  Following the recent resurgence in establishing linear control theoretic
benchmarks for reinforcement leaning (RL)-based policy optimization (PO) for
complex dynamical systems with continuous state and action spaces, an optimal
control problem for a continuous-time infinite-dimensional linear stochastic
system possessing additive Brownian motion is optimized on a cost that is an
exponent of the quadratic form of the state, input, and disturbance terms. We
lay out a model-based and model-free algorithm for RL-based stochastic PO. For
the model-based algorithm, we establish rigorous convergence guarantees. For
the sampling-based algorithm, over trajectory arcs that emanate from the phase
space, we find that the Hamilton-Jacobi Bellman equation parameterizes
trajectory costs -- resulting in a discrete-time (input and state-based)
sampling scheme accompanied by unknown nonlinear dynamics with continuous-time
policy iterates. The need for known dynamics operators is circumvented and we
arrive at a reinforced PO algorithm (via policy iteration) where an upper bound
on the $\mathcal{H}_2$ norm is minimized (to guarantee stability) and a
robustness metric is enforced by maximizing the cost with respect to a
controller that includes the level of noise attenuation specified by the
system's $H_\infty$ norm. Rigorous robustness analyses is prescribed in an
input-to-state stability formalism. Our analyses and contributions are
distinguished by many natural systems characterized by additive Wiener process,
amenable to \^Ito's stochastic differential calculus in dynamic game settings.
