A continuous space Monte Carlo method is used to simulate epitaxial growth of lattice matched and mismatched compound epilayers. For comparison with experimental findings we calculate the growth rate of our layer by layer epitaxy as a function of temperature. The method also reveals the incorporation of dislocations and the accommodation of strain to overcome the lattice mismatch between substrate and epilayer. Finally we show that continuous space models can be used to calculate rhombic distortion of a strained epilayer.