  The Method of Successive Approximations (MSA) is a fixed-point iterative
method used to solve stochastic optimal control problems. It is an indirect
method based on the conditions derived from the Stochastic Maximum Principle
(SMP), an extension of the Pontryagin Maximum Principle (PMP) to stochastic
control problems. In this study, we investigate the contractivity and the
convergence of MSA for a specific and interesting class of stochastic dynamical
systems (when the drift coefficient is one-sided-Lipschitz with a negative
constant and the diffusion coefficient is Lipschitz continuous). Our analysis
unfolds in three key steps: firstly, we prove the stability of the state
process with respect to the control process. Secondly, we establish the
stability of the adjoint process. Finally, we present rigorous evidence to
prove the contractivity and then the convergence of MSA. This study contributes
to enhancing the understanding of MSA's applicability and effectiveness in
addressing stochastic optimal control problems.
