Martingale property

We do not describe or prove this property here, but the martingale property is used to derive Equation (2.13) the price of an asset at time t ... 

A key element of the description of a stochastic process is a specification of the level of informatimi oti the behaviour of prices that is available to an observer at each point in time. As with the martingale property, a calculation of the expected future values of a price process requires information on current prices. Generally, fmancial valuation models require data on both the current and the historical security prices, but investors are only able to deal on the basis of current known information and do not have access to future information. In a stochastic model, this concept is captured via the process known as filtration. 

Property (1) ensures that the paths are continuous in t and T. The properties (2) and (3) imply that we have a martingale for each time t together with unit variance. Finally, property (4) ensures that we work with a deterministic function c T,V) = dW(t, T)dW(t,V) fulfilling the attributes of a correlation function. 

Brownian motion is very similar to a Wiener process, which is why it is common to see the terms used interchangeably. Note that the properties of a Wiener process require that it be a martingale, while no such constraint is required for a Brownian process. A mathematical property known as the Levy s theorem allows us to consider any Wiener process Z, with respect to an information set Ft as a Brownian motion Z, with respect to the same information set. 

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