I am studying Dirichlet processes at the moment (processes that can be written as a sum of a martingale and a process with zero quadratic variation). I am looking for an example of a Dirichlet process that is not a semimartingale, i.e. a function that has quadratic variation zero, but is not of bounded variation.

Any reading suggestions concerning Dirichlet processes is appreciated very muchly!




1 Answer 1


You can see a paper by R. Chitashvili and M. Mania, On functions transforming Brownian motion into a Dirichlet process. Probability theory and mathematical statistics (Tokyo, 1995), World Sci. Publishing, River Edge, NJ, (1996), pp. 20-27. There is an example you need: If W is a Brownian motion, then the process $|W_t|^a$, where $\dfrac{1}{2} < a < 1$, is a Dirichlet process, but not a semimartingale.


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