A Ruin Model with Compound Poisson Income and Dependence Between Claim Sizes and Claim Intervals
|
|
- Erik Hodges
- 6 years ago
- Views:
Transcription
1 Acta Mathematicae Applicatae Sinica, English Series Vol. 3, No. 2 (25) DOI:.7/s & Acta Mathema cae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 25 A Ruin Model with Compound Poisson Income and Dependence Between Claim Sizes and Claim Intervals Yuan-yuan HAO,2, Hu YANG, Department of Statistics and Actuarial Science, Chongqing University, Chongqing 433, China ( yh@cqu.edu.cn) 2 College of Mathematics, Chongqing Normal University, Chongqing 433, China Abstract We consider a ruin model with random income and dependence between claim sizes and claim intervals. In this paper, we extend the determinate premium income into a compound Poisson process and assume that the distribution of the time between two claim occurrences depends on the previous claim size. Given the premium size is exponentially distributed, the (Gerber-Shiu) discounted penalty functions is derived. Finally, we consider a similar model. Keywords discounted penalty function; laplace transform; ruin model; dependence 2 MR Subject Classification 9B3; 6J25 Introduction The classical compound Poisson risk model to describe the surplus process of an insurance portfolio relies on the assumption of independence between claim sizes and claim intervals. As the time went by, this assumption is often too restrictive and there is a need for more general models. Albrecher and Boxma 2 discussed a risk model in which the Poisson arrival rate of next claim is determined by the previous claim size. Essentially, it is a Markov-dependent risk model 3. In addition, Zhou and Cai 5 have considered the dependent risk model with diffusion. And they obtained the explicit expression of the ruin probability. Boudreault et.al. 8 have given a risk model with the reverse dependent structure (i.e. the distribution of the next claim size depends on the last interclaim time). For these risk models, it is explicitly that the premiums are assumed to be received at a constant rate over time. Bao 5 discussed a ruin model, in which the premium is no longer a linear function of time but a Poisson process. Essentially, it is a dependent risk model 3. Bao and Ye 7 studied the discounted penalty function in a class of delayed renewal risk model with random income. Labb and Sendova 2 considered a risk model where both premiums and claims follow compound Poisson processes. For a dependent ruin model, Zhang and Yang 4 derived the discounted penalty penalty function. With this extension, the surplus process has two-side jumps instead of one-side jump only in classic risk model. And for the risk model with two-side jumps, Chi 9 analyzed the discounted penalty function. In this paper a generalization of a ruin model is considered in Section 2, where the income is compound Poisson process and the distribution of the claim intervals depends on the previous Manuscript received January 9, 2. Revised June, 2. Supported by the National Natural Science Foundation of China (No.4265), National Social Science Fund of China (3BTJ8), Scientic and Technological Research Program of Chongqing Municipal Education Commission (No.KJ452, No.KJ3658) and the Fundamental Research Funds for the Central Universities (No. CDJXS8) Corresponding author.
2 446 Y.Y. Hao, H. YANG claim size. In Section 3, the Laplace transforms of the discounted penalty functions are discussed and an example has been shown. In Section 4, we consider a similar ruin model. 2 Model Let us consider the following ruin model U(t) which is an insurance portfolio M(t) N(t) U(t) = u + Y j X i, (2.) j= where u = U() is the initial capital and Y j is the jth premium income with distribution function G( ) and mean σ. M(t) is a Poisson process with intensity. N(t) is a claimnumber process representing the number claims up to time t with interclaim times {W i }. {X i } are assumed to be independent and identically distributed (i.i.d.) with distribution function F(x) = F(x), mean µ. We assume the claim occurrence process to be of the following Markov type: If a claimx i is not less than a threshold B i, then the time until the next claim W i+ is exponentially distributed with rate, otherwise it is exponentially distributed with rate 2. The quantities B i are assumed to be i.i.d. random variables with distribution function B( ) = B( ). Now define T = inf{t : U(t) < } to be the ruin time, Ψ(u) = P(T < U() = u) to be the ruin probability. From 4, We know that µ < Pr(X B) σ + Pr(X < B) σ 2, (2.2) to insure the positive safety loading condition. Let δ be a constant, and ω(x, x 2 ) be a nonnegative measurable function defined on, ) (, ). The discounted penalty function is of the following form m(u) = Ee δt ω(u(t ), U(T) )I(T < ) U() = u, u, (2.3) where I(A) is the indicator function of event A, U(T) is the deficit at ruin and U(T ) is the surplus immediately prior to ruin. If we assume the first claim occurs according to the exponential distribution with rate i, the discounted penalty function is denoted by m i (u). Let Ψ i (u) be the corresponding ruin probability. We will use f( ) to denote the Laplace transform of any function f( ). 3 The Discounted Penalty Function of the Exponential Premium Income In this section, we consider the integral function satisfied by the discounted penalty function. Given that the premium income is exponentially distributed, the Laplace transforms of the discounted penalty function can be obtained. Let V be the time for the first premium. By considering W is exponentially distributed with rate, we have m (u) = Pr(V < W, V dt)e δt m (u + y)dg(y)
3 A ruin model with compound Poisson income and dependence between claim sizes and claim intervals Pr(V > W, W dt)e δt m (u x)pr(x B)dF(x) + m 2 (u x)pr(x < B)dF(x) + ω(u, x u)df(x) u = e (++δ)t dt m (u + y)dg(y) + e (++δ)t u dt m (u x)b(x)df(x) + m 2 (u x) B(x)dF(x) + ω(u) = m (u + y)dg(y) + m (u x)b(x)df(x) + + δ + + δ + m 2 (u x) B(x)dF(x) + ω(u), (3.) where Let we have ω(u) = u ω(u, x u)df(x). γ (x) = B(x)f(x), γ 2 (x) = B(x)f(x), A i (u) = m i (u + y)dg(y), m (u) = + + δ A (u) + m (u x)γ (x)dx + + δ + m 2 (u x)γ 2 (x)dx + ω(u). (3.2) Similarly, we have m 2 (u) = δ A 2 2(u) + m (u x)γ (x)dx δ + m 2 (u x)γ 2 (x)dx + ω(u). (3.3) Taking Laplace transforms of (3.2) and (3.3), we obtain m (s) = + + δã(s) + m (s) γ (s) + m 2 (s) γ 2 (s) + ω(s), + + δ (3.4) 2 m 2 (s) = δã2(s) + m (s) γ (s) + m 2 (s) γ 2 (s) + ω(s) δ (3.5) Suppose that the premium incomes Y j s are exponentially distributed, i.e., G(y) = e y σ, y >. Now we introduce the Dickson-Hipp operator T r studied in. For r C having a nonnegative real part, the operator T r defined on an integral function h, T r h(x) = x e r(y x) h(y)dy, x. (3.6)
4 448 Y.Y. Hao, H. YANG When x =, it is the usual Laplace transform. The operator T r is commutative, i.e. T r T s h(x) = T s T r h(x), and furthermore T r T s h(x) = T rh(x) T s h(x), r s. s r Let Re(s) > σ and by using Dickson-Hipp operator in Ãi(s), we have à i (s) = = σ = σ = σ e su m i (u + y) y σ e σ dydu e su u m i (t)e (t u) σ dtdu e su T σ m i(u)du = σ T st σ m i() T s m i () T m i() σ σ s = m i(s) m i ( σs Applying the above result in (3.4) and (3.5), we have + + δ = ω(s) + + δ δ σs γ (s) m (s) + + δ + + δ m ( σs 2 γ 2 (s) δ. (3.7) γ 2 (s) + + δ m 2(s) σs, (3.8) m 2 (s) 2 γ (s) δ m (s) = 2 ω(s) δ δ m 2( σs. (3.9) Simplifying (3.8) and (3.9), we could obtain m (s) = h (s) ω(s) χ 2(s) m ( ( ++δ)( σs) γ 2(s) m 2( χ (s)χ 2 (s) ( ++δ)( 2++δ)( σs) 2 γ(s) γ2(s) ( ++δ)( 2++δ) m 2 (s) = h 2(s) ω(s) χ (s) m 2( ( 2++δ)( σs) 2 γ (s) m ( χ (s)χ 2 (s) ( ++δ)( 2++δ)( σs) 2 γ(s) γ2(s) ( ++δ)( 2++δ), (3.), (3.) where χ i (s) = ( i + + δ)( σs) i γ i (s), i =, 2, i + + δ h (s) = + + δ ( + + δ)( δ)( σs), 2 h 2 (s) = δ 2 ( + + δ)( δ)( σs). To solve m i (u), we need to find m i ( σ), i =, 2. Here we will consider the roots of the denominator of (3.) and (3.), or equally the roots of the following equation: χ (s)χ 2 (s) 2 γ (s) γ 2 (s) =. (3.2) ( + + δ)( δ)
5 A ruin model with compound Poisson income and dependence between claim sizes and claim intervals 449 Lemma. For δ >, the denominators of Equations (3.) and (3.) have exactly 2 roots, say, ρ (δ), ρ 2 (δ) in the right half complex plane, Re(ρ i (δ)) >, i =, 2. Proof. The left side of the Equation (3.2) can be rewritten as which is equivalent to 2 2 σs χ i (s)( σs) = 2 γ (s) γ 2 (s)( σs) 2 ( + + δ)( δ), i + + δ = γ (s)( σs) σs ( + + δ) + 2 γ 2 (s)( σs) ( δ) σs δ + + δ. (3.3) Let r > be a sufficiently large number, and C r denote a collection which contains a right semicircle with radius and the imaginary axis running from ir to ir. For s on the imaginary axis, we have and for s on the semicircle, we have for ε >, when r is sufficiently large. Specially, we use { + δ ε = min, + δ }, 2 i i σs <, i =, 2, i + δ ( i + + δ)σs i i σs i + δ ( i + + δ)σs = i i + + δ σ s i+δ < + ε, i =, 2, ( s i++δ)σ σ s i+δ ( s < i ( + ε). i + + δ i++δ)σ Now, we prove the rightside of Equation (3.3) is no more than the leftside. γ (s)( σs) σs + 2 γ 2 (s)( σs) δ σs ( + + δ) δ ( δ) = σs γ (s)( σs) i + + δ + δ ( + + δ)σs + 2 γ 2 (s)( σs) 2 + δ ( δ)σs 2 ( ) σs γ (s) σs i + + δ + δ ( + + δ)σs + γ 2 (s) 2 2 σs 2 + δ ( δ)σs 2 < ( σs γ (s) + γ 2 (s) ) i + + δ 2 ( σs γ () + γ 2 () ) i + + δ 2 = σs. i + + δ
6 45 Y.Y. Hao, H. YANG It is easy to say that both sides of (3.3) are analytic. So by Rouché s theorem, we know that (3.3) has the same number of roots as the following equation in C r, 2 σs =. i + + δ The above equation has two roots. Then Equation (3.2) also has two roots. The Lemma is proved. Remark. Denote the root with the smaller real part by ρ (δ), then it is easy to see that lim ρ (δ) =. We denote the two roots byρ, ρ 2, for simplicity. δ + If we put ρ, ρ 2, be the roots of Equation (3.2), it must also be zeros of the numerators of (3.) and (3.). Then we could find: h (ρ i ) ω(ρ i ) = χ 2 (ρ i ) m ( ( + + δ)( σρ i ) + γ 2 (ρ i ) m 2 (, i =, 2. (3.4) ( + + δ)( δ)( σρ i ) By solving (3.4), we can get m i (. Then m i(s) can also be obtained. Example. Let =, =.4, 2 =.5, σ =, claim sizes and the threshold distribution are exponentially distributed B(x) = e.5x, F(x) = e x. It is easy to check that the positive safety loading Condition (2.2) is obviously fulfilled. Let δ = and ω(u(t ), U(T) ) =, the ruin probability can be obtained. Equation (3.2) becomes = ( s).4 (.4 s + s +.5 ( s + ) s +.5 s +.5. ).5( s).5.5(s +.5) The roots of the above equation are, , , Then solving (3.4), we could obtain Ψ ( σ) =.43265, Ψ2 ( σ) = Finally, Taking inverse Laplace transform in equations (3.) and (3.) yields Ψ (u) = e u e u, Ψ 2 (u) = e u e u, which are the ruin probabilities in this special case. 4 Another Models In every exact t >, the risk process is in one of the two states i =, 2, corresponding to the rate i of the exponential distribution for the time until the first claim occurs. At the time of a claim occurrence the state of the system may change depending on the corresponding claim size. If a claim X i is smaller than a threshold B i, then the state of the risk process changes, otherwise it doesn t. Also, the thresholds {B i, i } are assumed to be i.i.d. random variables with distribution function B( ). Other conditions are the same as the above model. We assume ( 2µ < σ + ), (4.) 2
7 A ruin model with compound Poisson income and dependence between claim sizes and claim intervals 45 then m i (u) (which is the discounted penalty function with initial capital u, given that the system starts out in state i ) is analogous to the previous section. Then we obtain m (u) = + + δ A (u) δ m (u x)γ (x)dx+ m 2 (u) = δ A 2 2(u) δ m 2 (u x)γ (x)dx+ m 2 (u x)γ 2 (x)dx + ω(u), (4.2) m (u x)γ 2 (x)dx + ω(u). (4.3) We assume premium incomes are exponentially distributed with parameter σ. And then taking Laplace transform, we have where m (s) = h (s) ω(s) ξ 2(s) m ( ( ++δ)( σs) γ 2(s) m 2( ξ (s)ξ 2 (s) ( ++δ)( 2++δ)( σs) 2 γ2(s) γ2(s) ( ++δ)( 2++δ) m 2 (s) = h 2(s) ω(s) ξ (s) m 2( ( 2++δ)( σs) 2 γ 2(s) m ( ξ (s)ξ 2 (s) ( ++δ)( 2++δ)( σs) 2 γ2(s) γ2(s) ( ++δ)( 2++δ) ξ i (s) = i + + δ σs i γ (s), i =, 2, i + + δ h (s) = + + δ + 2 ( σs)( γ (s) γ 2 (s)) ( + + δ)( δ)( σs), h 2 (s) = δ 2 2 ( σs)( γ (s) γ 2 (s)) ( + + δ)( δ)( σs)., (4.4), (4.5) m i (u) is the Laplace transform of the discounted penalty function again. Note that the denominators on the right-hand side of (4.4) and (4.5) coincide too. As in Model, we need to prove the denominators of (4.4) and (4.5) have exactly 2 roots in the right half complex plane. Then, m i (u) also can be obtained. 5 Conclusion In this paper, we study the expected discounted penalty function of two general ruin models. These two models have the same random incomes, the same independent relationship between premiums and the claims process, and so on. Although there are minor differences concerning the dependence. The results derived in this paper can be generalized to similar dependence ruin models. References Adan, I., Kulkarni, V. Single-sever queue with Markov depent interarrival and service times. Queueing Systems, 45: (23) 2 Albrecher, H., Boxma, O.J. A ruin model with dependence between claim sizes and claim intervals. Insurances: Mathematics and Economics, 35: (24)
8 452 Y.Y. Hao, H. YANG 3 Albrecher, H., Boxma, O.J. On the discounted penalty function in a Markov-dependent risk model. Insurances: Mathematics and Economics, 37: (25) 4 Asmussen, S. Ruin probabilities. World Scientific, Singapore, 2 5 Bao, Z.H. The expected discounted penalty at ruin in the risk process with random income. Applied Mathematics and Computation, 79: (26) 6 Bao, Z.H. A note on the compound binomial model with randomized dividend strategy. Applied Mathematics and Computation, : (27) 7 Bao, Z.H., Ye, Z.X. The Gerber-Shiu discounted penalty function in the delayed renewal risk process with random income. Applied Mathematics and Computation, 2: (27) 8 Boudreault, M., Cossette, H., Landriault, D. On a risk of model with dependence between interclaim arrivals and claim sizes. Scandinavian Actuarial Journal, (26) 9 Chi, Y.C. Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance. Insurance: Mathematics and Economics, 46(2): (2) Dickson, D., Hipp, H. On the time for Erlang(2) risk process. Insurance: Mathematics and Economics, 29: (2) Gerber, H.U., Shiu, E.S.W. On the time value of ruin. North American Actuarial Journal, 2: 48 78(998) 2 Labbé, C., Sendova, K.P. The expected discounted penalty function under a risk model with stochastic income. Applied Mathematics and Computation., 25: (29) 3 Yang, H., Hao, Y.Y. A ruin model with random income and dependence between claim sizes and claim intervals. Acta Mathematica Applicatae Sinica, 26(4): (2) 4 Zhang, Z.M., Yang, H. On a risk model with stochastic premiums income and dependence between income and loss. Journal of Computational and Applied Mathematics, 234: (2) 5 Zhou, M., Cai, J. A perturbed risk model with dependence between premium rates and claim sizes. Insurance: Mathematics and Economics, 45: (29)
The finite-time Gerber-Shiu penalty function for two classes of risk processes
The finite-time Gerber-Shiu penalty function for two classes of risk processes July 10, 2014 49 th Actuarial Research Conference University of California, Santa Barbara, July 13 July 16, 2014 The finite
More informationA Note On The Erlang(λ, n) Risk Process
A Note On The Erlangλ, n) Risk Process Michael Bamberger and Mario V. Wüthrich Version from February 4, 2009 Abstract We consider the Erlangλ, n) risk process with i.i.d. exponentially distributed claims
More informationFinite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims
Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract
More informationarxiv: v1 [math.pr] 19 Aug 2017
Parisian ruin for the dual risk process in discrete-time Zbigniew Palmowski a,, Lewis Ramsden b, and Apostolos D. Papaioannou b, arxiv:1708.06785v1 [math.pr] 19 Aug 2017 a Department of Applied Mathematics
More informationA RISK MODEL WITH MULTI-LAYER DIVIDEND STRATEGY
A RISK MODEL WITH MULTI-LAYER DIVIDEND STRATEGY HANSJÖRG ALBRECHER AND JÜRGEN HARTINGER Abstract In recent years, various dividend payment strategies for the classical collective ris model have been studied
More informationNecessary and sucient condition for the boundedness of the Gerber-Shiu function in dependent Sparre Andersen model
Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 15 (214), No 1, pp. 159-17 OI: 1.18514/MMN.214.757 Necessary and sucient condition for the boundedness of the Gerber-Shiu function in dependent Sparre
More informationReinsurance and ruin problem: asymptotics in the case of heavy-tailed claims
Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims Serguei Foss Heriot-Watt University, Edinburgh Karlovasi, Samos 3 June, 2010 (Joint work with Tomasz Rolski and Stan Zachary
More informationOn a compound Markov binomial risk model with time-correlated claims
Mathematica Aeterna, Vol. 5, 2015, no. 3, 431-440 On a compound Markov binomial risk model with time-correlated claims Zhenhua Bao School of Mathematics, Liaoning Normal University, Dalian 116029, China
More informationOn a discrete time risk model with delayed claims and a constant dividend barrier
On a discrete time risk model with delayed claims and a constant dividend barrier Xueyuan Wu, Shuanming Li Centre for Actuarial Studies, Department of Economics The University of Melbourne, Parkville,
More informationRuin probabilities in multivariate risk models with periodic common shock
Ruin probabilities in multivariate risk models with periodic common shock January 31, 2014 3 rd Workshop on Insurance Mathematics Universite Laval, Quebec, January 31, 2014 Ruin probabilities in multivariate
More informationStochastic Areas and Applications in Risk Theory
Stochastic Areas and Applications in Risk Theory July 16th, 214 Zhenyu Cui Department of Mathematics Brooklyn College, City University of New York Zhenyu Cui 49th Actuarial Research Conference 1 Outline
More informationRuin probabilities of the Parisian type for small claims
Ruin probabilities of the Parisian type for small claims Angelos Dassios, Shanle Wu October 6, 28 Abstract In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For
More informationOn the probability of reaching a barrier in an Erlang(2) risk process
Statistics & Operations Research Transactions SORT 29 (2) July-December 25, 235-248 ISSN: 1696-2281 www.idescat.net/sort Statistics & Operations Research c Institut d Estadística de Transactions Catalunya
More informationUniversity Of Calgary Department of Mathematics and Statistics
University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, 2018 1 / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National
More informationOn the discounted penalty function in a discrete time renewal risk model with general interclaim times
On the discounted penalty function in a discrete time renewal risk model with general interclaim times Xueyuan Wu, Shuanming Li Centre for Actuarial Studies, Department of Economics The University of Melbourne,
More informationThe Diffusion Perturbed Compound Poisson Risk Model with a Dividend Barrier
The Diffusion Perturbed Compound Poisson Risk Model with a Dividend Barrier Shuanming Li a and Biao Wu b a Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Australia
More informationPractical approaches to the estimation of the ruin probability in a risk model with additional funds
Modern Stochastics: Theory and Applications (204) 67 80 DOI: 05559/5-VMSTA8 Practical approaches to the estimation of the ruin probability in a risk model with additional funds Yuliya Mishura a Olena Ragulina
More informationTHE DISCOUNTED PENALTY FUNCTION AND THE DISTRIBUTION OF THE TOTAL DIVIDEND PAYMENTS IN A MULTI-THRESHOLD MARKOVIAN RISK MODEL
THE DISCOUNTED PENALTY FUNCTION AND THE DISTRIBUTION OF THE TOTAL DIVIDEND PAYMENTS IN A MULTI-THRESHOLD MARKOVIAN RISK MODEL by Jingyu Chen B.Econ., Renmin University of China, 27 a Thesis submitted in
More informationDiscounted probabilities and ruin theory in the compound binomial model
Insurance: Mathematics and Economics 26 (2000) 239 250 Discounted probabilities and ruin theory in the compound binomial model Shixue Cheng a, Hans U. Gerber b,, Elias S.W. Shiu c,1 a School of Information,
More informationAsymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk. Model Perturbed by an Inflated Stationary Chi-process
Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process Enkelejd Hashorva and Lanpeng Ji Abstract: In this paper we consider the
More informationThreshold dividend strategies for a Markov-additive risk model
European Actuarial Journal manuscript No. will be inserted by the editor Threshold dividend strategies for a Markov-additive risk model Lothar Breuer Received: date / Accepted: date Abstract We consider
More informationA direct approach to the discounted penalty function
A direct approach to the discounted penalty function Hansjörg Albrecher, Hans U. Gerber and Hailiang Yang Abstract This paper provides a new and accessible approach to establishing certain results concerning
More informationUpper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process
Λ4flΛ4» ν ff ff χ Vol.4, No.4 211 8fl ADVANCES IN MATHEMATICS Aug., 211 Upper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process HE
More informationCharacterizations on Heavy-tailed Distributions by Means of Hazard Rate
Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 1 (23) 135 142 Characterizations on Heavy-tailed Distributions by Means of Hazard Rate Chun Su 1, Qi-he Tang 2 1 Department of Statistics
More informationThe Compound Poisson Risk Model with a Threshold Dividend Strategy
The Compound Poisson Risk Model with a Threshold Dividend Strategy X. Sheldon Lin Department of Statistics University of Toronto Toronto, ON, M5S 3G3 Canada tel.: (416) 946-5969 fax: (416) 978-5133 e-mail:
More informationIDLE PERIODS FOR THE FINITE G/M/1 QUEUE AND THE DEFICIT AT RUIN FOR A CASH RISK MODEL WITH CONSTANT DIVIDEND BARRIER
IDLE PERIODS FOR THE FINITE G/M/1 QUEUE AND THE DEFICIT AT RUIN FOR A CASH RISK MODEL WITH CONSTANT DIVIDEND BARRIER Andreas Löpker & David Perry December 17, 28 Abstract We consider a G/M/1 queue with
More informationMeasuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model
Measuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model Centeno, Maria de Lourdes CEMAPRE, ISEG, Technical University of Lisbon and Centre for Actuarial Studies,
More informationApplications of claim investigation in insurance surplus and claims models
Applications of claim investigation in insurance surplus and claims models by Mirabelle Huynh A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of
More informationAnalysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems
Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems Corina Constantinescu and Enrique Thomann Abstract The classical result of Cramer-Lundberg states that if the rate
More informationOPTIMAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY
Dynamic Systems and Applications 2 2) 93-24 OPTIAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY JINGXIAO ZHANG, SHENG LIU, AND D. KANNAN 2 Center for Applied Statistics,School of Statistics, Renmin
More informationAsymptotics of random sums of heavy-tailed negatively dependent random variables with applications
Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010 1. Preliminaries
More informationConditional Tail Expectations for Multivariate Phase Type Distributions
Conditional Tail Expectations for Multivariate Phase Type Distributions Jun Cai Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L 3G1, Canada Telphone: 1-519-8884567,
More informationarxiv: v2 [math.pr] 15 Jul 2015
STRIKINGLY SIMPLE IDENTITIES RELATING EXIT PROBLEMS FOR LÉVY PROCESSES UNDER CONTINUOUS AND POISSON OBSERVATIONS arxiv:157.3848v2 [math.pr] 15 Jul 215 HANSJÖRG ALBRECHER AND JEVGENIJS IVANOVS Abstract.
More informationRuin Theory. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at George Mason University
Ruin Theory A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at George Mason University by Ashley Fehr Bachelor of Science West Virginia University, Spring
More informationMULTIVARIATE COMPOUND POINT PROCESSES WITH DRIFTS
MULTIVARIATE COMPOUND POINT PROCESSES WITH DRIFTS By HUAJUN ZHOU A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department
More informationSome Approximations on the Probability of Ruin and the Inverse Ruin Function
MATIMYÁS MATEMATIKA Journal of the Mathematical Society of the Philippines ISSN 115-6926 Vol. 38 Nos. 1-2 (215) pp. 43-5 Some Approximations on the Probability of Ruin and the Inverse Ruin Function Lu
More informationf X (x) = λe λx, , x 0, k 0, λ > 0 Γ (k) f X (u)f X (z u)du
11 COLLECTED PROBLEMS Do the following problems for coursework 1. Problems 11.4 and 11.5 constitute one exercise leading you through the basic ruin arguments. 2. Problems 11.1 through to 11.13 but excluding
More informationABC methods for phase-type distributions with applications in insurance risk problems
ABC methods for phase-type with applications problems Concepcion Ausin, Department of Statistics, Universidad Carlos III de Madrid Joint work with: Pedro Galeano, Universidad Carlos III de Madrid Simon
More informationTechnical Report No. 10/04, Nouvember 2004 ON A CLASSICAL RISK MODEL WITH A CONSTANT DIVIDEND BARRIER Xiaowen Zhou
Technical Report No. 1/4, Nouvember 24 ON A CLASSICAL RISK MODEL WITH A CONSTANT DIVIDEND BARRIER Xiaowen Zhou On a classical risk model with a constant dividend barrier Xiaowen Zhou Department of Mathematics
More informationStability of the Defect Renewal Volterra Integral Equations
19th International Congress on Modelling and Simulation, Perth, Australia, 12 16 December 211 http://mssanz.org.au/modsim211 Stability of the Defect Renewal Volterra Integral Equations R. S. Anderssen,
More informationOn Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance
Applied Mathematical Sciences, Vol. 11, 217, no. 53, 269-2629 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ams.217.7824 On Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance
More informationDistribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion
Insurance: Mathematics and Economics 28 (21) 13 2 Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion Hanspeter Schmidli Laboratory of Actuarial Mathematics,
More informationThe equivalence of two tax processes
The equivalence of two ta processes Dalal Al Ghanim Ronnie Loeffen Ale Watson 6th November 218 arxiv:1811.1664v1 [math.pr] 5 Nov 218 We introduce two models of taation, the latent and natural ta processes,
More informationRuin Probability for Non-standard Poisson Risk Model with Stochastic Returns
Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns Tao Jiang Abstract This paper investigates the finite time ruin probability in non-homogeneous Poisson risk model, conditional
More informationRuin Probabilities of a Discrete-time Multi-risk Model
Ruin Probabilities of a Discrete-time Multi-risk Model Andrius Grigutis, Agneška Korvel, Jonas Šiaulys Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 4, Vilnius LT-035, Lithuania
More informationSubexponential Tails of Discounted Aggregate Claims in a Time-Dependent Renewal Risk Model
Subexponential Tails of Discounted Aggregate Claims in a Time-Dependent Renewal Risk Model Jinzhu Li [a];[b], Qihe Tang [b];, and Rong Wu [a] [a] School of Mathematical Science and LPMC Nankai University,
More informationWorst-Case-Optimal Dynamic Reinsurance for Large Claims
Worst-Case-Optimal Dynamic Reinsurance for Large Claims by Olaf Menkens School of Mathematical Sciences Dublin City University (joint work with Ralf Korn and Mogens Steffensen) LUH-Kolloquium Versicherungs-
More informationRuin probabilities in a finite-horizon risk model with investment and reinsurance
Ruin probabilities in a finite-horizon risk model with investment and reinsurance R. Romera and W. Runggaldier University Carlos III de Madrid and University of Padova July 3, 2012 Abstract A finite horizon
More informationA Dynamic Contagion Process with Applications to Finance & Insurance
A Dynamic Contagion Process with Applications to Finance & Insurance Angelos Dassios Department of Statistics London School of Economics Angelos Dassios, Hongbiao Zhao (LSE) A Dynamic Contagion Process
More informationRare event simulation for the ruin problem with investments via importance sampling and duality
Rare event simulation for the ruin problem with investments via importance sampling and duality Jerey Collamore University of Copenhagen Joint work with Anand Vidyashankar (GMU) and Guoqing Diao (GMU).
More informationPoisson Processes. Stochastic Processes. Feb UC3M
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written
More information1 Delayed Renewal Processes: Exploiting Laplace Transforms
IEOR 6711: Stochastic Models I Professor Whitt, Tuesday, October 22, 213 Renewal Theory: Proof of Blackwell s theorem 1 Delayed Renewal Processes: Exploiting Laplace Transforms The proof of Blackwell s
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 6
MATH 56A: STOCHASTIC PROCESSES CHAPTER 6 6. Renewal Mathematically, renewal refers to a continuous time stochastic process with states,, 2,. N t {,, 2, 3, } so that you only have jumps from x to x + and
More informationA LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES.
A LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES Riccardo Gatto Submitted: April 204 Revised: July 204 Abstract This article provides
More informationRuin problems for a discrete time risk model with non-homogeneous conditions. 1 A non-homogeneous discrete time risk model
Ruin problems for a discrete time risk model with non-homogeneous conditions ANNA CASTAÑER a, M. MERCÈ CLARAMUNT a, MAUDE GATHY b, CLAUDE LEFÈVRE b, 1 and MAITE MÁRMOL a a Universitat de Barcelona, Departament
More informationarxiv: v1 [q-fin.rm] 27 Jun 2017
Risk Model Based on General Compound Hawkes Process Anatoliy Swishchuk 1 2 arxiv:1706.09038v1 [q-fin.rm] 27 Jun 2017 Abstract: In this paper, we introduce a new model for the risk process based on general
More informationA Study of the Impact of a Bonus-Malus System in Finite and Continuous Time Ruin Probabilities in Motor Insurance
A Study of the Impact of a Bonus-Malus System in Finite and Continuous Time Ruin Probabilities in Motor Insurance L.B. Afonso, R.M.R. Cardoso, A.D. Egídio dos Reis, G.R Guerreiro This work was partially
More informationOptimal Dividend Strategies for Two Collaborating Insurance Companies
Optimal Dividend Strategies for Two Collaborating Insurance Companies Hansjörg Albrecher, Pablo Azcue and Nora Muler Abstract We consider a two-dimensional optimal dividend problem in the context of two
More informationThe Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber Shiu Discounted Penalty Function
Methodol Comput Appl Probab (29) 11:41 423 DOI 1.17/s119-7-95-6 The Compound Poisson Surplus Model with Interest Liquid Reserves: Analysis of the Gerber Shiu Discounted Penalty Function Jun Cai Runhuan
More informationA polynomial expansion to approximate ruin probabilities
A polynomial expansion to approximate ruin probabilities P.O. Goffard 1 X. Guerrault 2 S. Loisel 3 D. Pommerêt 4 1 Axa France - Institut de mathématiques de Luminy Université de Aix-Marseille 2 Axa France
More informationTime to Ruin for. Loss Reserves. Mubeen Hussain
Time to Ruin for Loss Reserves by Mubeen Hussain A Major Paper Submitted to the Faculty of Graduate Studies through the Department of Mathematics and Statistics in Partial Fulfillment of the Requirements
More informationAsymptotic Ruin Probabilities for a Bivariate Lévy-driven Risk Model with Heavy-tailed Claims and Risky Investments
Asymptotic Ruin robabilities for a Bivariate Lévy-driven Risk Model with Heavy-tailed Claims and Risky Investments Xuemiao Hao and Qihe Tang Asper School of Business, University of Manitoba 181 Freedman
More informationMinimization of ruin probabilities by investment under transaction costs
Minimization of ruin probabilities by investment under transaction costs Stefan Thonhauser DSA, HEC, Université de Lausanne 13 th Scientific Day, Bonn, 3.4.214 Outline Introduction Risk models and controls
More informationModern Mathematical Methods for Actuarial Sciences
Modern Mathematical Methods for Actuarial Sciences Thesis submitted for the degree of Doctor of Philosophy at the University of Leicester by Ahmet Kaya Department of Mathematics University of Leicester
More informationExtremes and ruin of Gaussian processes
International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Extremes and ruin of Gaussian processes Jürg Hüsler Department of Math. Statistics, University
More informationRecursive methods for a multi-dimensional risk process with common shocks. Creative Commons: Attribution 3.0 Hong Kong License
Title Recursive methods for a multi-dimensional risk process with common shocks Author(s) Gong, L; Badescu, AL; Cheung, ECK Citation Insurance: Mathematics And Economics, 212, v. 5 n. 1, p. 19-12 Issued
More informationResearch Article Strong Convergence Bound of the Pareto Index Estimator under Right Censoring
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 200, Article ID 20956, 8 pages doi:0.55/200/20956 Research Article Strong Convergence Bound of the Pareto Index Estimator
More informationType II Bivariate Generalized Power Series Poisson Distribution and its Applications in Risk Analysis
Type II Bivariate Generalized Power Series Poisson Distribution and its Applications in Ris Analysis KK Jose a, and Shalitha Jacob a,b a Department of Statistics, St Thomas College, Pala, Arunapuram, Kerala-686574,
More informationOn the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures
On the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures October 7, 2015 Boutsikas M.V. 1, Rakitzis A.C. 2 and Antzoulakos D.L. 3 1 Dept. of Statistics & Insurance Science,
More informationON THE LAW OF THE i TH WAITING TIME INABUSYPERIODOFG/M/c QUEUES
Probability in the Engineering and Informational Sciences, 22, 2008, 75 80. Printed in the U.S.A. DOI: 10.1017/S0269964808000053 ON THE LAW OF THE i TH WAITING TIME INABUSYPERIODOFG/M/c QUEUES OPHER BARON
More informationAsymptotic behavior for sums of non-identically distributed random variables
Appl. Math. J. Chinese Univ. 2019, 34(1: 45-54 Asymptotic behavior for sums of non-identically distributed random variables YU Chang-jun 1 CHENG Dong-ya 2,3 Abstract. For any given positive integer m,
More informationA unified analysis of claim costs up to ruin in a Markovian arrival risk model. Title
Title A unified analysis of claim costs up to ruin in a Marovian arrival ris model Author(s) Cheung, ECK; Feng, R Citation Insurance: Mathematics and Economics, 213, v. 53 n. 1, p. 98-19 Issued Date 213
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationarxiv: v1 [math.pr] 7 Aug 2014
A Correction Term for the Covariance of Renewal-Reward Processes with Multivariate Rewards Brendan Patch, Yoni Nazarathy, and Thomas Taimre. August 18, 218 arxiv:148.153v1 [math.pr] 7 Aug 214 Abstract
More informationA review of discrete-time risk models. Shuanming Li, Yi Lu and José Garrido
RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 103 (2), 2009, pp. 321 337 Matemática Aplicada / Applied Mathematics Artículo panorámico / Survey A review of discrete-time risk models Shuanming Li, Yi Lu
More informationLarge deviations for weighted random sums
Nonlinear Analysis: Modelling and Control, 2013, Vol. 18, No. 2, 129 142 129 Large deviations for weighted random sums Aurelija Kasparavičiūtė, Leonas Saulis Vilnius Gediminas Technical University Saulėtekio
More informationComplete q-moment Convergence of Moving Average Processes under ϕ-mixing Assumption
Journal of Mathematical Research & Exposition Jul., 211, Vol.31, No.4, pp. 687 697 DOI:1.377/j.issn:1-341X.211.4.14 Http://jmre.dlut.edu.cn Complete q-moment Convergence of Moving Average Processes under
More informationLecture Notes on Risk Theory
Lecture Notes on Risk Theory February 2, 21 Contents 1 Introduction and basic definitions 1 2 Accumulated claims in a fixed time interval 3 3 Reinsurance 7 4 Risk processes in discrete time 1 5 The Adjustment
More informationOn Finite-Time Ruin Probabilities for Classical Risk Models
On Finite-Time Ruin Probabilities for Classical Risk Models Claude Lefèvre, Stéphane Loisel To cite this version: Claude Lefèvre, Stéphane Loisel. On Finite-Time Ruin Probabilities for Classical Risk Models.
More informationBridging Risk Measures and Classical Risk Processes
Bridging Risk Measures and Classical Risk Processes Wenjun Jiang A Thesis for The Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements for the Degree of Master
More informationESTIMATING THE PROBABILITY OF RUIN FOR VARIABLE PREMIUMS BY SIMULATION. University of Lausanne, Switzerland
ESTIMATING THE PROBABILITY OF RUIN FOR VARIABLE PREMIUMS BY SIMULATION BY FREDERIC MICHAUD University of Lausanne, Switzerland ABSTRACT There is a duality between the surplus process of classical risk
More informationNonlife Actuarial Models. Chapter 5 Ruin Theory
Nonlife Actuarial Models Chapter 5 Ruin Theory Learning Objectives 1. Surplus function, premium rate and loss process 2. Probability of ultimate ruin 3. Probability of ruin before a finite time 4. Adjustment
More informationThe optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture
The optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture by Javid Ali A thesis presented to the University of Waterloo in fulfilment
More informationRandomly Weighted Sums of Conditionnally Dependent Random Variables
Gen. Math. Notes, Vol. 25, No. 1, November 2014, pp.43-49 ISSN 2219-7184; Copyright c ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in Randomly Weighted Sums of Conditionnally
More informationExperience Rating in General Insurance by Credibility Estimation
Experience Rating in General Insurance by Credibility Estimation Xian Zhou Department of Applied Finance and Actuarial Studies Macquarie University, Sydney, Australia Abstract This work presents a new
More informationarxiv: v2 [math.oc] 26 May 2015
Optimal dividend payments under a time of ruin constraint: Exponential claims arxiv:11.3793v [math.oc 6 May 15 Camilo Hernández Mauricio Junca July 3, 18 Astract We consider the classical optimal dividends
More informationHawkes Processes and their Applications in Finance and Insurance
Hawkes Processes and their Applications in Finance and Insurance Anatoliy Swishchuk University of Calgary Calgary, Alberta, Canada Hawks Seminar Talk Dept. of Math. & Stat. Calgary, Canada May 9th, 2018
More informationSimulation methods in ruin models with non-linear dividend barriers
Simulation methods in ruin models with non-linear dividend barriers Hansjörg Albrecher, Reinhold Kainhofer, Robert F. Tichy Department of Mathematics, Graz University of Technology, Steyrergasse 3, A-8
More informationRuin Probability for Dependent Risk Model with Variable Interest Rates
Applied Mathematical Sciences, Vol. 5, 2011, no. 68, 3367-3373 Ruin robability for Dependent Risk Model with Variable Interest Rates Jun-fen Li College of Mathematics and Information Science,Henan Normal
More informationRecursive Calculation of Finite Time Ruin Probabilities Under Interest Force
Recursive Calculation of Finite Time Ruin Probabilities Under Interest Force Rui M R Cardoso and Howard R Waters Abstract In this paper we consider a classical insurance surplus process affected by a constant
More informationScandinavian Actuarial Journal. Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks
Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks For eer Review Only Journal: Manuscript ID: SACT-- Manuscript Type: Original Article Date Submitted
More informationResearch Article Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation
Applied Mathematics Volume 2012, Article ID 436531, 12 pages doi:10.1155/2012/436531 Research Article Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation
More informationFigure 10.1: Recording when the event E occurs
10 Poisson Processes Let T R be an interval. A family of random variables {X(t) ; t T} is called a continuous time stochastic process. We often consider T = [0, 1] and T = [0, ). As X(t) is a random variable
More informationApplying the proportional hazard premium calculation principle
Applying the proportional hazard premium calculation principle Maria de Lourdes Centeno and João Andrade e Silva CEMAPRE, ISEG, Technical University of Lisbon, Rua do Quelhas, 2, 12 781 Lisbon, Portugal
More informationBand Control of Mutual Proportional Reinsurance
Band Control of Mutual Proportional Reinsurance arxiv:1112.4458v1 [math.oc] 19 Dec 2011 John Liu College of Business, City University of Hong Kong, Hong Kong Michael Taksar Department of Mathematics, University
More informationExcursions of Risk Processes with Inverse Gaussian Processes and their Applications in Insurance
Excursions of Risk Processes with Inverse Gaussian Processes and their Applications in Insurance A thesis presented for the degree of Doctor of Philosophy Shiju Liu Department of Statistics The London
More informationModelling the risk process
Modelling the risk process Krzysztof Burnecki Hugo Steinhaus Center Wroc law University of Technology www.im.pwr.wroc.pl/ hugo Modelling the risk process 1 Risk process If (Ω, F, P) is a probability space
More informationWaiting time characteristics in cyclic queues
Waiting time characteristics in cyclic queues Sanne R. Smits, Ivo Adan and Ton G. de Kok April 16, 2003 Abstract In this paper we study a single-server queue with FIFO service and cyclic interarrival and
More informationReadings: Finish Section 5.2
LECTURE 19 Readings: Finish Section 5.2 Lecture outline Markov Processes I Checkout counter example. Markov process: definition. -step transition probabilities. Classification of states. Example: Checkout
More informationEstimation of arrival and service rates for M/M/c queue system
Estimation of arrival and service rates for M/M/c queue system Katarína Starinská starinskak@gmail.com Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics
More information