A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles
Volume 7, Issue 1 (2020), pp. 43–60
Pub. online: 20 February 2020
Type: Research Article
Open Access
Open Access
Received
19 April 2019
19 April 2019
Revised
3 February 2020
3 February 2020
Accepted
3 February 2020
3 February 2020
Published
20 February 2020
20 February 2020
Abstract
Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process S under a probability measure P we characterize all probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound renewal process under Q. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.
References
Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd Ed. World Scientific Publishing, London (2010) MR2766220. https://doi.org/10.1142/9789814282536
Boogaert, P., De Waegenaere, A.: Simulation of ruin probabilities. Insur. Math. Econ. 9, 95–99 (1990) MR1084493. https://doi.org/10.1016/0167-6687(90)90020-E
Cohn, D.L.: Measure Theory, 2nd Ed. Birkhäuser Advanced Texts (2013) MR3098996. https://doi.org/10.1007/978-1-4614-6956-8
Delbaen, F., Haezendonck, J.: A martingale approach to premium calculation principles in an arbitrage free market. Insur. Math. Econ. 8, 269–277 (1989) MR1029895. https://doi.org/10.1016/0167-6687(89)90002-4
Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin Heidelberg (2006) MR2200584
Geman, H., Yor, M.: Stochastic time changes in catastrophe option pricing. Insur. Math. Econ. 21, 269–277 (1997) MR1614517. https://doi.org/10.1016/S0167-6687(97)00017-6
Haslip, G.G., Kaishev, V.K.: Pricing of reinsurance contracts in the presence of catastrophe bonds. ASTIN Bull. 40, 307–329 (2010) MR2758262. https://doi.org/10.2143/AST.40.1.2049231
Holtan, J.: Pragmatic insurance option pricing. Scand. Actuar. J. 1, 53–70 (2007) MR2345739. https://doi.org/10.1080/03461230601088213
Jacod, J.: Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 235–253 (1975) MR0380978. https://doi.org/10.1007/BF00536010
Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes. Springer, Berlin Heidelberg (2003) MR1943877. https://doi.org/10.1007/978-3-662-05265-5
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, New York (1998) MR3184878. https://doi.org/10.1007/978-3-642-31898-6
Schmidli, H.: Cramér–Lundberg approximations for ruin functions of risk processes perturbed by diffusion. Insur. Math. Econ. 16, 135–149 (1995) MR1347857. https://doi.org/10.1016/0167-6687(95)00003-B
Schmidt, K.D.: Lectures on Risk Theory. B.G. Teubner, Stuttgart (1996) MR1402016. https://doi.org/10.1007/978-3-322-90570-3
Serfozo, R.: Basics of Applied Stochastic Processes. Springer, Berlin Heidelberg (2009) MR2484222. https://doi.org/10.1007/978-3-540-89332-5
von Weizsäcker, H., Winkler, G.: Stochastic Integrals: An Introduction. Vieweg+Teubner Verlag (1990) MR1062600. https://doi.org/10.1007/978-3-663-13923-2
Wang, J.H., Chen, K.C., Lee, S.J., Huang, W.J., Yu, Y.H., Leu, P.L.: The frequency distribution of inter-event times of $m\ge 3$ earthquakes in the Taipei metropolitan area: 1973–2010. Terr. Atmos. Ocean. Sci. 23, 269–281 (2012) MR2898624
