A point process model based on a class of Cox processes is developed to analyse precipitation data at a point location. The model is constructed using state-dependent exponential pulses that are governed by an unobserved underlying Markov chain. The mathematical formulation of the model where both the arrival rate of the rain cells and the initial pulse depth are determined by the Markov chain is presented. Second-order properties of the rainfall depth process are derived and utilised in model assessment. A method of moment estimation is employed in model fitting. The proposed model is used to analyse 69 years of sub-hourly rainfall data from Germany and 15 years of English rainfall data. The results of the analysis using variants of the proposed model with fixed pulse lifetime and variable pulse duration are presented. The performance of the proposed model, in reproducing second-moment characteristics of the rainfall, is compared with that of two stochastic models where one has exponential pulses and the other has rectangular pulses. The proposed model is found to capture most of the empirical rainfall properties well and outperform the two alternative models considered in our analysis.