Advanced Wiener Process Calculator

Wiener Process Calculator

Process Mode

Initial Value X(0)

Drift μ

Volatility σ

Time Horizon t

Increment Length Δt

Target Level

Upper Barrier

Lower Barrier

Confidence Z Value

Simulation Steps

Random Seed

Use standard mode for pure Brownian motion. Use drifted mode when a deterministic trend exists.

Plotly Graph

Example Data Table

Time t	Observed Path X(t)	Theoretical Mean	Theoretical Variance
0.00	0.0000	0.0000	0.0000
0.25	0.1800	0.1250	0.0900
0.50	0.4100	0.2500	0.1800
0.75	0.2300	0.3750	0.2700
1.00	0.6200	0.5000	0.3600

This sample illustrates how the path changes over time while the theoretical mean and variance evolve according to the selected drift and volatility.

Formula Used

Drifted Wiener process: X(t) = X(0) + μt + σW(t)

Standard Wiener process: W(t) ~ N(0, t)

Mean: E[X(t)] = X(0) + μt

Variance: Var[X(t)] = σ²t

Standard deviation: SD[X(t)] = σ√t

Increment mean: E[X(t + Δt) - X(t)] = μΔt

Increment variance: Var[X(t + Δt) - X(t)] = σ²Δt

Target probabilities are estimated from the normal distribution because X(t) is normally distributed under the Wiener process model.

How to Use This Calculator

Select standard mode for pure Brownian motion or drifted mode for trend-based motion.
Enter the initial value, drift, volatility, and total time horizon.
Set the increment length, target level, and optional upper and lower barriers.
Choose simulation steps and a seed for reproducible sample paths.
Press the calculate button to display summary results above the form.
Review the graph, probabilities, interval estimate, and barrier-crossing indicators.
Download the generated path table as CSV or export the results as PDF.

Frequently Asked Questions

1. What does this calculator measure?

It estimates key Wiener process quantities such as mean, variance, increment behavior, confidence interval, threshold probabilities, and one simulated sample path.

2. What is the difference between standard and drifted modes?

Standard mode fixes drift at zero and volatility at one. Drifted mode lets you model a Brownian path with user-defined trend and scale.

3. Why is the final path value different each time?

A Wiener path is random. Changing the random seed creates a new trajectory, while using the same seed reproduces the same simulated path.

4. What does volatility control?

Volatility controls how spread out the process becomes over time. Larger volatility produces wider variance and more jagged simulated paths.

5. Are the target probabilities exact?

The target probabilities use the normal distribution of X(t), so they are theoretical for the process at time t, not only for the simulated sample path.

6. What are barriers used for?

Barriers help detect whether the simulated path crossed chosen upper or lower levels. This is useful in finance, reliability, and first-passage studies.

7. Can I use this for teaching statistics?

Yes. It is useful for demonstrating Brownian motion, Gaussian increments, stochastic trends, and the link between theory and simulation results.

8. Does the calculator support exports?

Yes. You can export the simulated path table as CSV and create a PDF summary containing the main inputs and output metrics.