How do I use Brownian motion code?

I know brownian motion code is bm(mu sigma)

I can get mu(mean), sigma(variance)from the data set.

I need a graph and other information about brownian motion from this code.(such as dW(t).

x(t) = x_0 + (mu(t)) + (sigma(t)*dW(t))

I looked at the mathwork website about brownian motion but all I get is

obj =

   Class BM: Brownian Motion
   ----------------------------------------
     Dimensions: State = 1, Brownian = 1
   ----------------------------------------
      StartTime: 0
     StartState: 0
    Correlation: 1
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
             Mu: 45.5
          Sigma: 24.8267

I need an actual number with equation. And a motion graph.

How do I get that?

Please explain step by step.

Thank you.

Answer 1

[Hari]

Hi,

I understand that you want to simulate and visualize Brownian motion using your own values of mu (mean drift) and sigma (volatility), and you’d like to see both the motion graph and the increment values (dW).

I assume you already have numerical values for mu and sigma based on your data, and you're looking for a step-by-step way to simulate and plot the Brownian motion in MATLAB, along with calculating the stochastic increments.

In order to simulate Brownian motion and analyze it, you can follow the below steps:

Step 1: Set up the time parameters

Define how long the simulation will run and how many points you want. This means setting a time range (T) and number of steps (N).

The time step (dt) is then calculated as T divided by N.

Step 2: Generate the Brownian increments (dW)

Create a vector of random values from a normal distribution with mean 0 and standard deviation sqrt(dt).

This models the random changes in the Brownian process at each time step.

Step 3: Simulate Brownian motion (W)

Calculate W by taking the cumulative sum of dW.

To apply drift and volatility, compute x as:

x = x0 + mu * t + sigma * W

where t is the vector of time values and x0 is the starting point (often 0).

Step 4: Plot the Brownian motion path

Plot x against time to visualize the Brownian motion with drift and diffusion.

Step 5: Output or analyze dW if needed

If you want to understand the stochastic component, plot dW separately or save it for analysis.

Refer to the documentation of "randn" function to generate normal random values:

https://www.mathworks.com/help/matlab/ref/randn.html

Refer to the documentation of "cumsum" function for cumulative sum:

https://www.mathworks.com/help/matlab/ref/cumsum.html

Refer to the documentation of "plot" function for plotting:

https://www.mathworks.com/help/matlab/ref/plot.html

Hope this helps!