Mean and variance
At a fixed time, an SDE solution \(X_t\) is a random variable. Its mean describes the expected state, while its variance measures dispersion around that mean.
\[m(t)=E[X_t],\qquad V(t)=E[(X_t-m(t))^2].\]Equivalently,
\[V(t)=E[X_t^2]-E[X_t]^2.\]Many simulated paths can estimate these quantities, but in some models differential equations for moments can be derived analytically.
Key idea. The mean alone does not describe stochastic uncertainty; variance quantifies how widely possible trajectories are distributed.