I start these notes on stochastic calculus with the definition of a continuous time stochastic process. Very simply, a stochastic process is a collection of random variables defined on a probability space . That is, for each time , is a measurable function from to the real numbers.
Stochastic processes may also take values in any measurable space but, in these notes, I concentrate on real valued processes. I am also restricting to the case where the time index runs through the non-negative real numbers , although everything can easily be generalized to other subsets of the reals.
A stochastic process can be viewed in either of the following three ways.
- As a collection of random variables, one for each time .
- As a path
one for each . These are referred to as the sample paths of the process.
- As a function from the product space
As is often the case in probability theory, we are not interested in events which occur with zero probability. Stochastic process theory is no different, and two processes are said to be indistinguishable if there is an event of probability one such that for all and all . This is the same as saying that they almost surely (i.e., with probability one) have the same sample paths. Alternative language which is often used is that and are equivalent up to evanescence. In general, when discussing any properties of a stochastic process, it is common to only care about what holds up to evanescence. For example, if a processes has continuous sample paths with probability one, then it is referred to as a continuous process and we don’t care if it actually has discontinuous paths on some event of zero probability.
It is important to realize that even if we have two processes satisfying almost surely, at each time , then it does not follow that they are indistinguishable. As an example, consider a random variable uniformly distributed over the interval , and define the process . For each time , . However, is not indistinguishable from the zero process, as the sample path always has one point at which takes the value . The problem here is the uncountability of nonnegative real numbers used for the time index. By countable additivity of measures, if almost surely, for each , then we can infer that the sample paths of and are almost surely identical on any given countable set of times , but cannot extend this to uncountable sets.
This necessitates a further definition. A process is a version or modification of if, for each time , . Alternative language sometimes used is that and are stochastically equivalent.
Whenever a stochastic process is defined in terms of its values at each individual time , or in terms of its joint distributions at finite times, then replacing it by any other version will still satisfy the definition. It is therefore important to choose a good version. Right-continuous (or left-continuous) versions are often used when possible, as this then defines the process up to evanescence.
Lemma 1 Let and be right-continuous processes (resp. left-continuous processes) such that almost surely, at each time . Then, they are indistinguishable.
Proof: By countable additivity, the set has probability one. By right-continuity (resp. left-continuity) of the sample paths, it follows that and have the same paths for all . ⬜
As an example, the definition of a standard Brownian motion consists of two parts. First, and, for , is normal with mean 0 and variance , independently of , which defines its finite distributions. The additional condition that it has continuous sample paths is required in order to select the correct version up to evanescence.
Viewing a stochastic process in the third sense mentioned above, as a function on the product space , it is often necessary to impose a measurability condition. The process is said to be jointly measurable if it is measurable with respect to the product sigma-algebra . Here, denotes the Borel sigma-algebra. Another benefit of choosing left or right-continuous versions is that it ensures joint measurability.
Lemma 2 All right-continuous and left-continuous processes are jointly measurable.
Proof: If is right-continuous, then it is the limit of the following sequence of processes
Clearly are jointly measurable functions and, therefore, is also jointly measurable.
The proof for left-continuous processes follows along the same lines, but using in place of . ⬜
Finally, it is often useful to sample a stochastic process at a random time. The value of a process at the measurable time is . This need not even be a measurable quantity unless a good version of the process is used. Fortunately, joint measurability of is enough.
Lemma 3 If is jointly measurable and is a random time then is measurable.
Proof: Using the functional monotone class theorem, it is only necessary to prove the result for processes with and . However, is clearly measurable. ⬜