Continuous Ocone martingales as weak limits of rescaled martingales (Q1860598)

The martingale \(N\) is called Ocone martingale if for every predictable process \(a\) taking values in \(\{-1,1\}\), the martingales \(\int a dN\) and \(N\) have the same law. Let \(M\) be a martingale with bounded jumps and let \(a_n,\) \(b_n\) be sequences of positive numbers, both increasing to infinity. The author shows that if the rescaled martingales \(M^n_t=M_{b_nt}/\sqrt{a_n}\) converge weakly, then the limit is necessarily a continuous Ocone martingale. Moreover, if \(N\) is a continuous Ocone martingale, then \(M^n\) converge weakly to \(N\) if and only if the quadratic variations \([M^n]\) converge weakly to the quadratic variation \([N]\) in \(D[0,\infty)\).