Mimicking self-similar Markov martingales and extensions

As stochastic processes are not uniquely defined by their marginal distributions, it is of interest to construct different processes that match the marginal distributions of a given process — we call this mimicking process. In this thesis, we provide a construction scheme to mimic any self-similar, Markov, martingale process. Given such a process, we can construct a family of processes that are also self-similar and Markovian, and can be chosen to be martingales under certain condition. This mimicking can be done by randomising the transition function, or by time-changing the process together with an appropriate scaling. We obtain also the infinitesimal generator and some properties of the resulting processes. Some examples are also provided, together with the computations of the infinitesimal generators and the predictable quadratic variations of the resulting processes. We then extend our construction to mimic a broader class of processes, including some non-Markov processes.