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quadratic variation of Brownian motion
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(Theorem)
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As Brownian motion is a martingale and, in particular, is a semimartingale then its quadratic variation must exist. We just need to compute its value along a sequence of partitions.
If $P=\{0=t_0\le t_1\le\cdots\le t_m=t\}$ is a partition of the interval $[0,t]$ , then the quadratic variation on $P$ is \begin{equation*} [W]^P=\sum_{k=1}^m(W_{t_k}-W_{t_{k-1}})^2. \end{equation*}Using the property that the increments $W_{t_k}-W_{t_{k-1}}$ are independent normal random variables with mean zero and variance $t_k-t_{k-1}$ , the mean
and variance of $[W]^P$ are
Here, $|P|=\max_k(t_k-t_{k-1})$ is the mesh of the partition. If $(P_n)_{n=1,2,\ldots}$ is a sequence of partitions of $[0,t]$ with mesh going to zero as $n\rightarrow \infty$ then, \begin{equation*} \mathbb{E}\left[([W]^{P_n}-t)^2\right]\le 3|P_n|t\rightarrow 0 \end{equation*}as $n\rightarrow\infty$ . This shows that $[W]^{P_n}\rightarrow t$ in the $L^2$ norm and, in particular, converges in probability. So, $[W]_t=t$ .
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"quadratic variation of Brownian motion" is owned by gel.
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Cross-references: converges in probability, norm, variance, mean, normal random variables, independent, partitions, sequence, semimartingale, martingale, quadratic variation, Brownian motion
This is version 2 of quadratic variation of Brownian motion, born on 2009-01-02, modified 2009-01-02.
Object id is 11442, canonical name is QuadraticVariationOfBrownianMotion.
Accessed 752 times total.
Classification:
| AMS MSC: | 60J65 (Probability theory and stochastic processes :: Markov processes :: Brownian motion) | | | 60H10 (Probability theory and stochastic processes :: Stochastic analysis :: Stochastic ordinary differential equations) |
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Pending Errata and Addenda
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![$\displaystyle \mathbb{E}\left[[W]^P\right]$](http://images.planetmath.org:8080/cache/objects/11442/js/img1.png "$\displaystyle \mathbb{E}\left[[W]^P\right]$"){kind=small}
![$\displaystyle =\sum_{k=1}^m\mathbb{E}\left[(W_{t_k}-W_{t_{k-1}})^2\right]=\sum_{k=1}^m(t_k-t_{k-1})=t,$](http://images.planetmath.org:8080/cache/objects/11442/js/img2.png "$\displaystyle =\sum_{k=1}^m\mathbb{E}\left[(W_{t_k}-W_{t_{k-1}})^2\right]=\sum_{k=1}^m(t_k-t_{k-1})=t,$")
![$\displaystyle \operatorname{Var}\left[[W]^P\right]$](http://images.planetmath.org:8080/cache/objects/11442/js/img3.png "$\displaystyle \operatorname{Var}\left[[W]^P\right]$"){kind=small}
![$\displaystyle =\sum_{k=1}^m\operatorname{Var}\left[(W_{t_k}-W_{t_{k-1}})^2\right]=\sum_{k=1}^m3(t_k-t_{k-1})^2$](http://images.planetmath.org:8080/cache/objects/11442/js/img4.png "$\displaystyle =\sum_{k=1}^m\operatorname{Var}\left[(W_{t_k}-W_{t_{k-1}})^2\right]=\sum_{k=1}^m3(t_k-t_{k-1})^2$")
