Submit your work to tleise@amherst.edu (and/or turn in by hand) by 4pm Friday.

In this lab we will do some further exploration of Brownian motion.

Multi-Dimensional Brownian Motion

Multi-dimensional Brownian motion can be constructed using independent one-dimensional Brownian motions for the coordinates of a vector-valued function \(X_t=(X_t^{(1)}, \dots, X_t^{(n)})\).

Exercise 1

Update the R code from last week’s lab to run simulations of 2-dimensional Brownian motion by generating two Brownian motions \(X_t\) and \(Y_t\). Plot the resulting path \((X_t,Y_t)\) in the \(xy\)-plane.

Exercise 2

Add a 3rd Brownian motion \(Z_t\) to run simulations of 3-dimensional Brownian motion. Plot the resulting path using plot3D, which requires loading the plot3D package: in the RStudio window, click on Install in the Packages tab, then choose plot3D. It should then download and install the package; you may need to click on the box next to plot3D in the list of packages to actually load it.

library("plot3D")
## Warning: package 'plot3D' was built under R version 3.2.3
x<-(0:100)/6 # dummy vectors to illustrate 3D plotting function
y<-cos(x)
z<-sin(x)
lines3D(x,y,z,ticktype="detailed",main="3D Brownian motion simulation",colkey=FALSE)

Martingales and Brownian Motion

For continuous-time stochastic processes, we say that \(Y_t\) is a martingale with respect to filtration \(\mathcal{F}_t=\{X_s: 0\le s\le t\}\) if it is \(\mathcal{F}_t\)-measurable, \(\mathbb{E}|Y_t|<\infty\), and \(E(Y_t|\,\mathcal{F}_s)=Y_s\) for all \(s<t\).

Geometric Brownian motion can be defined as \[Y_t=e^{\mu t+\sigma X_t},\] where \(X_t\) is a standard Brownian motion.

Exercise 3

Calculate \(\mathbb{E}[e^{\sigma X_t}]=\int_{-\infty}^\infty{e^{\sigma x}f(x)dx}\), given that \(X_t\) is a standard Brownian motion, so \(f(x)\) here is the density function for the normal distribution with mean 0 and variance \(t\). As it’s not an easy integral to do by hand, feel free to use Mathematica or Wolfram Alpha.

Exercise 4

Calculate \(E(Y_t|\,\mathcal{F}_s)\) for \(t>s\ge0\), using the result from Exercise 3.

Exercise 5

What condition on \(\mu\) and \(\sigma\) is required for a geometric Brownian motion \(Y_t\) to be a martingale?