Stochastic Processes¶
Definitions¶
The idea now is to study sequences of random variables. A stochastic process is similar to a random variable, with the difference that it also depends on time. Adding the time dimension adds notation, but it does not change any of the main ideas. For convenience we will first go over the definition of a random variable.
Definition 26.1: Random variable
Let \(\left(\Omega,\mathcal{A},P\right)\) be a probability space and \(x:\Omega\to\mathbb{R}\) a real valued function. \(x\) is a random variable if and only if \(x\) is measurable, that is, if and only if \(x^{-1}\left(B\right)\in\mathcal{A}\) for all \(B\in\mathcal{B}\), where \(\mathcal{B}\) is the Borel \(\sigma\)-algebra on \(\mathbb{R}\). We further establish the same notation:
- An outcome is an element \(\omega\in\Omega\).
- An event is a measurable subset of \(\Omega\): \(A\in\mathcal{A}\).
- The real number \(x\left(\omega\right)\) is a realization of the random variable.
- The probability measure for \(x\) is then: \(\mu\left(B\right)=P\left(x^{-1}\left(B\right)\right)=P\left(\left\lbrace \omega\in\Omega|x\left(\omega\right)\in B\right\rbrace \right)\), for \(B\in\mathcal{B}\).
- The distribution function for \(f\) is: \(G\left(b\right)=\mu\left(\left(-\infty,b\right]\right)\), for \(b\in\text{\mathbb{R}}\).
Now we can work on adding the time dimension to the definition of a random variable. In general time can be discrete or continuous, but in what follows we will assume that time is continuous starting at 0 and going on forever, so \(t\in\left[0,\infty\right)\). Intuitively a stochastic process is formed by function \(x:\left[0,\infty\right)\times\Omega\to\mathbb{R}\) that gives a realization for every outcome and time. At every point in time the random variable takes a variable, the sequence of those values forms the realization (path) of the stochastic process.
The question is on how to measure the possible outcomes of the random variable through time. We need a way of determining where the random variable is at a certain point in time, and where it has been, but that does not provide information about the value of future realizations. This is achieved using a filtration.
Definition 26.2: Filtration
Let \(\mathcal{A}\) be a \(\sigma\)-algebra. The set \(\mathbb{A}=\left\lbrace \mathcal{A}_{t}|t\geq0\right\rbrace\) is a filtration if \(\mathcal{A}_{t}\subseteq\mathcal{A}\) and \(\mathcal{A}_{s}\subseteq\mathcal{A}_{t}\) for all \(t\geq0\) and \(s\leq t\). \(\mathcal{A}_{t}\) is the set of events known at time \(t\).
Now we can define a stochastic process as a function that is measurable in a filtered space.
Definition 26.3: Stochastic Process
Let \(\left(\Omega,\mathbb{A},P\right)\) be a filtered probability space with a time index \(t\in\mathbb{R}_{+}\), and let \(\mathcal{B}_{+}\) be the Borel sets of \(\mathbb{R}_{+}\). A stochastic process is a function \(x:\left[0,\infty\right)\times\Omega\to\mathbb{R}\) that is measurable with respect to \(\mathcal{B}_{+}\times\mathcal{A}\) (that is, \(x\) is jointly measurable in \(\left(t,\omega\right)\)). Moreover:
- For all \(t\in\mathbb{R}_{+}\) and \(\omega\in\Omega\), \(x\left(t,\omega\right)\) is measurable with respect to \(\mathcal{A}_{t}\), where \(\mathcal{A}_{t}\) is in the filtration \(\mathbb{A}\).
- For all \(t\in\mathbb{R}_{+}\), \(x\left(t,\cdot\right):\Omega\to\mathbb{R}\) is an ordinary random variable on the probability space \(\left(\Omega,\mathcal{A}_{t},P_{t}\right)\).
- For all \(\omega\in\Omega\), \(x\left(\cdot,\omega\right):\mathbb{R}_{+}\to\mathbb{R}\) is a Borel measurable function. This is called the sample path of \(x\).
Discrete time examples¶
It is not hard to come up with examples of discrete time stochastic processes. They are often used to model the behavior of many stationary economic variables by means of ARMA(p,q) representations, as well non-stationary variables usually related to random walks.
To fix ideas we start with the simple example of a (fair) coin toss. There are two possible outcomes, so \(\Omega=\left\lbrace H,T\right\rbrace\), when tossing the coin is always possible to know which outcome occurred, and whether or not the coin was tossed, this gives: \(\mathcal{A}=\left\lbrace \left\lbrace H\right\rbrace ,\left\lbrace T\right\rbrace ,\emptyset,\Omega\right\rbrace\). Finally the probability distribution \(P\) assigns values to sets in the \(\sigma\)-algebra \(\mathcal{A}\):
Now we can define a random variable \(\epsilon:\Omega\to\mathbb{R}\) as: \(\epsilon\left(H\right)=1\) and \(\epsilon\left(T\right)=-1\). \(\epsilon\) is a random variable with respect to the probability space \(\left(\Omega,\mathcal{A},P\right)\). As will be the case almost always we can dispense of the outcome space \(\Omega\) for most applications and just refer to the random variable and the probability distribution induced over its values. In this way we have: \(\epsilon\in\left\lbrace -1,1\right\rbrace\) with \(\Pr\left(\epsilon=1\right)=\Pr\left(\epsilon=-1\right)=\frac{1}{2}\).
Furthermore we can extend this example to define the stochastic process that comes up from the repeated coin toss. In this case time is discrete and finite \(t\in\left\lbrace 1,2,3\right\rbrace\) and at each time a coin is tossed, then the random variable variable \(\epsilon_{t}\) is defined as the value of \(\epsilon\) given the outcome of the \(t^{th}\) coin toss. The sequence \(\left\lbrace \epsilon_{t}\right\rbrace _{t=1}^{3}\) is a stochastic process with respect to the filtered probability space \(\left(\Omega,\mathbb{A},P\right)\), where:
The filtration is established taking into account that at each point in time only the outcome of current and past tosses is known:
So, in the first \(\sigma\)-algebra all outcomes for which the first toss comes up heads are indistinguishable from each other, in the second \(\sigma\)-algebra one can distinguish between outcomes that have the sequence \(\left\lbrace H,T\right\rbrace\) and \(\left\lbrace H,H\right\rbrace\), but no information is given about the outcome of third toss. This same ideas apply if time goes on forever, so we can define our stochastic process over \(t\in\mathbb{N}\).
In the previous example the stochastic process obtained satisfies the property of being iid (identically and independently distributed). The values of the stochastic process at each point in time are independent from its previous values, and they all have the same probabilities of occurring.
We now use our stochastic process \(\left\lbrace \epsilon_{t}\right\rbrace\) to define a random walk. Random walks are particularly useful to understand the behavior of continuous time stochastic processes. As we will see the building block of most of them is the continuous time approximation of a random walk.
Example 26.1: Random Walk Process
Consider a stochastic process \(x\). Denote by \(x_{t}\) the value of \(x\) at time \(t\), and fix the initial value \(x_{0}\). \(x_{t}\) is assumed to evolve according to:
\(\epsilon_{t}\) is a random variable that can take two values \(\left\lbrace -1,1\right\rbrace\), and its probability distribution is independent of time, so that: \(\Pr\left(\epsilon_{t}=1\right)=\Pr\left(\epsilon_{t}=-1\right)=\frac{1}{2}\).
Given the starting value \(x_{0}\) the variable \(x_{t}\) can only take on discrete values. For instance, for \(x_{0}=0\) and \(t\) odd they are \(\left\lbrace -t,\ldots,-1,0,1,\ldots,t\right\rbrace\), and for \(t\) even they are \(\left\lbrace -t,\ldots,-2,0,2,\ldots,t\right\rbrace\). These values tell you which paths of the process cam be known at time \(t\).
Finally, this process has no drift. Given an initial value \(x_{0}\) the expected value of \(x_{t}\) for any \(t\) is \(x_{0}\left(E\left[x_{t}\right]=x_{0}\right)\), this follows from the expected value of each change being \(E\left[x_{t}-x_{t-1}\right]=E\left[\epsilon_{t}\right]=0\).
This process can be generalized in many ways. The most useful one for our purposes is to allow for drift, which can be done by changing the probabilities of the random variable \(\epsilon_{t}\), letting \(\Pr\left(\epsilon_{t}=1\right)=p\) and \(\Pr\left(\epsilon_{t}=-1\right)=1-p\) achieves the desired result. If \(p>\frac{1}{2}\) the process will have positive drift.
Brownian motion (Wiener processes)¶
A Brownian motion, or Weiner process, is a continuous time stochastic process \(\left(W\left(t\right)\right)\) that satisfies three properties:
- \(W\left(t\right)\) has continuous sample paths.
- \(W\left(t\right)\) has stationary independent increments.
- Increments of \(W\left(t\right)\) over a finite interval of time are normally distributed with variance that increases linearly in time.
The first property implies that a Brownian motion has no jumps, so as the time interval goes to zero the change in the process must also go to zero. The second and third properties imply that the change in \(W\left(t\right)\) over some interval of length \(\Delta t\) must satisfy:
which we write as \(dW=\epsilon_{t}\sqrt{dt}\) as \(\Delta t\to0\). This implies that:
Moreover we assume that \(\epsilon_{t}\) is serially uncorrelated, i.e., \(E\left[\epsilon_{t}\epsilon_{s}\right]=0\) for \(t\neq s\), so the values of \(dW\) for any two different time intervals are independent.
Its easy to note the relation between the Brownian motion and the random walk processes. In discrete time we had \(\Delta x_{t}=x_{t}-x_{t-1}=\epsilon_{t}\Delta t\), where \(\Delta t=1\). We will use this fact when approximating Brownian motions using random walks as \(\Delta t\to0\).
To see that this representation implies the third property consider a time interval that starts at \(t\) and ends at \(T\), and divide into \(n\) intervals of length \(\Delta t=\frac{T}{n}\). Then we have:
What we want to show is that \(W\left(t+T\right)-W\left(t\right)\sim N\left(0,t\right)\). To prove this we can use the Central Limit Theorem:
Theorem 26.1: Central Limit Theorem
If \(\left\lbrace \epsilon_{1},\epsilon_{2},\epsilon_{3},\ldots\right\rbrace\) are iid (but not necessarily normal) with \(E\left[\epsilon_{i}\right]=\mu<\infty\) and \(V\left[\epsilon_{i}\right]=\sigma^{2}<\infty\), then \(Z_{n}=\frac{1}{\sqrt{n}}\frac{\underset{i=1}{\overset{n}{\sum}}\epsilon_{i}-n\mu}{\sigma}\to N\left(0,1\right)\) as \(n\to\infty\).
Note that \(\epsilon_{i}\) already satisfies being iid and \(E\left[\epsilon_{i}\right]=0\) and \(V\left[\epsilon_{i}\right]=1\), so \(Z_{n}=\sqrt{n}\underset{i=1}{\overset{n}{\sum}}\epsilon_{i}\). Then we can write:
By the CLT this converges to a \(N\left(0,T\right)\) as \(n\to\infty\).
A Brownian motion can be generalized to have drift \(\mu\) and variance \(\sigma^{2}\). This is done by adjusting the way the increments of the stochastic process work:
In this case the increments are given by a non-stochastic component \(\mu dt\), which indicates that the process will drift by \(\mu\) per unit of time deterministically if there are no shocks, and by a stochastic component \(\sigma dW\), where \(\sigma\) is scaling the variance of the increments of the Weiner process \(W\). This process satisfies:
Random walk approximation of a Brownian motion¶
As mentioned above we can use the similarities between the increments of a Brownian motion and the increments of a random walk to approximate continuous time processes using discrete time ones. This is important because of two reasons: it helps explain the mechanics of the continuous time model, and it provides an algorithm for simulation in the computer.
Our objective is to approximate the a Brownian motion with drift:
We will approximate with a discrete time process \(y\) whose increments are \(h\) with probability \(p\) and \(-h\) with probability \(1-p\). This gives:
In order to get the approximation we need to choose values for \(h\), \(p\) and \(\Delta t\) so that:
Solving for \(p\) we get:
The roots of these equation are:
where the approximation follows if \(\Delta t\) is small enough relative to \(\frac{\sigma^{2}}{\mu^{2}}\), because we are taking \(\Delta t\) close to zero this assumption is satisfied. We further choose only the "+" root because that way \(p\geq\frac{1}{2}\) when \(\mu\geq0\).
Now we can find a value for \(h\):
As before we can disregard the term \(\left(\frac{\mu}{\sigma}\right)^{2}\Delta t\) as long as \(\Delta t\) is small enough relative to \(\frac{\sigma^{2}}{\mu^{2}}\).
As an exercise can verify that the first equation also holds:
In order to simulate a Brownian motion with parameters \(\left(\mu,\sigma\right)\) we can do as follows:
- Set a \(\Delta t\) small relative to \(\frac{\sigma^{2}}{\mu^{2}}\).
- Set \(p=\frac{1}{2}\left(1+\frac{\mu}{\sigma}\sqrt{\Delta t}\right)\) and \(h=\sigma\sqrt{\Delta t}\).
- Simulate the increments of \(x\) by drawing realization of a random variable \(\epsilon_{t}\) that takes value \(h\) with probability \(p\) and \(-h\) with probability \(1-p\).
Ito processes¶
Ito processes are the generalization of Brownian motions. Their drift and variance is allowed to depend on the level of the process and the time:
where the functions \(\mu\) and \(\sigma\) give the value of the mean and standard deviations of the increments of the process \(x\):
For future reference note that an Ito process can also be represented as:
where the last term is a stochastic integral. Stochastic integrals play an important role in the theory of stochastic processes, for now it suffices to state the following result.
Proposition 26.1
Let \(x\left(t\right)\) be an integrable function, then \(E\left[\int_{0}^{t}x\left(s\right)dW\left(s\right)\right]=0\).
This proposition states that the expected value of a stochastic integral is identically zero. The derivation of the result, along with other properties can be found in Stokey (2009, Sec. 3.2).
Two Ito process are of particular importance. They are presented in the examples below.
Example 26.2: Geometric Brownian notion
A Geometric Brownian motion is an Ito process with \(\mu\left(x,t\right)=\mu x\) and \(\sigma\left(x,t\right)=\sigma x\), so:
A geometric Brownian motion can be thought of as a Brownian motion where the properties apply to percentage increments instead of increments:
So the percentage increment, \(\frac{dx}{x}\), are normally distributed with mean \(\mu\Delta t\) and variance \(\sigma^{2}\Delta t\).
Example 26.3: Ornstein-Uhlenbeck process
Unlike the previous processes an OU process is mean reverting, similar to an AR(1) process in discrete time. An OU process is an Ito process with \(\mu\left(x,t\right)=\mu\left(\overline{x}-x\right)\) and \(\sigma\left(x,t\right)=\sigma\). If \(x>\overline{x}\) then the process drifts down, and if \(x<\overline{x}\) the process drifts up.
Jump processes - Poisson Processes¶
Jump processes are a type of stochastic process that has discontinuous paths. Jump process change by discrete amounts when a certain outcome occurs. The most important Jump process is the Poisson process, which is just a jump process such that the time of the jumps follows a Poisson distribution. To define it let \(\lambda\) be the mean arrival rate of a jump and \(u\) the size of the change of the process (usually \(u=1\), but in general \(u\) can be itself a random variable). Then for some process \(q\) we have:
We can now define a more general process that depends on the Jump process \(q\):
where absent a jump \(x\) evolves deterministically according to the function \(f\), and when there is a jump it moves according to function \(g\),