Brownian Motion from Gaussian Increments and Random Walks

Summary

The document compares two proposed ways to simulate a standard one-dimensional Brownian path over a fixed time interval. The first sums independent, mean-zero Gaussian increments with variance equal to the time step. The second traces a symmetric random walk whose step size is the square root of the time step. The answer explains that Brownian increments over a time interval have variance equal to that interval, and that the random walk needs equally likely upward and downward moves to have the matching mean and variance.

The response treats both constructions as discretized Brownian motion, but it does not give a complete account of convergence. Gaussian increments produce Brownian motion sampled on a time grid; the linearly connected path is an interpolation. The symmetric walk has discrete increments and approaches Brownian motion in an appropriate scaling limit, rather than having Gaussian increments at finite resolution. The answer’s final statement about convergence is cut off, so it does not specify the convergence mode or provide a proof. Its main useful point is the increment scaling and probability condition for the walk.

Key ideas

• Brownian motion increments over a time step have mean zero and variance equal to the step length.
• Independent Gaussian increments with variance matching the time step give Brownian motion values on a discrete grid.
• A symmetric random walk needs equal probabilities for its positive and negative steps to have zero mean.
• The random walk uses steps scaled by the square root of the time step and converges in a limiting sense.
• The response does not finish its discussion of convergence mode or provide a proof.

Tags

# Simulations of (standard, one-dimensional) Brownian motion


# Simulations of (standard, one-dimensional) Brownian motion












Consider the following two proposed simulations of paths of standard, one-dimensional Brownian motion between time $0$ and time $1$.

- Normal Increments Roll out a large sequence of, say $M$, independent Gaussian variables $X_1, X_2, X_3, \dots, X_M$ such that $X_n \sim N\left(\mu=0, \sigma^2=\frac{1}{M}\right)$, and plot a scatter plot of the points $(0.001n, \sum_{i = 1}^n X_i)$ with straight lines connecting consecutive points.

- Random Walk Trace out the path of a one-dimensional random walk that starts at $0$ and progresses by steps of $\pm \sqrt{\frac{1}{M}}$ every $\frac{1}{M}$th of a second.

Does any of these methods converge to a Brownian motion path? If so, what type of convergence is it?

## Answer by Neeraj (score 3)

https://quant.stackexchange.com/a/24399

Your first case is nothing but simulation of Brownian motion process.

Your second case is just an alternative view of your first case and hence also Brownian motion. Under Brownian motion, $dX_t \sim N(0, dt)$. So, your increments must follow $N(0, \frac1M)$. You have not stated probability of your increment, but for this process to be Brwonian motion, the probability of up and down step must be $0.5$.

The variance of your increment in $\frac1M$ second is: $$0.5 \left(\sqrt\frac1M-0 \right)^2 + 0.5\left(-\sqrt\frac1M-0\right)^2 = 0.5\frac1M + 0.5 \frac1M = \frac1M$$

EDIT : Your both case is discretized form of standard Brownian motion and they both are equivalent. Your second case only converge to Brownian motion only if $lim_{m \to \infty}$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.