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 motion not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules, making one of his first big contributions to science. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist, and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work