100 A. A. Pogorui and Ram´ on M. Rodr´ ıguez-Dagnino Evolution process as an alternative to diffusion process and Black-Scholes Formula A. A. POGORUI 1 and Ram´on M. RODR ´ IGUEZ-DAGNINO 2 1 Tecnol´ ogico de Monterrey (ITESM), M´ exico E-mail: pogorui@itesm.mx 2 Tecnol´ ogico de Monterrey (ITESM). Sucursal de correos “J” C.P. 64849, Monterrey, N.L., M´ exico E-mail: rmrodrig@itesm.mx Received for ROSE Month X, 2008 Abstract—In this paper, we study the one-dimensional transport process in the case of disbalance. In the hydrodynamic limit, this process approximates the diffusion process on a line. By using this property, we propose to apply transport processes instead of diffusion processes in some economical models, particularly in Black-Scholes formula. This application manages to avoid some drawbacks of diffusion processes. Key words and phrases: Random evolutions; Black-Scholes formula; diffusion process; Brown- ian motion; transport process 1. INTRODUCTION Wiener process is the well-known mathematical model used to study random movement of a particle, called the Brownian motion. It is also prominent in the mathematical the- ory of finance, in particular the Black-Scholes option pricing model. However, in some recent papers, for instance [3], considered alternative models of the Brownian motion based on the telegrapher process. This model manages to avoid such drawbacks of the Wiener process as a fractal trajectory of the process and others. Since the diffusion process used in the Black-Scholes formula, we propose to use Markovian evolutions in the case of disbalance instead of the diffusion process. Unlike diffusion process trajecto- ries, Markovian evolutions have trajectories which are differentiable almost everywhere. Because of this, we think that the Markovian evolution model is more accurate in both the particle physics and economical theory applications. 2. TRANSPORT PROCESS AS AN ALTERNATIVE TO A DIFFUSION MODEL Let {ξ (t),t ≥ 0} be a Markov process on the phase space {0, 1} with the infinitesimal matrix Q = λ   −1 1 1 −1   .