Brownian model of financial markets

The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes. Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes. This model requires an assumption of perfectly divisible assets and a frictionless market (i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models. Consider a financial market consisting of financial assets, where one of these assets, called a bond or money market, is risk free while the remaining assets, called stocks, are risky. A financial market is defined as that satisfies the following: A probability space . A time interval . A -dimensional Brownian process where adapted to the augmented filtration . A measurable risk-free money market rate process . A measurable mean rate of return process . A measurable dividend rate of return process . A measurable volatility process , such that . A measurable, finite variation, singularly continuous stochastic . The initial conditions given by . Let be a probability space, and a be D-dimensional Brownian motion stochastic process, with the natural filtration: If are the measure 0 (i.e. null under measure ) subsets of , then define the augmented filtration: The difference between and is that the latter is both left-continuous, in the sense that: and right-continuous, such that: while the former is only left-continuous. A share of a bond (money market) has price at time with , is continuous, adapted, and has finite variation. Because it has finite variation, it can be decomposed into an absolutely continuous part and a singularly continuous part , by Lebesgue's decomposition theorem.