# Maximum likelihood estimation of continuous-time non-homogeneous Markov chains with time-dependent intensity and covariates

This paper presents elementary proofs on distributional properties of sample paths of continuous-time non-homogeneous Markov chain stated in Section 8.9 of Iosifescu(1980). The results are used to develop likelihood function of continuously observed realizations of Markov chains for general transition intensity matrix. It verifies and elaborates the formula (2) in Andersen and Keiding(2002) for the likelihood function of non-homogeneous Markov chain. Using finite-difference discretization of the Kolmogorov backward equation, an application of implicit Euler method shows that the transition probability matrix is explicit in terms of the intensity matrix. The solution coincides with the product integration of Aalen and Johansen(1978) and Andersen et al.(1995). In particular, it takes an explicit form of an exponential matrix. Score function and observed information matrices are derived explicitly in terms of matrix tensor notations. Thanks to explicit forms of score function and information matrix a fast convergence Newton-Raphson method is presented for ML recursive estimation of piecewise constant transition intensity and multivariate regression coefficient.