## The Annals of Probability

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

Secondary: 60J55: Local time and additive functionals

**Keywords**

Quadratic variation functionals of Brownian motion

#### Citation

Wang, Albert T. Quadratic Variation of Functionals of Brownian Motion. Ann. Probab. 5 (1977), no. 5, 756--769. doi:10.1214/aop/1176995717. https://projecteuclid.org/euclid.aop/1176995717