Stochastic Processes

What are Stochastic Processes?

Stochastic processes, also called random processes, are a way to describe how random systems change with respect to time. This can be viewed as a collection of random variable, one for each time. This can be used within for example the financial industry to describe the value of stocks, funds, commodities or other financial goods. This can be viewed as an extension of random variables, since the concept of random variables did not allow for the mean and variance to change with time, rather, they were fixed.

Example of a Stochastic Process

Start by imagining a Gaussian random variable with mean $latex \mu $ and variance $latex \sigma$. Denote this variable by $latex X$, and then we have

$latex X \sim N( \mu, \sigma) $

Now assume that we have a random variable, with mean and variance changing with time. Let’s denote this by $latex X_t$, we then have

$latex X \sim N( \mu_t, \sigma_t) $

We have now created stochastic process. For example, assume that the variance $latex \sigma_t$ decreases with time. We can then plot an example with the help of Matlab, the result can be seen in the figure below

This is what is called on realization of the process. The process had a expected mean and variance at each time, and when we ran a simulation, this time, these values were chosen. However if we ran the process again, new values would appear. At each time, in this case the process is described by

$latex X \sim N( \mu_t, \sigma_t) $

Where $latex \sigma_t = t$. Hence the variance is equal to the time! Here, the mean of the process is zero. This is a pretty simple example, however it shows the characteristics of a stochastic┬áprocess. Let’s consider a case where the mean is varying with time

As can be seen in the figure, the mean in of the process is changing slowly with time. In this gave, the stochastic process is specified by

$latex X_t \sim N(\mu_t, \sigma_t) $

Where $latex \mu_t = sin(0.005 t)$ and $latex \sigma_t = 15$. Notice here that the variance is held constant, and only the mean is changing. These two examples show how the stochastic process can change over time. In simple terms, a random process is like a time-depended random variable, which can take on a unique value at each time.

Description of Stochastic Processes

To conclude, we can say that a Stochastic Process is kind of like a time-dependent random variable, it’s distribution changing with time.

Correlation Stochastic Processes

Correlation is an important subject within Stochastic Processes. It simply described, between two times, how correlated the variables are. In the completely non-correlated case, the random process is completely independent from itself at different times. This is sometimes refereed to as a white process. Another useful concept is the correlation function, it is defined as follows

C(t,s) = E[X(t) * X(s)]

Where E[.] is the expected value operator, and X(t) is the stochastic process at time t, and X(s) is the stochastic process at time s. Hence this metric is an indicator of how correlated the random process is at two times.