Optimal estimation and numerical methods for conditional diffusion processes

Abstract

Stochastic differential equation(SDE) is a main tool representing the continuous time stochastic process. It is used in modeling diffusion phenomena and widely applicable in pure and applied sciences. SDE is closely related to partial differential equation (PDE) in such a way that the probability density function of $X_t$ is represented as the solution of a parabolic PDE. In many cases, the purpose of such density functions is to evaluate the expected value of some functionals of $X_t$ and there is an alternative numerical approach. In this thesis, we study on the two different approaches for SDEs i.e. direct time discrete approximations and PDE methods in particular for conditional diffusion processes. In Chapter 2, we give a brief review on the numerical approximations for SDEs and we evaluate the exact strong convergence rate of the linearly interpolated numerical solutions and give an error bound of strong Euler scheme for `random` discretizations. For weak approximation, we suggest a simple and new random number generation method for systems of SDEs. In Chapter 3, we study on boundary crossing time densities for 1 dimensional diffusion processes, the problem of which is closely related to the optimal estimation problems in Chapter 4 and 5. In Chapter 4 and 5, we consider the excursion type conditional diffusion processes and evaluate the conditional densities by solving related parabolic PDEs. We derive the formulas for the forward and backward estimators as well as the interpolator. We give a numerical algorithm for general 1-dimensional diffusion process. We also suggest several different methods for Brownian motion process and compare the results by numerical simulations.