Saturday, January 03, 2009

Stochastic processes

Mathematicians define a stochastic process as a time-ordered family of random variables Xt upon a probability space Ω. In itself, this is not particularly illuminating. However, the basic idea is that Ω is the path-space for the system under consideration. In other words, each point in this probability space, ω ∈ Ω, represents a possible history of the system. (In the case of a stochastic process, these histories will typically be non-differentiable).

By definition, a random variable X is a function on a probability space Ω which possesses a probability distribution over its range of possible values by virtue of the probability measure on the subsets of the probability space Ω. In the case of a stochastic process, Xt is a function on the path-space of the system, which represents the position of the system at time t. ('Position' here can be taken to be spatial position, or any sort of state-defining value, such as the price of a financial stock). Thus Xt(ω), the value of the random variable Xt at the point ω ∈ Ω, is the position of the system at time t in the history ω. Xt takes different values at different points because the different points in Ω correspond to different histories of the system. The probability measure on Ω, the space of histories, determines the probability distribution over the range of each random variable Xt, and thereby determines a probability distribution over position at each time t. Different positions at time t have different probabilities because different histories have different probabilities.

A stochastic process can also be defined by a function G(x,x'; t) which specifies the probability of a transition from x to x' over a time interval t. Given an initial probability distribution ρ(x,0), this determines the probability distribution ρ(x',t) at a future time t.

ρ(x',t) = ∫ G(x,x'; t)ρ(x,0) dx

In fact, given the transition probabilities and an initial probability distribution, a probability measure is determined on the path-space, and the time evolution of the probability distribution over position x is determined. In the special case of a discrete stochastic process, with the transition probability of going from x to y in one time-step denoted as T(x,y), the probability p(γ) of a path γ defined by the sequence of positions (x0,...,xn) is defined to be

p(γ) = ρ(x0,0)T(x0,x1)...T(xn-1,xn)

It is these concepts which tacitly lie behind the mathematicians' definition of a stochastic process.

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