On intrinsic ultracontractivity of perturbed Levy processes and applications of Levy processes in actuarial mathematics

In this thesis, we study certain aspects of Levy processes and their applications. In the first part of this thesis, we study the applications of Levy processes in actuarial mathematics. Our topics are closely related to the generalized Ornstein-Uhlenbeck processes. We investigate their intimate relationships with the exponential functionals of Levy processes, which enable us to develop efficient semi-analytical algorithms to solve the pricing and risk management problem of certain exotic variable annuity products. In particular, we consider two variable annuity products with guaranteed benefits, the Guaranteed Minimum Accumulation Benefit (GMAB) and the Guaranteed Minimum Withdrawal Benefit (GMWB). For the first one, we develop efficient semi-analytical algorithms to compute its risk measures and hedging costs to solve the risk management problem of the rider. For the other one, we consider pricing the rider. We identify the Laplace transforms of the GMWB rider's risk-neutral values analytically, which leads to efficient solutions to its pricing problem. In the second part, we consider the intrinsic ultracontractivity of certain Levy processes under nonlocal perturbations. More precisely, we establish the intrinsic ultracontractivity of the Laplacian (corresponding to Brownian motions) and the fractional Laplacian (corresponding to symmetric $\alpha$-stable processes) perturbed by a class of nonlocal operators. Conditions on the nonlocal perturbations are given in order to guarantee that the perturbed operators are intrinsically ultracontractive in general bonded open sets. The methods we use are probabilistic. Essentially, the methods rely on the heat kernel estimates of the fundamental solutions of the operators as well as the Levy systems of the corresponding processes