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A backward scalar or matrix recurrence for the quadratic coefficients in a value function for a linear-quadratic optimal control problem. It results from completing the control square in the Bellman equation. Multiplicative noise modifies the quadratic coefficients through its second moments.

Completing the square in the Hamilton-Jacobi-Bellman equation shows that the minimizing feedback isThis is a linear-quadratic optimal control problem, so set $F(x,t)=a(t)x^2$. The equation and terminal condition becomeThis Riccati equation is separable and givesConsequently the general optimal feedback law isWhen $c_t=c$ is constant,Along the corresponding optimal trajectory, $x_s=x(c+h-s)/(c+h-t)$, so the open-loop control is constant: $u_s^{\ast }=-x/(c+h-t)$ for $s\in [t,h]$.