TL;DR
This paper introduces a novel continuous martingale called fake exponential Brownian motion that shares the same marginal distributions as classical exponential Brownian motion but differs in its path properties.
Contribution
It constructs the first known fake exponential Brownian motion, extending previous work on fake Brownian motions to exponential diffusions and other types.
Findings
Successfully constructed a fake exponential Brownian motion
Extended the concept of fake Brownian motions to exponential diffusions
Demonstrated the existence of continuous martingales with identical marginals but different paths
Abstract
We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.
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Taxonomy
TopicsStochastic processes and financial applications · Financial Risk and Volatility Modeling · Stochastic processes and statistical mechanics