Stochastic processes

Mathematicians define a stochastic process as a time-ordered family of random variables X_t 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, X_t 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 X_t(ω), the value of the random variable X_t at the point ω ∈ Ω, is the position of the system at time t in the history ω. X_t 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 X_t, 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,...,x_n) is defined to be

p(γ) = ρ(x₀,0)T(x₀,x₁)...T(x_n-1,x_n)

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

Stochastic Processes