title: Exact probability distribution functions for Parrondo's games.
abstract: We study the discrete time dynamics of Brownian ratchet models and 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. In certain cases we find strong oscillations near the maximum of the probability distribution with two limiting distributions for odd and even number of rounds of the game. Indications of such oscillations first appeared in the analysis of real financial data, but now we have found this phenomenon in model systems and a theoretical understanding of the phenomenon. The method of our work can be applied to Brownian ratchets, molecular motors, and portfolio optimization.
authors.affiliation: Max Planck Institute for the Physics of Complex Systems, Dresden, Germany.
authors.forename: Rubina
authors.initials: R
authors.lastname: Zadourian
authors.affiliation: Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.|A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers St., Yerevan 375036, Armenia.
authors.forename: David B
authors.initials: DB
authors.lastname: Saakian
authors.affiliation: Wuppertal University, D-42119 Wuppertal, Germany.
authors.forename: Andreas
authors.initials: A
authors.lastname: Klümper