Single jump martingales and the Skorokhod embedding

In the talk we consider the class of local martingales that start from zero and such that their running maximum is continuous and the global maximum is a.s. finite. For such processes, the change of time generated by the running maximum process transforms a local martingale into a process of finite variation with a single jump down. At the same time, this change of time retains the terminal value of the process and its global maximum, the joint distribution of which is the subject of further study. Next, we consider a large subclass of the class under consideration. It is shown that the property to belong to this subclass is determined by the joint distribution of the terminal value of the process and its global maximum. This property is also necessary and sufficient for the time-changed local martingale to be a proper martingale. The set of joint distributions of the terminal value of the process and its global maximum corresponding to this subclass is fully described. Finally, it is shown how each joint distribution from this set can be realized as a solution to the Skorokhod emedding problem with a minimal stopping time.