  Recent studies have extended the use of the stochastic
Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for
deriving quantum mechanical equations. However, these studies often assume that
it is valid to apply the HJB equation directly to complex numbers, an approach
that overlooks the fundamental problem of comparing complex numbers to find
optimal controls. This paper explores how to correctly apply the HJB equation
in the context of complex variables. Our findings significantly reevaluate the
stochastic movement of quantum particles within the framework of stochastic
optimal control theory. We derived the complex diffusion coefficient in the
stochastic equation of motion using the Cauchy-Riemann theorem, considering
that the particle's stochastic movement is described by two perfectly
correlated real and imaginary stochastic processes. We demonstrated that the
derived diffusion coefficient took a form that allowed the HJB equation to be
linearized, thereby leading to the derivation of the Dirac equations. These
insights deepen our understanding of quantum dynamics and enhance the
mathematical rigor of the framework for applying stochastic optimal control to
quantum mechanics.
