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.

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.

Stock prices are modeled as being similar to that of bonds, except with a randomly fluctuating component called its volatility. As a premium for the risk originating from these random fluctuations, the mean rate of return of a stock is higher than that of a bond.

The solution to this is:.

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We say that the portfolio is self-financed if:. To avoid the case of insider trading i.

The standard theory of mathematical finance is restricted to viable financial markets, i. If such opportunities exists, it implies the possibility of making an arbitrarily large risk-free profit.

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Also, according to Girsanov's theorem ,. A complete financial market is one that allows effective hedging of the risk inherent in any investment strategy.

However, in a complete market it is possible to set aside less capital viz. Karatzas, Ioannis; Shreve, Steven E. Methods of mathematical finance. Korn, Ralf; Korn, Elke Option pricing and portfolio optimization: The Review of Economics and Statistics.

Stochastic Modeling

Journal of Economic Theory. From Wikipedia, the free encyclopedia.

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