### Simulation solutions of the Stochastic Differential Equation

*dW = dt + dX *

This is the simplest stochastic differential equation -- its solution is a
Wiener process (Brownian motion) with drift coefficient
and diffusion coefficient .

By the analysis in the preceding lecture notes, we know to expect the solution
W(t) to be a random variable, normally distributed with mean
*t* and standard deviation sqrt(t).

In this simulation, you get to use the slider bars to set
and , and then
have the simulation repeatedly construct a (Brownian) path with the
parameters you have chosen. You can compare the
stochastic simulation (blue graph) with the solution of the
deterministic differential equation dW=dt
(pink graph). At the right of the graph, the number of times
the path ends in each of the intervals delineated by the black lines is
tallied (after you set
and , you can choose how many intervals you would like
to consider).

After you have done a number of simulations (you may need
to do at least one or two hundred to get good statistics) you can see
the results of the simulation (for t=10) summarized against the theoretical
probability distribution. In order to help you complete a large number of
simulations quickly, you may click on the "many" button to do them 25 at
a time.

#### Step-by-Step Instructions:

- Use slider bars to set
and .
- Choose the number of W-axis subdivisions (4, 12, 20 or 40).
- Click on the "once" button to run the simulation once, or the
"many" button for 25 runs at a time. Repeat this until you feel you have
enough samples.
- Click "summarize" to compare your simulation to the theoretical
probability distribution for t=10.
- Click "reset" (at any time) to start over.