In this paper a weak stability property of a random rotational motion of a ship with a two position automatic pilot, is dealt with. This is done by choosing two different sets of stochastic differential equations as mathematical models for the ships motion. For each model, a partial differential equation over a bounded domain in the state space is derived. The value of the solution to the PDE at a pointPin the domain provides the conditional expectation of the first time that the system's trajectory in the state space hits the boundary given that P is the starting point of the trajectory. The partial differential equations derived for the two models are of two and three independent variables, respectively. Numerical procedures for solving these equations are suggested, and their efficiency and applicability are demonstrated with examples. For each model, a functional is defined on the PDE solutions. The values of the functionals serve as a measure of the system's stochastic stability.