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

  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.

  2. 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?

  • $\begingroup$ I think in your first part, it must be $\sum_{i=1}^{n}$ rather than $\sum_{i=1}^{M}$ $\endgroup$ – Neeraj Feb 19 '16 at 13:33
  • $\begingroup$ @Neeraj: You're right. I've corrected it. Thanks. $\endgroup$ – Evan Aad Feb 19 '16 at 14:01

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 $lim_{m \to \infty}$

  • $\begingroup$ Thanks. What do you mean by " Your second case only converge to Brownian motion only if $lim_{m \rightarrow \infty}$?" What about the first case? $\endgroup$ – Evan Aad Feb 19 '16 at 14:12
  • $\begingroup$ @EvanAad You asked about convergence. Brownian motion process is continuous in time and increments have normal distribution (other characteristics are too) . In your second case, increments takes only two possible value. Statistics shows that process only converge to Brownian motion if $lim_{M \to \infty}$. In your first case, increments already have normal distribution. $\endgroup$ – Neeraj Feb 19 '16 at 14:20
  • $\begingroup$ Thanks. But what about the paths? Does the first simulation's path converge to a Brownian motion's path? $\endgroup$ – Evan Aad Feb 19 '16 at 14:23
  • $\begingroup$ @EvanAad Remember one thing, Brownian motion is continuous in time and you can never simulate process in continuous time. To simulate you have to discretized the process. $\endgroup$ – Neeraj Feb 19 '16 at 14:29
  • $\begingroup$ @EvanAad so your first process is just discretized Brownian motion. $\endgroup$ – Neeraj Feb 19 '16 at 16:56

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