Simulation of Brownian Motion Generation

Simulating Brownian motion in MATLAB provides an intuitive understanding of random walk characteristics. The essence of Brownian motion lies in the irregular movement of microscopic particles caused by numerous molecular collisions in a fluid medium. Through numerical simulation, we can observe trajectories formed by the cumulative effect of random step increments.

The core of Brownian motion simulation involves constructing cumulative sums of random steps. At each time increment, particle displacement can be modeled as an independent normal random variable with zero mean and variance proportional to the time step. MATLAB's `randn` function efficiently generates these random increments using Box-Muller transformation for normal distribution sampling.

The simulation process typically follows these algorithmic steps: 1. Define the number of time steps and step size parameters, initialize starting position 2. At each iteration, generate random displacements using normally distributed random numbers to determine direction and magnitude 3. Accumulate displacements through vectorized operations to form the particle's trajectory over time

By adjusting time step parameters and the quantity of random steps, different Brownian motion characteristics become observable, including trajectory smoothness and diffusion range. This simulation methodology finds applications beyond physical phenomenon research, extending to financial mathematics for asset price modeling and biostatistics for analyzing random processes in biological systems.