Brownian Motion Multiplication and Quadratic Variation
Summary
The document explains the stochastic-calculus shorthand that the square of a Brownian motion increment behaves like a time increment. It identifies this rule as a lemma associated with Itô calculus, rather than an arbitrary definition, and connects it to Brownian motion’s quadratic variation.
Quadratic variation can be characterized through the limiting sum of squared increments over increasingly fine partitions, or through the martingale property of the process obtained by subtracting it from squared Brownian motion. For standard Brownian motion, that limit is time. More generally, the covariation of stochastic integrals driven by the same Brownian motion is the time integral of the product of their integrands. This mathematical result underlies the differential multiplication rules used in Itô’s formula; the document gives the conceptual derivation but not a full proof of convergence.
Key ideas
- The rule that a Brownian increment squared corresponds to a time increment expresses quadratic variation.
- Standard Brownian motion has quadratic variation equal to elapsed time.
- Quadratic variation is obtained as a probability limit of sums of squared increments.
- For stochastic integrals driven by the same Brownian motion, covariation integrates the product of their integrands.
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# Is the Brownian motion multiplication rule a definition or is it a theorem?
# Is the Brownian motion multiplication rule a definition or is it a theorem?
Is the Brownian motion multiplication rule a definition or is it a theorem?
Refer to the highlight part of https://i.sstatic.net/doQuT.png where $dw_1(t)dw_1(t)=dt$
## Answer by Mark Joshi (score 7, accepted)
https://quant.stackexchange.com/a/18111
It's a lemma! Ito's Lemma gives the change of coordinates rule for stochastic calculus. The multiplication rule is a shorthand way of expressing it.
## Answer by Gordon (score 5)
https://quant.stackexchange.com/a/18116
Formally, this is a shorthand for the quadratic variation. For a more rudimentary definition, $\langle W, W\rangle$ is a process such that $W^2-\langle W, W\rangle$ is a martingale. Moreover, $\langle W, W\rangle_t$ is a limit, in probability, of the variation \begin{align*} \sum_{i=1}^n|W_{t_{i}}-W_{t_{i-1}}|^2, \end{align*} over the partition \begin{align*} 0=t_0 < t_1 < \cdots < t_n = t. \end{align*} From the property of a standard Brownian motion, it can be shown that the above limit equals $t$. That is, $\langle W, W\rangle_t = t$.
In general, if $X_t = \int_0^t \xi_s dW_s$ and $Y_t = \int_0^t \eta_s dW_s$, then \begin{align*} \langle X, Y \rangle_t &= \int_0^t \xi_s \eta_s ds, \end{align*} which we also write as \begin{align*} \langle dX_t, dY_t\rangle = \xi_t\eta_tdt. \end{align*} For $\xi_t=\eta_t = 1$, we then have \begin{align*} \langle dW_t, dW_t\rangle = dt, \end{align*} which is usually write as $dW_t dW_t = dt$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.