  A trust-region algorithm is presented for finding approximate minimizers of
smooth unconstrained functions whose values and derivatives are subject to
random noise. It is shown that, under suitable probabilistic assumptions, the
new method finds (in expectation) an $\epsilon$-approximate minimizer of
arbitrary order $ q \geq 1$ in at most $\mathcal{O}(\epsilon^{-(q+1)})$ inexact
evaluations of the function and its derivatives, providing the first such
result for general optimality orders. The impact of intrinsic noise limiting
the validity of the assumptions is also discussed and it is shown that
difficulties are unlikely to occur in the first-order version of the algorithm
for sufficiently large gradients. Conversely, should these assumptions fail for
specific realizations, then "degraded" optimality guarantees are shown to hold
when failure occurs. These conclusions are then discussed and illustrated in
the context of subsampling methods for finite-sum optimization.
