Decomposing correlated random walks on common and counter movements

Random walk is one of the most classical models in probability theory which has got extensive applications in many areas and is still of great interest in practice. For those problems that require modelling two random walks on lattice, correlation of the random walks is non-ignorable. This paper presents a new method to study the dependency structure of two generally correlated random walks. By introducing a change-of-time process, two correlated random walks can be decomposed into sum/difference of two independent random walks with time change, where the two independent random walks present respectively the common movements and counter movements of the original random walks. A sufficient and necessary condition is given for the mutual independence of the change-of-time process and the two independent random walks. For the prospective applications of the decomposition method in theory and practice, we consider the calculations of the characteristic functions for Markovian and non-Markovian random walks and an empirical example in futures trading is given.