Penalisation of the Symmetric Random Walk by Several Functions of the Supremum

Call (Omega, F-infinity, P, X, F) the canonical space for the standard random walk on Z. Thus, Omega denotes the set of paths phi : N -> Z such that vertical bar phi(n + 1) - phi(n)vertical bar = 1, X = (X-n, n >= 0) is the canonical coordinate process on Omega; F = (F-n, n >= 0) is the natural filtration of X, F-infinity the sigma-field V-n >= 0 F-n, and P-0 the probability on (Omega, F-infinity) such that under P-0, X is the symmetric nearest neighbour random walk started from 0. Let G : N x Omega -> R+ be a positive, adapted functional. For several types of functionals G, we show the existence of a positive F-martingale (M-n, n >= 0) such that, for all n and all A(n) is an element of F-n, E-0[1{Lambda(n)}G(p)]/E-0[G(p)] -> E-0[1{Lambda(n)}M-n] when p -> infinity. Thus, there exists a probability Q on (Omega, F-infinity) such that Q(A(n)) = E-0[1{Lambda(n)}M-n] for all Lambda(n) is an element of F-n. We describe the behavior of the process (Omega, X, F) under Q. In this paper, we penalised the standard random walk by several functions of its maximum. The aim is to show that in spite of very close penalisation functions, under the new probabilities, the canonical process behaves very differently. We study here five kinds of G: G(p) is a function of S-p where S-p is the unilateral supremum of X. G(p) is a function of S-gp where g(p) is the last 0 at the left of p. G(p) is a function of S-dp where d(p) is the first 0 at the right of p. G(p) is a function of S*(gp) where S*(p) is the bilateral supremum of X. G(p) is a function of S*(p). A similar study has been realized for other kinds of G (cf. [2]).