  We here adapt an extended version of the adaptive cubic regularisation method
with dynamic inexact Hessian information for nonconvex optimisation in [3] to
the stochastic optimisation setting. While exact function evaluations are still
considered, this novel variant inherits the innovative use of adaptive accuracy
requirements for Hessian approximations introduced in [3] and additionally
employs inexact computations of the gradient. Without restrictions on the
variance of the errors, we assume that these approximations are available
within a sufficiently large, but fixed, probability and we extend, in the
spirit of [18], the deterministic analysis of the framework to its stochastic
counterpart, showing that the expected number of iterations to reach a
first-order stationary point matches the well known worst-case optimal
complexity. This is, in fact, still given by O(epsilon^(-3/2)), with respect to
the first-order epsilon tolerance. Finally, numerical tests on nonconvex
finite-sum minimisation confirm that using inexact first and second-order
derivatives can be beneficial in terms of the computational savings.
