Statistical Inference And Numeric Methods For Some Stochastic Diffusion Models

Asset pricing and risk management is core of the modern financial theory. Since the diffusion process defined by the stochastic diffusion equation is a good approximation model of the price process, the essence of these two process is the statistical inference for diffusion processes and the numeric methods for the stochastic differential equation. This paper aims to study these two issues for some diffusion models. The contents include the following four aspects.The first part of this paper concerns the second-order diffusion process with jumps. Based on infinitesimal moment conditions we propose the local linear estimators for the drift coefficient and diffusion coefficient. This estimator enjoys the theoretical advantages of design adaptation, automatic boundary correction and minimax efficiency. Moreover, we prove the consistency and asymptotic normality of these estimators under mild conditions.The second part of this paper concerns stochastic volatility models. This model can resolve empirical biases associated with the Black-Scholes model, such as volatility smiles. We apply the empirical likelihood methods to construct confidence interval for nonparametric diffusion coefficient in the volatility process. Compared with the confidence interval constructed by normal approximation, the shape and orientation of these intervals are determined by data. In addition, the confidence intervals are obtained without the estimation of the volatility. In this chapter, the empirical likelihood ratio statistic is constructed through the estimating equation satisfied by the Fourier estimator. And the limit theory of the empirical likelihood ratio statistic is developed under some mild conditions.The third part of this paper concerns nonlinear stochastic-volatility jump-diffusion models. Relative to the stochastic-volatility diffusion models considered in chapter 2, this model is more reasonable to characterize the financial phenome-na by introducing Levy process which describes the jumps in the financial market. We construct empirical likelihood ratio statistic for the unknown coefficients of the volatility process, and prove that, under some mild conditions, the empirical likelihood ratio statistic is asymptotically chi-square distributed with one degree of freedom.The last part of this paper concerns the numeric methods for solving s-tochastic differential equation. Milstein scheme is a modification of Euler scheme to improve the rate of convergence. Non-equidistant time-nets also have posi-tive effect on weak convergence. This chapter considers how to combine Milstein scheme and random sampling to improve the rate of convergence. We present the asymptotic behavior (weak convergence) of the error processes of the Milstein scheme for stochastic differential equations driven by a vector of continuous local martingales, for non-equidistant and stochastic evaluation times.