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Actually, the it does not satisfy sufficient conditions for existence and uniqueness when drift and diffusion terms has to be of at most linear growth (Lipschitz everywhere), so the solution of this SDE may not exist at all.
I see. Since mean (in whatever framework) is natural to define as a minimizer of a measure of spread of data/points/... it justifies the fact that the minimal such spread is called variance, as Kjetil wrote already. So the std.dev would indeed be natural to defined as a square root of the minimals sum of distances, however maybe you could say why are you interested in std.dev related to the Frechet mean? Perhaps, concentration theorem for random elements taking values in metric spaces?
@MichaelGreinecker: thanks, I just followed the terminology of the Bertsekas and Shreve's book. Have you seen such a statement as in OP somewhere? I haven't find it in BS.
By I also would like to find the corresponding standard deviation do you mean, that you are looking for a natural way of defining a std.dev. for the Frechet mean?
Btw, one usually assumes that strategies in controlled stochastic processes are allowed to be randomized (which beings convexity e.g.). In such a case there are uncounbly many strategies. If you allow for non-randomized only, just read the definition of the sequential decision policy, and you'll get the number of them immediately, as those are functions from the histories to the control space.
