Editing Martingale (probability theory) (section)

=== Connection to Stochastic Differential Equations ===

Solutions to <math>M(a, b)</math> correspond (in a weak sense) to solutions of the SDE <math>dX_t = b(X_t),dt + \sigma(X_t),dB_t</math>, where <math>\sigma\sigma^\top = a</math>. One sees this by applying the generator <math>A</math> to simple functions such as <math>x_i</math> or <math>x_i,x_j</math>, thereby recovering the drift <math>b</math> and the diffusion matrix <math>a</math>.