Some Properties Of Skew Ornstein-Uhlenbeck Processes With Two Barriers

The goal of this work is to study the properties of Skew Ornstein-Uhlenbeck processes with two control barriers in terms of Sturm-Liouville eigenfunction ex-pansion and characterize the spectrum from the behaviour of infinitesimal param-eters near the boundaries[r0,r1].We assume that the process starts from a random position x(0)?[r0,r1]and it is confined to remain within the boundaries.Gener-ally,stochastic processes do not stay within boundaries,neither are they allowed to cross a certain boundary.The main contribution is to give an explicit spectral representation for the hit-ting time densities of Skew Ornstein-Uhlenbeck processes in terms of their asso-ciated Sturm-Liouville eigenfunction expansion between the two control barriers.The Sturm-Liouville eigenfunction expansion provides a good alternative to nu-merical Laplace approach when it comes to recovering the density.We will classify feller's natural boundaries and give large-n eigenvalue and eigenfunction asymptotics in terms of elementary functions.It has already been shown see e.g[23]that eigenfunction expansion method is more suitable than numerical Laplace transform inversion algorithms.This is basically because numerical integration is not required in eigenfunction expansion thus the cost of computation is low.Numerical Laplace transform is exact and accurate but because each wavelength must be resolved,this makes its cost to be higher.Basically they are limited to low frequency.At the end of this thesis,we will give a numerical example of Skew Ornstein Uhlenbeck process on[-2,2].We reveal that eigenvalues have a linear growth as n increases and that the series converges slower.