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Fragility of arbitrage and bubbles in local martingale diffusion models. (English) Zbl 1310.91068

Non uniformly integrable local martingales, or strict local martingales, provide a natural model of asset bubbles, either if we take the price process to belong to such class under the risk neutral measure or if we replace risk neutral measures with martingale densities, a more natural object. The present paper is based on a core result, Theorem 5, asserting that a \(d\)-dimensional diffusion is close in some due sense to a process which is a true martingale under some equivalent measure. In other words, although a martingale measure for the original price process may not exist, it always exists for a modified price process in the given class. The authors interpret the price approximation quite nicely in terms of transaction costs and also, they claim, in terms of model uncertainty. Given that under a martingale measure neither arbitrage nor bubbles exist they suggest that for this class of price processes, the existence of bubbles is a fragile phenomenon.

MSC:

91B25 Asset pricing models (MSC2010)
91G10 Portfolio theory
60G44 Martingales with continuous parameter
62P05 Applications of statistics to actuarial sciences and financial mathematics
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