  This paper introduces a novel operator, termed the Y operator, to elevate
control performance in Actor-Critic(AC) based reinforcement learning for
systems governed by stochastic differential equations(SDEs). The Y operator
ingeniously integrates the stochasticity of a class of child-mother system into
the Critic network's loss function, yielding substantial advancements in the
control performance of RL algorithms.Additionally, the Y operator elegantly
reformulates the challenge of solving partial differential equations for the
state-value function into a parallel problem for the drift and diffusion
functions within the system's SDEs.A rigorous mathematical proof confirms the
operator's validity.This transformation enables the Y Operator-based
Reinforcement Learning(YORL) framework to efficiently tackle optimal control
problems in both model-based and data-driven systems.The superiority of YORL is
demonstrated through linear and nonlinear numerical examples showing its
enhanced performance over existing methods post convergence.
