The maximal variation of a bounded martingale

Abstract

Let$\chi _0^n = \left\{ {X_t } \right\}_0^n$ be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ℳ ⁿ _p the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ⁿ₀ is denoted byV ⁿ₀ and defined by$V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)$. It is proved that

$$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$

, where ϕ(p) is the well known normal density evaluated at itsp-quantile, i.e.

$$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$

. A sequence of martingales χ ⁿ₀ ,n=1,2, … is constructed so as to satisfy$\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)$.