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Stochastic Differential Equations: Some Risk and Insurance Applications

In this dissertation, we have studied diffusion models and their applications in risk theory and insurance. Let Xt be a d-dimensional diffusion process satisfying a system of Stochastic Differential Equations defined on an open set G Rd, and let Ut be a utility function of Xt with U0 = u0. Let T be the first time that Ut reaches a level u^*. We study the Laplace transform of the distribution of T, as well as the probability of ruin, psileft(u_{0}right)=Prleft{ T<inftyright} , and other important probabilities. A class of exponential martingales is constructed to analyze the asymptotic properties of all probabilities. In addition, we prove that the expected discounted penalty function, a generalization of the probability of ultimate ruin, satisfies an elliptic partial differential equation, subject to some initial boundary conditions. Two examples from areas of actuarial work to which martingales have been applied are given to illustrate our methods and results: 1. Insurer's insolvency. 2. Terrorism risk. In particular, we study insurer's insolvency for the Cram'{e}r-Lundberg model with investments whose price follows a geometric Brownian motion. We prove the conjecture proposed by Constantinescu and Thommann. / Mathematics

Identiferoai:union.ndltd.org:TEMPLE/oai:scholarshare.temple.edu:20.500.12613/3866
Date January 2011
CreatorsXiong, Sheng
ContributorsYang, Wei-shih, 1954-, Powers, Michael R., Chen, Hua, Berhanu, Shiferaw
PublisherTemple University. Libraries
Source SetsTemple University
LanguageEnglish
Detected LanguageEnglish
TypeThesis/Dissertation, Text
Format76 pages
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Relationhttp://dx.doi.org/10.34944/dspace/3848, Theses and Dissertations

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