 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim If b (years) N + is a period of λ p, then t t/b b [0, b) is the time of the season, where t is the integer part of t. The function defined by Λ p (t) = t 0 λ p (x)dx, called the cumulative intensity function of the process {N p (t), t 0}, has the almost linear property (Dimitrov et al. (1997)): Λ p (t) = t b Λ p(b) + Λ p (t t b), t 0. b

16 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Then, Λ p (t) lim = Λ p(b) not = λ 0. t t b λ 0 is called average arrival rate (Asmussen and Rolski (1994)). The net profit condition becomes c i > λ 0 µ i, for i = 1, 2,..., m.

17 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Let us consider the case of light-tailed marginal claim size distributions: if M Xi (r i ) = E[e r i X i ] is the moment generating function (m.g.f.) of X i (1 i m), then there exists 0 < ri 0 such that It results that M Xi (r i ) < for all r i < r 0 i and lim r i ri 0 M Xi (r i ) =. M = {(r 1,.., r m ) [0, r 0 1 ).. [0, r 0 m) M X1,..,X m (r 1,.., r m ) < } {(0,.., 0)}, where M X1,..,X m (r 1,..., r m ) is the joint m.g.f. of (X 1,..., X m ).

18 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Approach: Assume that the vector process {(U 1 (t),..., U m (t)), t 0} is defined on the filtered probability space (Ω, F, {F t } t 0, P), where F t = F U 1 t... Ft Um and F U i t is the natural filtration of the process {U i (t), t 0}. Martingale technique via piecewise deterministic Markov processes (Davis (1984)): Z(t) = e m i=1 r i U i (t) tg(r 1,...,r m) [ (t t b b)λ 0+Λ(t t b b)][m X 1,...,Xm (r 1,...,r m) 1], is a martingale with respect to the filtered probability space (Ω, F, {F t } t 0, P),

19 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim where g(r 1,..., r m ) = Let us define l = m c i r i + λ 0 [M X1,...,X m (r 1,..., r m ) 1]. i=1 sup v [0,b) ( v + Λ p(v) ) and λ 0 S = {(r 1,..., r m ) M g(r 1,..., r m ) = 0}. Theorem If sup g(r 1,..., r m ) > 0, then (r 1,...,r m) M ψ sim (u 1,..., u m ) provided that u i c i l 0 for i = 1,..., m. m r i (u i c i l) inf (r 1,...,r e i=1, m) S

20 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Corollary If N p (t) Poisson(λ) and ψ sim (u 1,..., u m ) Asmussen and Albrecher (2010) sup g(r 1,..., r m ) > 0, then (r 1,...,r m) M inf m r i u i (r 1,...,r e i=1. m) S

21 Lundberg-type upper bound for the infinite time ruin probability An upper bound for the ruin probability ψ sim Corollary (Trigonometric form) If λ p (t) = λ 0 + λ 1 sin[ 2π T (t + t 0)] and sup g(r 1,..., r m ) > 0, (r 1,...,r m) M then ψ sim (u 1,..., u m ) m r i (u i c i l) inf (r 1,...,r e i=1, m) S where l = λ 1T 2πλ 0 [1 + cos 2πt 0 T ] and u i c i l 0 for i = 1,..., m. Dassios and Embrechts (1989): univariate case.

22 Lundberg-type upper bound for the infinite time ruin probability Numerical illustrations Beta-shape periodic intensity model (Garrido and Lu (2004)) In this sense, if we assume that the period is 1 year, then the intensity function is given by λ p (t) = λ 0λ 1 (t t ), for t 0, where λ 0 > 0 is the (constant) peak level of this intensity and λ 1 is a beta-type function defined on [0, 1], such that λ 1 (t 1 ) = 1, where t 1 [0, 1] is the mode of the function: λ 1 (t) = ( t m 1 ) q 1 1 d ) p1 1 (1 t m 1 d for 0 m 1 t m 2 1, α 1 0 otherwise, where d = m 2 m 1, α 1 = ( t 1 m 1 ) p1 1 (1 t 1 m 1 ) q1 1 is a d d p 1 1 scale factor, while t 1 = m 1 + d p 1 + q 1 2.

23 Lundberg-type upper bound for the infinite time ruin probability Numerical illustrations The corresponding cumulative intensity function is Λ(t) = λ 0 d [ t B(p 1, q 1 ) + B(p 1, q 1 ; t t m ] 1 ), t 0, α 1 d where B(p, q) = 1 0 v p 1 (1 v) q 1 dv = Γ(p)Γ(q) Γ(p + q) is the beta function at p, q > 0, while 0 if t 0, t B(p, q, ; t) = v p 1 (1 v) q 1 dv if t (0, 1), is the usual incomplete beta function. 0 B(p, q) if t 1

24 Lundberg-type upper bound for the infinite time ruin probability Numerical illustrations m 1 = 5/12, m 2 = 11/12, p 1 = 3, q 1 = 2, λ 0 = 5; It results λ 0 = 1.40, l = ; X 1 Exp(α 1 ) and X 2 Exp(α 2 ) and bivariate Farlie-Gumbel-Morgenstern (FGM) copula: F (x 1, x 2 ) = (1 e α 1x 1 )(1 e α 2x 2 ) ( 1 + ρe α 1x 1 e α 2x 2 ), where ρ [ 1, 1] and the linear correlation coefficient is Corr(X 1, X 2 ) = ρ/4; The mean claim sizes: µ 1 = 1/α 1 = 5, µ 2 = 1/α 2 = 1; The safety loading coefficients: θ 1 = 0.5, θ 2 = 0.4 The premium rates, computed as c i = (1 + θ i )λ 0 µ i, have the following values c 1 = 7.35 and c 2 = 1.46 lc 1 = 0.69, lc 2 = 0.14; (u 1, u 2 ) = (100, 50).

25 Lundberg-type upper bound for the infinite time ruin probability Numerical illustrations The effect of the correlation coefficient of the claims The upper bound of ψ sim (u 1, u 2 ) is increasing in the correlation parameter ρ.

26 Lundberg-type upper bound for the infinite time ruin probability Numerical illustrations Values of e 2 r i (u i c i l) r2 =kr 1 i=1 ρ k = 0.01 k = 1 k =

27 Lundberg-type upper bound for the infinite time ruin probability Multivariate risk model perturbed by a diffusion process Multivariate risk model perturbed by a diffusion process N p(t) Ui d (t) = u i + c i t X ik + σ i B i (t), t 0, i = 1, 2,.., m, k=1 where B(t) = (B 1 (t),..., B m (t)) denotes a standard correlated m-dimensional Brownian motion with constant correlation coefficients ρ ii = 1, ρ ij = ρ ji [ 1, 1], i, j = 1,..., m, and σ i 0 are the diffusion volatility coefficients of B i (t). The diffusion term accounts for some market uncertainties such as uncertainty of the aggregate claims or of the premium income.

28 Lundberg-type upper bound for the infinite time ruin probability Multivariate risk model perturbed by a diffusion process Let us denote S d = {(r 1,..., r m ) M g d (r 1,..., r m ) = 0}, where and l = g d (r 1,..., r m ) = m i=1 m c i r i + λ 0 [M X1,...,X m (r 1,..., r m ) 1] i=1 sup ( v + Λ(v) ). v [0,b) λ 0 σ 2 i r 2 i + 1 i<j m σ i σ j ρ ij r i r j,

29 Lundberg-type upper bound for the infinite time ruin probability Multivariate risk model perturbed by a diffusion process Theorem Assume that sup (r 1,...,r m) M g d (r 1,..., r m ) > 0 and m σi 2 ri σ i σ j ρ ij r i r j 0 for m 3. Then i=1 1 i<j m ψ sim (u 1,..., u m ) inf e (r 1,...,r m) S d provided that u i c i l 0 for i = 1,..., m. m i=1 r i (u i c i l),

30 Asymptotic behavior of the finite-time ruin probabilities Some classes of heavy-tailed distributions Let F (x) be a distribution function on [0, ) such that F (x) = 1 F (x) > 0 for all x 0. Then 1. F is said to belong to the dominant variation class D if lim sup x F (tx) F (x) < holds for some (or, for all) 0 < t < F is said to belong to the subexponential class S if F lim n (x) = n x F (x) holds for some (or, equivalently, for all) n = 2, 3,...

31 Asymptotic behavior of the finite-time ruin probabilities Some classes of heavy-tailed distributions 3. F is said to belong to the long tailed class L if the relation F (x + t) lim = 1 holds for all real t. x F (x) Proposition (Embrechts et al. (1997)) i. D L S L. ii. If X 1, X 2,...are i.i.d. random variables with common distribution function F S then, for every n = 2, 3,..., P(X 1 + X X n > x) P(max{X 1, X 2,..., X n } > x) nf (x).

32 Asymptotic behavior of the finite-time ruin probabilities Li, Liu and Tang (2007) : asymptotic estimate of ψ sim (u 1, u 2, t) (independent and subexponentially distributed claims) in a bivariate perturbed risk model; Asmussen and Albrecher (2010) : asymptotic estimate of ψ sim (u 1,..., u m, t) (independent and subexponentially distributed claims).

33 Asymptotic behavior of the finite-time ruin probabilities Theorem Consider the multivariate risk model defined by (1.1) with the assumption that the claim sizes X 1,...,X m (m 2) are independent and subexponentially distributed. Then, as u 1,...,u m, we have ψ sim (u 1,..., u m, t) E[N p (t) m ]F 1 (u 1 )...F m (u m ).

34 Asymptotic behavior of the finite-time ruin probabilities Dependent heavy-tailed claims Assumption 1. [Ko and Tang (2008)] Let X 1,...,X n be n (n 2) random variables with distributions F 1,...,F n concentrated on [0, ), respectively. Assume that there exists some large x 0 > 0 such that, for every j = 2,.., n, the relation P(X X j 1 > x t X j = t) P(X X j 1 > x t) = O(1) holds uniformly for all t [x 0, x], meaning that lim sup x sup x 0 t x P(X X j 1 > x t X j = t) P(X X j 1 > x t) <.

35 Asymptotic behavior of the finite-time ruin probabilities Theorem (Ko and Tang (2008)) Let X 1,...,X n be n (n 2) random variables with distributions F 1,...,F n concentrated on [0, ), respectively, such that Assumption 1 holds for all j = 2,..., n. Then the relations P(X X n > x) P(max{X 1,..., X n } > x) n F k (x) k=1 hold for each of the following two cases: (i) F k S for all k = 1,.., n, and either F i (x) = O(F j (x)) or F j (x) = O(F i (x)) for all i, j = 1,..., n; (ii) F k D L for all k = 1,..., n. f (x) g(x) means that lim sup f (x)/g(x) = 1 as x.

36 Asymptotic behavior of the finite-time ruin probabilities Theorem Assume that the claim sizes X 1,...,X m (m 2) follow a dependence structure given by Assumption 1, with F 1,...,F m satisfying the conditions of cases (i) or (ii) of the above theorem. 1. If u i for some i = 1, 2,..., m, then ψ sim (u 1,..., u m, t) E[N p (t)] ψ or (u 1,..., u m, t) 1 m E[N p(t)] m F i (u u m ); i=1 m F i (u u m ); i=1 f (x) g(x) means that lim sup f (x)/g(x) 1 as x f (x) g(x) means that lim inf f (x)/g(x) 1 as x.

37 Asymptotic behavior of the finite-time ruin probabilities Theorem (continued) 2. If u i for all i = 1, 2,..., m, then ψ and (u 1,..., u m, t) E[N p (t)] m i=1 F i ( min 1 i m u i).

38 References Ambagaspitiya R. S., On the distribution of a sum of correlated aggregate claims, Insurance: Mathematics and Economics, 23, 15-19, Asmussen S. and Rolski T., Risk theory in a periodic environment: the Cramér-Lundberg approximation and Lundberg inequality, Mathematics of Operations Research, 19, , Asmussen S. and H. Albrecher H., Ruin probabilities, Second Ed. World Scientific, New Jersey, Beard R.E., Pentikäinen T. and M. Pesonen M., Risk theory, 3rd ed., Chapman and Hall, London, Cai J. and Li H., Multivariate risk model of phase type, Insurance: Mathematics and Economics, 36, , Cai J. and Li H., Dependence properties and bounds for ruin probabilities in multivariate compound risk models, Journal of Multivariate Analysis, 98, , 2007.

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41 References Ko B. and Tang Q., Sums of dependent nonnegative random variables with subexponential tails, Journal of Applied Probability, 45, 85-94, Lu Y. and Garrido J., Doubly periodic non-homogeneous Poisson models for hurricane data, Statistical Methodology, 2, 17-35, Li J., Liu Z. and Tang Q., On the ruin probabilities of a bidimensional perturbed risk model, IME, 41, , M. Morales, On a surplus process under a periodic environment: a simulation approach, North American Actuarial Journal, 8, 4, 76-89, Rolski T., Schmidli H., Schmidt V. and Teugels J.L., Stochastic Processes for Insurance and Finance, Wiley, New York, 1999.

42 References Wang G. and Yuen K., On a correlated aggregate claims model with thinning-dependence structure, Insurance: Mathematics and Economics, 36, , Yuen K., Guo J. and Wu X., On the first time of ruin in the bivariate compound Poisson model, Insurance: Mathematics and Economics, 38, , 2006.

43 References Thank you for your attention.

44 References