  This paper considers the problem of constructing finite-dimensional state
space realizations for stochastic processes that can be represented as the
outputs of a certain type of a causal system driven by a continuous
semimartingale input process. The main assumption is that the output process is
infinitely differentiable, where the notion of differentiability comes from the
functional It\^o calculus introduced by Dupire as a causal (nonanticipative)
counterpart to Malliavin's stochastic calculus of variations. The proposed
approach builds on the ideas of Hijab, who had considered the case of processes
driven by a Brownian motion, and makes contact with the realization theory of
deterministic systems based on formal power series and Chen-Fliess functional
expansions.
