Strong consistency of the nonparametric kernel estimator of the transition density for the second-order diffusion process

The integrals of diffusion processes are of significant importance in the field of finance, particularly in relation to stochastic volatility models, which are frequently employed to represent the temporal variability of stock prices. In this paper, we consider the strong consistency of the nonparametric kernel estimator of the transition density for second-order diffusion processes, using the moment inequalities of $\rho$-mixing sequences to demonstrate the strong consistency under some regularity conditions. Furthermore, the asymptotic mean square error is provided based on the deviation and variance of the transition density kernel estimator. The optimal bandwidth is found and thus the convergence rate of the kernel estimator is obtained. At the same time, our results are compared with the conclusions of the univariate density function.