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 ratchet for example lifting a weight ("m") against gravity. The energy necessary to do this work apparently would come from the heat bath, without any heat gradient. Were such a machine to work successfully, its operation would violate the second law of thermodynamics, one form of which states: "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." Although at first sight the Brownian ratchet seems to extract useful work from Brownian motion, Feynman demonstrated that if the entire device is at the same temperature, the