Local and Occupation Time of a Time Elapsed Process of Brownian Motion

We consider a renewal process Ut, defined as the time elapsed since the last zero of BM. Within the framework of a piecewise-deterministic Markov process involving Ut, where we assume the jump arrivals are driven by a general... [ view full abstract ]

We consider a renewal process U_t, defined as the time elapsed since the last zero of BM. Within the framework of a piecewise-deterministic Markov process involving U_t, where we assume the jump arrivals are driven by a general function of U_t, we derive an explicit expression for the Laplace transform of the Markov process, the marginal densities and simulation schemes at the first jumping time. Furthermore, we prove that the local time of a scaled U_t converges to that of BM, as well as the difference of the two quantities converges to a distribution of a rate of ½.