Probability distribution of financial returns in a model of multiplicative Brownian motion with stochastic diffusion coefficient

It is well-known that the mathematical theory of Brownian motion was first developed in the Ph.~D.\ thesis of Louis Bachelier for the French stock market before Einstein [1]. In Ref.\ [2] we studied the so-called Heston model, where the stock-price dynamics is governed by multiplicative Brownian motion with stochastic diffusion coefficient. We solved the corresponding Fokker-Planck equation exactly and found an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula interpolates between the exponential (tent-shaped) distribution for short time lags and the Gaussian (parabolic) distribution for long time lags. The theoretical formula agrees very well with the actual stock-market data ranging from the Dow-Jones index [2] to individual companies [3], such as Microsoft, Intel, etc. \\[4pt] [1] Louis Bachelier, ``Th\'eorie de la sp\'eculation,'' Annales Scientifiques de l'\'Ecole Normale Sup\'erieure, III-17:21-86 (1900).\\[0pt] [2] A. A. Dragulescu and V. M. Yakovenko, ``Probability distribution of returns in the Heston model with stochastic volatility,'' Quantitative Finance {\bf 2}, 443--453 (2002); Erratum {\bf 3}, C15 (2003). [cond-mat/0203046] \\[0pt] [3] A. C. Silva, R. E. Prange, and V. M. Yakovenko, ``Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact,'' Physica A {\bf 344}, 227--235 (2004). [cond-mat/0401225]