zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Markovian regime-switching risk model with a threshold dividend strategy. (English) Zbl 1163.91438
Summary: We study a regime-switching risk model with a threshold dividend strategy, in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying (external) Markov jump process. The purpose of this paper is to study the unified Gerber-Shiu discounted penalty function and the moments of the total dividend payments until ruin. We adopt an approach which is akin to the one used by X. Lin und K. P. Pavlova [Insur. Math. Econ. 38, No. 1, 57–80 (2006; Zbl 1157.91383)] to extend the results for the classical risk model with a threshold dividend strategy to our model. The matrix form of systems of integro-differential equations is presented and the analytical solutions to these systems are derived. Finally, numerical illustrations with exponential claim amounts are also given.
MSC:
 91B30 Risk theory, insurance 91B28 Finance etc. (MSC2000)
References:
 [1] Albrecher, H.; Boxma, O. J.: On the discounted penalty function in a Markov-dependent risk model, Insurance: mathematics and economics 37, 650-672 (2005) · Zbl 1129.91023 · doi:10.1016/j.insmatheco.2005.06.007 [2] Albrecher, H.; Hartinger, J.: A risk model with multilayer dividend strategy, North American actuarial journal 11, No. 2, 43-64 (2007) [3] Asmussen, S.: Risk theory in a Markovian environment, Scandinavian actuarial journal, 69-100 (1989) · Zbl 0684.62073 [4] Asmussen, S.: Ruin probabilities, (2000) [5] Badescu, A.; Drekic, S.; Landriault, D.: Analysis of a threshold dividend strategy for a MAP risk model, Scandinavian actuarial journal, 227-247 (2007) · Zbl 1164.91024 · doi:10.1080/03461230701396474 [6] Burton, T. A.: Volterra integral and differential equations, (2005) [7] Dickson, D. C. M.; Hipp, C.: On the time to ruin for $Erlang\left(2\right)$ risk process, Insurance: mathematics and economics 29, 333-344 (2001) · Zbl 1074.91549 · doi:10.1016/S0167-6687(01)00091-9 [8] Gerber, H. U.; Shiu, E. S. W.: On the time value of ruin, North American actuarial journal 2, No. 1, 48-78 (1998) · Zbl 1081.60550 [9] Gerber, H. U.; Shiu, E. S. W.: The time value of ruin in a sparre andersen model, North American actuarial journal 9, No. 2, 49-69 (2005) · Zbl 1085.62508 [10] Gerber, H. U.; Shiu, E. S. W.: On optimal dividend strategies in the compound Poisson model, North American actuarial journal 10, No. 2, 76-93 (2006) [11] Li, S.; Garrido, J.: On a class of renewal risk models with a constant dividend barrier, Insurance: mathematics and economics 35, 691-701 (2004) · Zbl 1122.91345 · doi:10.1016/j.insmatheco.2004.08.004 [12] Li, S.; Lu, Y.: Moments of the dividend payments and related problems in a Markov-modulated risk model, North American actuarial journal 11, No. 2, 65-76 (2007) [13] Li, S., Lu, Y., 2008. The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model. ASTIN Bulletin 38 (in press) · Zbl 1169.91390 · doi:10.2143/AST.38.1.2030402 [14] Lin, X. S.; Pavlova, K. P.: The compound Poisson risk model with a threshold dividend strategy, Insurance: mathematics and economics 38, 57-80 (2006) · Zbl 1157.91383 · doi:10.1016/j.insmatheco.2005.08.001 [15] Lin, X. S.; Sendova, K. P.: The compound Poisson risk model with multiple thresholds, Insurance: mathematics and economics 42, 617-627 (2008) · Zbl 1152.91592 · doi:10.1016/j.insmatheco.2007.06.008 [16] Lu, Y.; Tsai, Cary C. L.: The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion, North American actuarial journal 11, No. 2, 136-152 (2007) [17] Ng, A. C. Y.; Yang, H.: On the joint distribution of surplus before and after ruin under a Markovian regime switching model, Stochastic processes and their applications 116, 244-266 (2006) · Zbl 1093.60051 · doi:10.1016/j.spa.2005.09.008 [18] Reinhard, J. M.: On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment, ASTIN bulletin 14, 23-43 (1984) [19] Yang, H.; Zhang, Z.: Gerber–shiu discounted penalty function in a sparre andersen model with multi-layer dividend strategy, Insurance: mathematics and economics (2008) [20] Zhou, X.: Classical risk model with multi-layer premium rate, Actuarial research clearing house (2007) [21] Zhu, J.; Yang, H.: Ruin theory for a Markov regime-switching model under a threshold dividend strategy, Insurance: mathematics and economics 42, 311-318 (2008) · Zbl 1141.91558 · doi:10.1016/j.insmatheco.2007.03.004