A study of time homogeneous, real valued Markov processes with a special property and a non-atomic initial distribution is provided. The new notion of a function of evolution of distribution which determines the dependency between one dimensional distributions of a process is introduced. This, along with the notion of bridge operators which determine the backward structure, as opposed to the forward structure determined by the usual semi-group operators, paves a way to the new approach for dealing with finite-dimensional distributions of Markov processes. This, in particular, produces explicit formulas which effectively simplify the computations of finite-dimensional distributions, giving an alternative to the standard approach based on computations using the chain rule of transition densities. This is a joint work with Jacek Jakubowski and Maciek Wisniewolski.

In many applications, a probabilistic model for a given system is defined implicitly by a simulation algorithm. Such implicit models can be simulated at any parameter value, but often the probability density function cannot be evaluated. We consider the case where the system described by an implicit model is partially observed. Parameter estimation for such partially observed, implicit models can in principle be carried out by repeated Monte Carlo simulations at various parameter values and comparing the log measurement densities of the observed data. However, the average of log measurement densities has a downward Jensen bias, which increases with increasing model dimensions. Under certain asymptotic assumptions, we develop methods for constructing Monte Carlo confidence intervals for the log likelihood of data and the maximum likelihood estimate given the data. Furthermore, for models that satisfy local asymptotic normality (LAN), we develop a method for estimating the unknown parameter and quantifying uncertainty. Our methods, for instance, can be used to enable likelihood-based inference for high-dimensional, partially observed, stochastic dynamic models.