Statistical Inference For Diffusion Processes

This thesis is devoted to a study of statistical inference for diffusion processes and some applications in finance. Three aspects of work are considered:The first aspect: we considered the error bound of the maximum likelihood estimation estimator for a class of nonstationary diffusion processes with parameters in both drift and diffusion parts. By transformation for one dimensional case, the diffusion process can be transferred into the case that the parameter is only in the drift part. A fundamental inequality based on the bounds for transitional densities of recent work is obtained. And then another important inequality, which is important to prove the main theorem, is showed by the some techniques in stochastic process. In the main theorem, we obtained the error bound between the maximum likelihood estimator and the true parameter. The interesting thing is that the bound is precise without any unknown constants, which is very important for statistical inference.The second aspect: we considered the local polynomial estimation for time homogenous diffusion processes. Local polynomial estimation method is a kind of a nonparametric estimation methods which are based on data-analytic approaches. We propose local polynomial estimation methods for drift and diffusion coefficients in time homogenous diffusion process. For simplicity and from the properties of local fitting, we only show the asymptotic theory of the local linear estimations for the drift and diffusion functions respectively. And a criteria to choose the smoothing parameter of the bandwidth is suggested.The third aspect: we considered risk measures in finance. Two kinds of risk measures are considered, volatility and Value at Risk(VaR). Some methods based on our local polynomial estimator and some other recent work to calculate the volatility based on discrete observations are given. A time dependent Value at Risk(TVaR) based on stochastic processes is defined. Some examples of TVaR for diffusionprocesses are given. We obtain some bounds of TVaR by recent improvements on the inequalities of transitional density of diffusion processes in our main theorems. The interesting thing is that there have no random parts and no unknown constants in these bounds. Moreover, these bounds can be used to predicate the range of future risk if we know now state of the diffusion process. And the other part is a sufficient condition for existence of growth optimal portfolio in a general market defined by multi jumps and multi risky assets.