A general stochastic maximum principle for singular control problems.
A stochastic integral equation corresponding to a probability space is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable . One constructs stochastic processes , connected with a Markov chain and with the space . The expected values of (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative .
We prove, by means of Malliavin calculus, the convergence in of some properly renormalized weighted quadratic variations of bi-fractional Brownian motion (biFBM) with parameters and , when and .
Limiting laws, as t→∞, for brownian motion penalised by the longest length of excursions up to t, or up to the last zero before t, or again, up to the first zero after t, are shown to exist, and are characterized.