Brownian model of financial markets as: Note that formula_21 and formula_139 are absolutely continuous with respect to each other, i.e. they are equivalent. Also, according to Girsanov's theorem, is a formula_6-dimensional Brownian motion process on the filtration formula_146 with respect to formula_139. A complete financial market is one that allows effective hedging of the risk inherent in any investment strategy. Let formula_89 be a standard financial market, and formula_149 be an formula_140-measurable random variable, such that: The market formula_89 is said to be "complete" if every such formula_149 is "financeable", i.e. if there is an formula_155-financed portfolio process formula_156, such that its associated wealth process
Brownian motor bias the motion of a particle (like an ion or polypeptide); the result would essentially be diffusion of a particle whose net motion is strongly biased in one direction. The translocation of the particle would only be loosely coupled to hydrolysis of ATP. The dynamics and activity of Brownian motors are current topics of study in theoretical and experimental biophysics. Brownian motors are sometimes modeled using the Fokker-Planck equation or with Monte Carlo methods. Many researchers are presently engaged in understanding how molecular-scale motors operate in environments with non-negligible thermal noise. The thermodynamics of such motors is constrained by the