Stochastic-calculus reference
Brownian motion: \(W_t\), with \(W_0=0\), independent increments and
\[W_{t+\Delta t}-W_t\sim N(0,\Delta t).\]Hence a simulated increment can be written
\[\Delta W=\sqrt{\Delta t}\,Z,\qquad Z\sim N(0,1).\]SDE
\[dX_t=a(X_t,t)dt+b(X_t,t)dW_t.\]\(a\) is the drift and \(b\) controls diffusion.
Itô multiplication rules
\[(dW_t)^2=dt,\qquad dt\,dW_t=0,\qquad (dt)^2=0.\]Itô's lemma
If \(dX_t=a\,dt+b\,dW_t\) and \(Y_t=f(X_t,t)\), then
\[df=\left(f_t+af_x+\frac12b^2f_{xx}\right)dt+bf_xdW_t.\]