  We present simple new examples of pure-jump strict local martingales. The
examples are constructed as exponentials of self-exciting affine Markov
processes. We characterize the strict local martingale property of these
processes by an integral criterion and by non-uniqueness of an associated
ordinary differential equation. Finally we show an alternative construction for
our examples by an absolutely continuous measure change in the spirit of
(Delbaen and Schachermayer, PTRF 1995).
