On the problem of stochastic integral representations of functionals of the brownian motion. II

In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl., 48 (2004), pp. 304-313], a method of obtaining stochastic integral representations of functionals S(ω) of Brownian motion B = (B t)t≧0 was stated. Functionals maxt≦T Bt and maxt≦T-a Bt, where T-a = inf{t: Bt = -a}, a > 0, were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional max t≦gT Bt, where (non-Markov time) gT = sup{0 ≦ t ≦ T: Bt = 0} are given. © 2007 Society for Industrial and Applied Mathematics.