Stochastic integral representation of some martingales

AbstractLet {Xt} be a continuous square integrable martingale. Denote its increasing (natural) process by {At}. Let St, Tt be the left and right inverses of At, respectively. Then for any square integrable martingale {Yt} defined on {Xt}, Yt = ∝0tψsdXs, R0 < t < S∞ where S∞ = limt→∞ St, R0 = inf {t: Xt ≠ 0} provided that Y(T(t)) is σ(X(T(s)): s ⩽ t)-measurable. All martingales are assumed to be zero at t = 0. Brownian motion and Poisson processes are considered also