Backward stochastic differential equations and generalized risk models
Since the early 1900s, a major topic in the field of risk theory has been the approximation of ruin probabilities. This thesis considers the ruin probabilities for two different generalized risk reserve models, employing different techniques for each one. In addition, the analysis of the first model inspired the study of backward stochastic differential equations with quadratic growth and jumps. 1. BSDEs with quadratic growth and jumps and the ruin probability for a reserve involving investments. The main part of this thesis shows the existence and uniqueness of BSDEs with quadratic growth and jumps. This problem is a generalization of Kobylanski  which considers BSDEs with the quadratic growth assumption; but continuous paths. An exponential change of variable is used to control the quadratic growth, which is a counterpart of the Hopf-Cole transformation used in PDEs. However, due to the presence of the jumps and the lack of information on the extra martingale integrand, the analysis is much more complicated than in the continuous case. This problem was inspired by the study of generalized risk reserve models, for which Ma and Sun constructed a new type of exponential martingale parameterized by a general rate function in order to calculate various Cramér-Lundberg bounds. Since this method is in general contingent upon the solution of a parabolic Integro-PDE with quadratic growth in the gradient parameterized by the rate function though, we are led to the study of BSDEs with quadratic growth and jumps. 2. Ruin probability for reserve involving α-stable processes . The last part characterizes the ruin probability in terms of a convolution formula for a classical risk process perturbed by both a Brownian motion B(t) and an α-stable Lévy motion Zα(t) with only downward jumps. Similar problems had been studied by Dufresne and Gerber  and Furrer  who considered models perturbed by either a Brownian motion or an α-stable Lévy process, respectively. It was found that the ruin probability can still be expressed as a convolution, but it is in terms of the distribution function of a stable random variable for which there is no known closed formula.
Ma, Purdue University.
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