68 research outputs found
Simple examples of pure-jump strict local martingales
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)
Forward-Invariance and Wong-Zakai Approximation for Stochastic Moving Boundary Problems
We discuss a class of stochastic second-order PDEs in one space-dimension
with an inner boundary moving according to a possibly non-linear, Stefan-type
condition. We show that proper separation of phases is attained, i.e., the
solution remains negative on one side and positive on the other side of the
moving interface, when started with the appropriate initial conditions. To
extend results from deterministic settings to the stochastic case, we establish
a Wong-Zakai type approximation. After a coordinate transformation the problems
are reformulated and analysed in terms of stochastic evolution equations on
domains of fractional powers of linear operators.Comment: 46 page
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