On computing ruin probabilities

Abstract

The objective of this thesis is to develop for an effective numerical scheme to calculate the finite-time ruin probabilities (equivalently the finite-time survival probabilities) under classical risk model. Ruin theory of this model has been widely studied in literatures especially those related to ruin probabilities. However, in a lot of cases, numerical solutions are needed and so an efficient numerical scheme is in great demand. In this thesis, the survival probability is going to be evaluated via a very effective wavelets scheme.

In 1997, Picard and Lefèvre derived an explicit formula for survival probabilities in finite-time horizon for general Lèvy processes. However, in a lot of risk models, this formula involves infinitely many convolutions of a compound Poisson density function. Hence, evaluating it becomes very difficult. We shall combine a discretization with a wavelets expansion to achieve the evaluation task.

Wavelets is a function basis that possesses a number of nice properties including compact supportness and this facilitates very efficient computations. Since its introduction, wavelets has attracted many researches and has been popular in solving PDEs and option pricing. As far as we know, wavelets method has not been applied to risk theory. It is new that wavelets expansion is used in computing survival probabilities. Our wavelets numerical scheme is direct and simple in computations. It also has a computational complexity of O(n) compared to that of O(n log n) via the typical methods, like Fast Fourier Transforms. An explicit error bound for our wavelets scheme is given with the help of Jackson's inequality.

In Chapter 1, a brief review on the development of risk theory and the Picard- Lefèvre formula on survival probability in finite-time horizon is presented, followed by a brief introduction of wavelets expansion and multi-resolution analysis in Chapter 2. An explicit error bound for the numerical approximation is provided in Chapter 3. Finally, numerical illustrations of the wavelets scheme are exhibited in Chapter 4.