  We consider discrete time Brownian ratchet models: Parrondo's games. Using
the Fourier transform, we calculate the exact probability distribution
functions for both the capital dependent and history dependent Parrondo's
games. We find that in some cases there are oscillations near the maximum of
the probability distribution, and after many rounds there are two limiting
distributions, for the odd and even total number of rounds of gambling. We
assume that the solution of the aforementioned models can be applied to
portfolio optimization.
