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COMMON STOCHASTIC PROCESSES

The most basic process is the Wiener process. Other processes of interest include the generalized Wiener process and the Ito process. The stock price process is a special case of the Ito process. Other processes, such as interest rate processes and jump processes are relatively straightforward extensions. In what follows, the process is called S and the innovation term is Az( = £a(t)^At. The differences between processes result from the definitions of the coefficients, a and b, applied, respectively, to the drift and the diffusion terms. Some basic processes are described to illustrate simple cases. But the most common processes include the Ito process which applies to asset returns, the mean-reverting process which applies to interest rates, and the jump process because it applies to rare events.

The Wiener Process

The Wiener process, also called Brownian motion, is the random process followed by a variable z(t), such that the random change is Az(t) — z(t + At) - z(t) — e, where £ follows a normal distribution 0(0,1) or 0(0, a). Its drift is zero. Its variance is constant per unit of time. Because all successive random changes are independent and follow the same distribution, the process is said to be "identically independently distributed, or "i.i.d." The distribution is stationary when time passes, meaning that it is identical when starting from any time point.

Dividing the horizon t into n equal sub-periods At — Tin, and considering as time interval [0, t], we observe that cumulating the Az(t) from 0 to T, we have a summation of n random normal variables, starting from some value z(0) known as of time point 0.

The expectation of each increment of z is 0. The expectation of the cumulated increments is the summation of the expectations of each change between 0 and T, each of them being 0, and is equal to zero: E[z(/)] 0. The Brownian motion follows an erratic path around the initial value.

At each time point, we move from the current value to the next "abruptly," always starting from the previous point. The path is continuous but not smooth. It can be obtained by generating independent standard normal variables. Cumulating the random standard normal changes over some period generates the time path of the process. Because the process has no drift term, the time path varies randomly around 0. Other processes combine a drift plus a function of the basic Wiener process.

 
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