STOCHASTIC PROCESSES

So far, we have considered a single random variable. However, more often than not, we have to deal with multiple random variables. There are two cases in which we have to do so:

• We might observe different random variables, say, X_i, i = 1,…,n, at the same time. In such a case, we speak of cross-sectional data. As practical examples, think of the return of several financial assets over an investing horizon; alternatively, consider the demand for several items, which could be complementary or substitute goods.
• We might observe a single random variable, over multiple time periods, say, X_t, t = 1,…, T. In such a case, we speak of longitudinal data. For instance, we may observe the weekly return of a financial asset over a timespan of a few months, or daily demand for an item.

In practice, we may also have the two views in combination, i.e., multiple variables observed over a timespan of several periods. In such a case, we speak of panel data. Given the scope of this book, we will not consider panel data. If we observe cross-sectional data, and the corresponding random variables are independent, then we may just study one variable at a time, and that’s it. However, this is just a very special and simple case. We need more sophisticated tools to deal with dependence.

DEFINITION 7.10 (Stochastic process) A time-indexed collection of random variables is called a stochastic process. If time is discretized, we have a discrete-time process:

If time is continuous, we have a continuous-time (also known as continuous-parameter) process:

In general, one could consider a collection of random variables depending on space, rather than time. Then, we could speak of discrete- or continuous-parameter processes. In the applications we consider in the book, the parameter will always be time. In some loose sense, the stochastic process is a generalization of deterministic functions of time, in that for any value of t it yields a random variable (which is a function itself) rather than a number. If we observe a sequential realization of the random variables over time, we get a sample path of the process. In this introductory book, we will essentially deal with discrete-time processes, but it is a good idea to consider at least a simple example of a relevant continuous-time process.

Example 7.15 (Poisson process) The Poisson process is an example of a counting process, i.e., a stochastic process N(t) counting the number of events that occurred in the time interval [0, t]. Such a process starts from zero and has unit increments over time. We may use such a process to model order or customer arrivals. The Poisson process is obtained when we make specific assumptions about the interarrival times of customers. Let X_k, k = 1,2,3,4,…, be the interarrival time between customer k − 1 and customer k; by convention, X₁ is the arrival time of the first customer after the start time t = 0. We obtain a Poisson process if we assume that variables X_k are mutually independent and all exponentially distributed with parameter λ, which is in this case the arrival rate, i.e., the average number of customers arriving per unit time. A sample path is illustrated in Fig. 7.23; we see that the process “jumps” whenever a customer arrives, so that sample paths are piece wise constant.

Fig. 7.23 Sample path of the Poisson process.

We have already mentioned the link between Poisson and exponential distributions and the Poisson process. If we consider a time interval [t₁,t₂], with t₁ < t₂, then the number of customers who arrived in this interval, i.e., N(t₂) − N(t₁), has Poisson distribution with parameter λ(t₂ − t₁). Furthermore, if we consider another time interval [t₃, t₄], where t₃ < t₄, which is disjoint from the previous one, i.e., (t₂ < t₃), then the random variables N(t₂) − N(t₁) and N(t₄) − N(t₃) are independent. We say that the Poisson process has stationary and independent increments.

The Poisson process is a useful model for representing the random arrival of customers who have no mutual relationships at all. This is a consequence of the lack of memory of the exponential distribution.

The model can be generalized to better fit reality. For instance, if we observe the arrival process of customers at a big retail store, we easily observe variations in the arrival rate. If we introduce a time-varying rate λ(t), we get the so-called inhomogeneous Poisson process. Furthermore, if we consider not only customer (or order) arrivals, but the demanded quantities as well, we see the opportunity of associating another random variable, the quantity per order, with each customer. The cumulative quantity demanded D(t) in the time interval [0, t] is another stochastic process, which is known as a compound Poisson process. The sample paths of this process would be qualitatively similar to those in Fig. 7.23, but the size of the jumps would be random. This is a possible model for demand, when sale volumes are not large enough to warrant use of a normal distribution.

Naive thinking would draw us to the conclusion that, in order to characterize a stochastic process, we should give the distribution of X_t for all the relevant time instants t. This is what we call the marginal distribution. The following example shows that marginal distributions do not tell the whole story.

Fig. 7.24 Sample paths of the stochastic process .

Example 7.16 (A Gaussian process) A common class of stochastic processes consists of sequences of random variables whose marginal distribution is normal, which is why they are termed Gaussian processes. To be precise, we should say that a Gaussian process requires that the random variables X_tl, X_t2,…, X_tm have a jointly normal distribution for any possible choice of time instants t₁, t₂, …, t_m, but for the sake of simplicity we will put in the same bag any process for which the marginal distribution of X_t is normal. However, it is important to realize that in doing so we are considering processes that may be very different in nature. Consider the stochastic process

where  is standard normal variable. In our loose sense, we may say that this is a Gaussian process, since X_t is normal with expected value 0 and variance t². However, it is a somewhat degenerate process, since uncertainty is linked to the realization of a single random variable. If we know the value of X_t for a single time instant, then we can figure out the whole sample path. Figure 7.24 illustrates this point by showing a few sample paths of this process. A quite different process is obtained if all variables X_t are normal with parameters μ and σ² and mutually independent. Figure 7.25 shows a sample path of the process , where . However, the marginal distributions of the individual random variables X_t are exactly the same for both processes.

Fig. 7.25 Sample path of process , where .

The example shows that we really need some way to characterize the interdependence of jointly distributed random variables, which is the topic. Discrete-time stochastic processes will be discussed further on time series models.