26,618 research outputs found
Single Jump Processes and Strict Local Martingales
Many results in stochastic analysis and mathematical finance involve local
martingales. However, specific examples of strict local martingales are rare
and analytically often rather unhandy. We study local martingales that follow a
given deterministic function up to a random time at which they jump
and stay constant afterwards. The (local) martingale properties of these single
jump local martingales are characterised in terms of conditions on the input
parameters. This classification allows an easy construction of strict local
martingales, uniformly integrable martingales that are not in , etc. As an
application, we provide a construction of a (uniformly integrable) martingale
and a bounded (deterministic) integrand such that the stochastic
integral is a strict local martingale.Comment: 21 pages; forthcoming in 'Stochastic Processes and their
Applications
Analysis of continuous strict local martingales via h-transforms
We study strict local martingales via h-transforms, a method which first
appeared in Delbaen-Schachermayer. We show that strict local martingales arise
whenever there is a consistent family of change of measures where the two
measures are not equivalent to one another. Several old and new strict local
martingales are identified. We treat examples of diffusions with various
boundary behavior, size-bias sampling of diffusion paths, and non-colliding
diffusions. A multidimensional generalization to conformal strict local
martingales is achieved through Kelvin transform. As curious examples of
non-standard behavior, we show by various examples that strict local
martingales do not behave uniformly when the function (x-K)^+ is applied to
them. Implications to the recent literature on financial bubbles are discussed.Comment: Significantly revised version. 28 page
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