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Tuesday, January 17, 2017

**Abstract:** In classical risk theory, the surplus process of an insurance company is modeled by the compound Poisson or the general compound renewal risk process. For both applied and theoretical investigations, calculation of the ruin probabilities for such models is of particular interest. In order to avoid technical issues and to allow for dependence among the claim sizes, these risk models are often approximated by the classical Brownian motion (or diffusion) (e.g., [1,2]) or the fractional Brownian motion risk model (e.g., [3,4]). Calculation of ruin probabilities and other ruin related quantities for Brownian motion and more general Gaussian risk models has been the subject of study of numerous contributions (e.g., [4-7]). This talk focuses on the most recent findings for Gaussian risk models with financial constraints such as inflation, interest and tax. In particular, three Gaussian risk models and their ruin probabilities will be discussed in detail. Finally, some future research directions on this topic will also be discussed. References: [1] Iglehart, D. L. 1969. Diffusion approximations in collective risk theory, Journal of Applied Probability 6: 285–292. [2] Grandell, J. 1991. Aspects of Risk Theory. New York: Springer. [3] Michna, Z. 1998. Self-similar processes in collective risk theory, J. Appl. Math. Stochastic Anal., 11(4): 429-448. [4] Asmussen, S. and Albrecher, H. 2010. Ruin Probabilities. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition. [5] Cai, J., Gerber, H.U. and Yang, H.L. 2006. Optimal dividends in an Ornstein–Uhlenbeck type model with credit and debit interest, North American Actuarial Journal 10 (2): 94–119. [6] Debicki, K. 2002. Ruin probability for Gaussian integrated processes, Stochastic Process. Appl., 98(1): 151-174. [7] Husler, J. and Piterbarg, V.I. 2008. A limit theorem for the time of ruin in a Gaussian ruin problem, Stochastic Process. Appl., 118(11): 2014-2021. See video of talk at https://www.youtube.com/watch?v=dYUhaq2j4CQ&feature=youtu.be