  The understanding of adaptive algorithms for SDEs is an open area where many
issues related to both convergence and stability (long time behaviour) of
algorithms are unresolved. This paper considers a very simple adaptive
algorithm, based on controlling only the drift component of a time-step. Both
convergence and stability are studied.
  The primary issue in the convergence analysis is that the adaptive method
does not necessarily drive the time-steps to zero with the user-input
tolerance. This possibility must be quantified and shown to have low
probability.
  The primary issue in the stability analysis is ergodicity. It is assumed that
the noise is non-degenerate, so that the diffusion process is elliptic, and the
drift is assumed to satisfy a coercivity condition. The SDE is then
geometrically ergodic (converges to statistical equilibrium exponentially
quickly). If the drift is not linearly bounded then explicit fixed time-step
approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In
this work, it is shown that the simple adaptive time-stepping strategy cures
this problem. In addition to proving ergodicity, an exponential moment bound is
also proved, generalizing a result known to hold for the SDE itself.
