We present an analytical policy update rule that is independent of parametric function approximators. The policy update rule is suitable for optimizing general stochastic policies and has a monotonic improvement guarantee. It is derived from a closed-form solution to trust-region optimization using calculus of variation, following a new theoretical result that tightens existing bounds for policy improvement using trust-region methods. The update rule builds a connection between policy search methods and value function methods. Moreover, off-policy reinforcement learning algorithms can be derived from the update rule since it does not need to compute integration over on-policy states. In addition, the update rule extends immediately to cooperative multi-agent systems when policy updates are performed by one agent at a time.