Plain 1D Fokker–Planck (forward Kolmogorov) equation for a wealth variable w ≥ 0:
Let \(P(w,t)\) be the density of wealth at time \(t\). Let \(A(w,t)\) be the drift (mean rate of change of \(w\)). Let \(B(w,t) ≥ 0\) be the diffusion (variance rate of change of \(w\)).
Then:
|eq ∂P(w,t)/∂t = - ∂/∂w [ A(w,t) P(w,t) ] + (1/2) ∂2/∂w2 [ B(w,t) P(w,t) ].
(One typically supplements this with a boundary condition at w = 0, e.g. reflecting/no-flux, to keep wealth nonnegative.)