## The Annals of Probability

### Quadratic Variation of Functionals of Brownian Motion

Albert T. Wang

#### 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.

#### Article information

Source
Ann. Probab., Volume 5, Number 5 (1977), 756-769.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995717

Digital Object Identifier
doi:10.1214/aop/1176995717

Mathematical Reviews number (MathSciNet)
MR445622

Zentralblatt MATH identifier
0382.60089

JSTOR