  We consider a nonlinear differential equation under the combined influence of
small state-dependent Brownian perturbations of size $\varepsilon$, and fast
periodic sampling with period $\delta$; $0<\varepsilon, \delta \ll 1$. Thus,
state samples (measurements) are taken every $\delta$ time units, and the
instantaneous rate of change of the state depends on its current value as well
as its most recent sample. We show that the resulting stochastic process
indexed by $\varepsilon,\delta$, can be approximated, as $\varepsilon,\delta
\searrow 0$, by an ordinary differential equation (ODE) with vector field
obtained by replacing the most recent sample by the current value of the state.
We next analyze the fluctuations of the stochastic process about the limiting
ODE. Our main result asserts that, for the case when $\delta \searrow 0$ at the
same rate as, or faster than, $\varepsilon \searrow 0$, the rescaled
fluctuations can be approximated in a suitable strong (pathwise) sense by a
limiting stochastic differential equation (SDE). This SDE varies depending on
the exact rates at which $\varepsilon,\delta \searrow 0$. The key contribution
here involves computing the effective drift term capturing the interplay
between noise and sampling in the limiting SDE. The results essentially provide
a first-order perturbation expansion, together with error estimates, for the
stochastic process of interest. Connections with the performance analysis of
feedback control systems with sampling are discussed and illustrated
numerically through a simple example.
