Quadratic Variation of Functionals of Brownian Motion

Abstract

The quadratic variation of functionals $F(t)$ of $n$-dimensional Brownian motion is investigated. Let $\Pi_n = \{t_1^n, t_2^n, \cdots, t^n_{l(n)}\}$ with $a = t_1^n < t_2^n < \cdots < t^n_{l(n)} = b$ be a family of partitions of the interval $\lbrack a, b\rbrack$. The limiting behavior of $Q^2(F, \Pi_n) = \sum^{l(n)-1}_{k=1} (F(t^n_{k+1}) - F(t_k^n))^2$ as $n \rightarrow \infty$, assuming $\|\Pi_n\| \rightarrow 0$, is studied. And the existence of this limit is obtained for a fairly general class of functionals of Brownian motion.