\[
\Delta W_i \sim \mathcal{N}(0,\Delta t).
\]
Any normal \(\mathcal{N}(0,\Delta t)\) random variable can be written as \(\sqrt{\Delta t}\,Z_i\),
where \(Z_i\sim\mathcal{N}(0,1)\). Therefore a standard simulation rule is:
\[
W_{i+1}=W_i+\sqrt{\Delta t}\,Z_i,
\qquad Z_i\sim\mathcal{N}(0,1).
\]
This is exactly what the calculator does: it draws independent standard normals \(Z_i\), scales by \(\sqrt{\Delta t}\),
and accumulates the increments to form a full path.
3) Why the endpoint is \(\mathcal{N}(0,T)\)
Since \(W_T\) is a sum of independent normal increments,
it is normal itself. The mean is the sum of means (zero), and the variance is the sum of variances:
\[
W_T=\sum_{i=0}^{n-1}\Delta W_i,
\qquad
E[W_T]=0,
\qquad
\mathrm{Var}(W_T)=\sum_{i=0}^{n-1}\Delta t = T.
\]
With many simulated paths, the collection of endpoints \(W_T\) should look like samples from \(\mathcal{N}(0,T)\).
The calculator displays simulated endpoint mean/variance alongside the theoretical values.
4) Covariance structure (university detail)
Brownian motion has a clean covariance function:
\[
\mathrm{Cov}(W_s,W_t)=\min(s,t).
\]
In particular, if \(s\le t\) then \(W_t=W_s+(W_t-W_s)\) and the increment \(W_t-W_s\) is independent of \(W_s\),
which leads to \(\mathrm{Cov}(W_s,W_t)=\mathrm{Var}(W_s)=s\).
5) Choosing \(\Delta t\) in practice
- Smaller \(\Delta t\) gives a finer time grid (more points) and a smoother visual path.
- Very small \(\Delta t\) means many steps, which can slow drawing/animation.
- This tool snaps \(\Delta t\) so that an integer number of steps ends exactly at \(T\).
6) Geometric Brownian motion (stocks tease)
A common finance extension is geometric Brownian motion (GBM), used in the Black–Scholes model:
\[
dS_t=\mu S_t\,dt+\sigma S_t\,dW_t,
\qquad S_0>0.
\]
One convenient discretization is the log form:
\[
S_{t+\Delta t}
=S_t\exp\!\Big(\big(\mu-\tfrac12\sigma^2\big)\Delta t+\sigma\sqrt{\Delta t}\,Z\Big),
\quad Z\sim\mathcal{N}(0,1).
\]
GBM keeps \(S_t>0\) and produces multiplicative randomness (paths look “wiggly” but on a positive scale).
7) What this calculator shows
- Multiple simulated paths on a coordinate system with numeric time and value axes.
- An animation that reveals the paths over time (Play/Pause + progress scrub).
- Endpoint sanity checks: theoretical vs. simulated mean and variance at time \(T\).
