With Two Boundary Diffusion Process Over Time, For The First Time

In the thesis we consider a one-dimensional time-homogeneous regular diffusion between two constant elastic barriers, and the special cases with pure absorbing and/ or reflecting barriers. We use the Kolmogorov backward equation with corresponding boundary conditions to derive the recursive relation for moments of the first passage time,and to give the Laplace transform and the distribution function of that. In particular, we give the explicit closed-form expressions for the mean time to reach the boundary. Finally, we give the Laplace transform of the hitting time of one-dimensional time-homogeneous regular diffusion with two reflecting barriers.The thesis is divided into three chapters according to contents:In Chapter 1, we derive the recursive formula on the moments of the first passage time of the one-dimensional diffusion process between two barriers (not both reflecting). Some examples of the results are computed. Then, we give the recursive formula on the moments of the first passage time of the one-dimensional diffusion process with two constant elastic barriers. As examples, we consider several popular diffusions. Finally, some applications to risk theory are considered.In Chapter 2, we obtain the distribution and the Laplace transform of the first passage time of the one-dimensional diffusion process between two barriers (not both reflecting). Then, we derive the formula on the Laplace transform of the first passage time of the one-dimensional diffusion process with two constant elastic barriers. As examples, we consider several popular diffusions. Finally, the distribution function is considered.In Chapter 3, we will consider the problem of hitting time of the one-dimensional diffusion process between two reflecting barriers. Next, we give the Laplace transform of the hitting time. Finally, we present some results on several special cases.