The Play Operator, the Truncated Variation and the Generalisation of the Jordan Decomposition

The play operator minimizes the total variation on intervals [0; T]; T > 0, of functions approximating uniformly given
regulated function with given accuracy and starting from a given point. It is closely related to the Skorohod problem encountered in Probability Theory. We link the play operator with so called truncated variation functionals, and provide a semi-explicit expression for the play operator in terms of these functionals. Generalisation for time-dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan-like decomposition.