A Nonuniformly Integrable Martingale Bubble with a Crash
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Date
2019-06
Publication Type
Journal Article
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yes
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Abstract
We investigate a deterministic criterion to determine whether a diffusive local martingale with a single jump (“crash”) is a uniformly integrable martingale. We allow the jump hazard rate and the relative jump size to depend on the state and prove that the process is a uniformly integrable martingale if and only if the relative jump size is bounded away from one. The result helps to classify seemingly explosive behavior in diffusive local martingales compensated by the existence of a jump and provides natural examples of nonuniformly integrable martingales. Local martingales that fail to be uniformly integrable martingales have been used to model financial bubbles in stock prices as deviation from the fundamental value. Our result extends this classification to a comprehensive and relevant model class that explicitly models the financially relevant situation of a crash.
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published
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Journal / series
Volume
10 (2)
Pages / Article No.
615 - 631
Publisher
SIAM
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Subject
uniformly integrable martingales; local martingales; single jump diffusions; explosive diffusion processes; financial bubbles
Organisational unit
03738 - Sornette, Didier (emeritus) / Sornette, Didier (emeritus)