This review article introduces two recent advances in stochastic
simulation: the construction of efficient algorithms for estimating
rare events and the generation of samples from a stationary
distribution that has no closed form. Estimating a very small
quantity requires extreme accuracy to form a useful confidence
interval. This makes the slowly convergent rare-event simulation a
challenge task in both efficiency and accuracy. In this report, we
introduce the examples of rare events of interest and the
difficulties in estimating them. Various approaches to pursue robust
and efficient estimators along the development are discussed and
evaluated. Numerical experiments on estimating ruin probability are
provided to show the quality of these approaches.
In steady-state
simulation, how to generate samples from a stationary stochastic
process has long been the key subject. The common practice is to
discard the data gathered during the initial transient period.
However, how long the warm-up period must be raises another problem
that has no satisfactory answer. Fortunately, by the development in
the past two decades, exact simulation has become possible for
certain stochastic models. In this report, we will introduce two
important methods and related applications.

Examining the conditions of positively or negatively associated
sequences of random variables obeying the strong law of large numbers provided by
Alexander, the sequences of Gaussian random variables, nonnegative and uniformly bounded
sequences of random variables with general dependent structure were studied, and the
sufficient conditions for they obeying the strong law of large numbers were given. At
last, an example for Gaussian sequence satisfying the strong law of large numbers was
given.

Inspired by intuitive meanings of truncated power basis's
coefficients, the local penalization based on range's linear decreasing function is given
in penalized spline regression model. This method gives less penalization to fitting curve
where data is with more volatility, which makes fitted curve controls tradeoff between
goodness-of-fit and smoothness better. Simulations show that regression models with local
penalized spline obtain lower information rules' scores than global penalized spline when
the data is with heteroskedasticity.

In this paper, various concepts of states for Markov chains in
random environments are introduced into the research field, and the relationships between
these states are discussed. Especially, a partition of the state space is given. Also,
the properties of those states are investigated. At last, some examples are given to
illustrate the rationality of those states.

Brownian motion and normal distribution have been widely used
in Cox-Ingersoll-Ross interest rate framework to model the instantaneous interest rate
dynamics. However, empirical studies have also shown that the return distribution of
interest rate has a higher peak and two fatter tails than those of the normal distribution.
Meanwhile, when the rare catastrophic shocks occur or the regime shifts in the economy
and finance, the money market may have jumps. In this paper, we will consider a class
of reflected Cox-Ingersoll-Ross interest rate models with  noise. Furthermore,
we shall continue to supply the Laplace transform of the stationary distribution about
this reflected diffusion process with jumps.

This paper investigates the test of significance for the binary
choice model with stochastic trend process. The results show that when the true parameter
vector is zero, the limiting distribution of the t statistic follows standard normal
distribution. The joint significance test statistics Wald, LM and LR are asymptotically
equivalent and have a Chi-square limiting distribution.

In the last few decades, longitudinal data was deeply research
in statistics science and widely used in many field, such as finance, medical science,
agriculture and so on. The characteristic of longitudinal data is that the values are
independent from different samples but they are correlate from one sample. Many
nonparametric estimation methods were applied into longitudinal data models with development
of computer technology. Using Cholesky decomposition and Profile least squares estimation,
we will propose a effective spline estimation method pointing at nonparametric model of
longitudinal data with covariance matrix unknown in this paper. Finally, we point that
the new proposed method is more superior than Naive spline estimation in the covariance
matrix is unknown case by comparing the simulated results of one example.