  This paper presents a simulation study on turnpike phenomena in stochastic
optimal control problems. We employ the framework of Polynomial Chaos
Expansions (PCE) to investigate the presence of turnpikes in stochastic LQ
problems. Our findings indicate that turnpikes can be observed in the evolution
of PCE coefficients as well as in the evolution of statistical moments.
Moreover, the turnpike phenomenon can be observed for optimal realization
trajectories and with respect to the optimal stationary distribution. Finally,
while adding variance penalization to the objective alters the turnpike, it
does not destroy the phenomenon.
