Standard Brownian Motion

Standard Brownian Motion (SBM) Calculator

Initial value

W_{0}

Time horizon

T

Steps

n

Simulations

N

Decimal Places: 2 3 5 8

Introduction

This calculator simulates and analyses Standard Brownian Motion, also known as a Wiener process. It is used in probability theory to study the behaviour of random paths where increments are independent and normally distributed. By specifying an initial value $W_{0}$ , time $T$ , and the number of steps $n$ , users can explore stochastic trajectories and their statistical properties.

What this calculator does

The tool performs a Monte Carlo simulation of multiple random paths. It requires the starting value, total time duration, the number of discrete steps per path, and the total number of simulations. It produces comprehensive outputs including average final values, standard deviation, skewness, kurtosis, quadratic variation, and the coefficient of determination $R^{2}$ for the variance trajectory. It also provides a 2D random walk visualisation for diffusion analysis.

Formula used

The simulation uses the iterative discrete-time approximation where each step increment is calculated using Gaussian random variables. The change in the process $ΔW$ is determined by the square root of the time step $Δt$ multiplied by a standard normal variable $Z$ . Variance validation is performed by comparing empirical results to the theoretical variance $t$ at time $t$ .

W_{t + Δt} = W_{t} + \sqrt{Δt} Z

Δt = \frac{T}{n}

How to use this calculator

1. Enter the initial value $W_{0}$ to set the starting point of the simulation.
2. Input the time horizon $T$ and the number of discrete steps $n$ to define the path resolution.
3. Specify the number of simulations $N$ to determine the size of the statistical sample.
4. Select the desired decimal precision and execute the calculation to view the trajectories and summary statistics.

Example calculation

Scenario: A student in a physics laboratory is modelling the one-dimensional diffusion of a particle over a specific duration to understand the variance of its final displacement.

Inputs: $W_{0} = 0$ , $T = 4$ , $n = 100$ , and $N = 1000$ .

Working:

Step 1: $Δt = \frac{T}{n}$

Step 2: $Δt = \frac{4}{100} = 0.04$

Step 3: $W_{i} = W_{i - 1} + \sqrt{0.04} Z$

Step 4: $ExpectedVariance = T = 4$

Result: Average Final Value approx 0, Variance approx 4.00.

Interpretation: The result confirms that the process is centred at the origin with a spread that scales linearly with time.

Summary: The simulation successfully matches the theoretical properties of the Wiener process.

Understanding the result

The outputs reveal the distribution of final positions. The Average Final Value should remain near the initial value, while the Standard Deviation should approximate $\sqrt{T}$ . High $R^{2}$ values for the variance trajectory indicate that the empirical simulation closely follows the theoretical linear growth of variance over time.

Assumptions and limitations

The model assumes that increments are independent and that the underlying random variables follow a standard normal distribution $N (0, 1)$ . Numerical accuracy is limited by the number of steps $n$ and the total runs $N$ used for averaging.

Common mistakes to avoid

One common error is using too few simulations $N$ , which leads to high sampling error and results that deviate significantly from theoretical expectations. Another mistake is confusing the number of steps with the time horizon, which alters the calculation of the incremental variance $Δt$ .

Sensitivity and robustness

The stability of the mean is robust across large simulation counts, but individual path extremes are highly sensitive to the random seed. Increasing the number of steps $n$ improves the approximation of the continuous process, particularly for the quadratic variation, which theoretically equals $T$ as $n$ approaches infinity.

Troubleshooting

If the distribution appears skewed or the variance is incorrect, ensure the number of runs is sufficient to satisfy the Law of Large Numbers. Inputs exceeding the permitted range, such as a negative time horizon or excessive steps, will trigger validation errors to prevent computational instability.

Frequently asked questions

What is quadratic variation in this context?

It is the sum of the squared differences between consecutive steps, which for a Wiener process over interval T, should converge to T.

Why is the Mean Square Displacement (2D) calculated?

It measures the average distance a particle travels in a 2D plane, illustrating the diffusion characteristics of the random walk.

What does a negative Kurtosis imply?

A negative excess kurtosis indicates that the distribution of final values is flatter than a standard normal distribution.

Where this calculation is used

Standard Brownian Motion is fundamental in academic curricula involving stochastic calculus and advanced probability. It is widely used in environmental modelling to simulate the dispersal of pollutants in fluids, in social research to model random fluctuations in population dynamics, and in physics to describe the erratic movement of particles suspended in a medium. Educational exercises often use these simulations to demonstrate the Central Limit Theorem and the properties of Gaussian processes in real-time modelling scenarios.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.