← Stochastic Differential Equations

Moment equations

Moment equations describe the evolution of quantities such as \(E[X_t]\), \(E[X_t^2]\) and variance.

For

\[dX_t=a(X_t)dt+b(X_t)dW_t,\]

taking expectation gives, when the required conditions hold,

\[\frac{d}{dt}E[X_t]=E[a(X_t)].\]

Applying Itô's lemma to \(X_t^2\) gives

\[\frac{d}{dt}E[X_t^2]=E[2X_ta(X_t)+b(X_t)^2].\]

These equations may involve higher-order moments when \(a\) or \(b\) is nonlinear.

Key idea. Moment equations can describe expected behaviour and uncertainty without following every individual stochastic trajectory.