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We consider functional equations (Cauchy’s, Abel’s and some other functional equations) and show that finding the general solution of these equations is equivalent to establishing that a space-transformation of a Brownian Motion by a suitable function (or functions) is a martingale.
It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international...
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