In the previous post I started by introducing the concept of a stochastic process, and their modifications. It is necessary to introduce a further concept, to represent the information available at each time. A filtration on a probability space is a collection of sub-sigma-algebras of satisfying whenever . The idea is that represents the set of events observable by time . The probability space taken together with the filtration is called a filtered probability space.
Given a filtration, its right and left limits at any time and the limit at infinity are as follows
Here, denotes the sigma-algebra generated by a collection of sets. The left limit as defined here only really makes sense at positive times. Throughout these notes, I define the left limit at time zero as . The filtration is said to be right-continuous if .
A probability space is complete if contains all subsets of zero probability elements of . Any probability space can be extended to a complete probability space (its completion) in a unique way by enlarging the sigma-algebra to consist of all sets such that for satisfying . Similarly, a filtered probability space is said to be complete if the underlying probability space is complete and contains all zero probability sets.
Often, in stochastic process theory, filtered probability spaces are assumed to satisfy the usual conditions, meaning that it is complete and the filtration is right-continuous. Note that any filtered probability space can be completed simply by completing the underlying probability space and then adding all zero probability sets to each . Furthermore, replacing by , any filtration can be enlarged to a right-continuous one. By these constructions, any filtered probability space can be enlarged in a minimal way to one satisfying the usual conditions.
Throughout these notes I assume a complete filtered probability space, although many of the results can be extended to the non-complete case without much difficulty. However, for the sake of a bit more generality, I don’t assume that filtrations are right-continuous.
One reason for using filtrations is to define adapted processes. A stochastic process process is adapted if is an -measurable random variable for each time . This is just saying that the value is observable by time . Conversely, the filtration generated by any process is the smallest filtration with respect to which it is adapted. This is given by , and referred to as the natural filtration of .
As mentioned in the previous post, it is often necessary to impose measurability constraints on a process considered as a map . Right-continuous and left-continuous processes are automatically jointly measurable. When considering more general processes, it is useful to combine the measurability concept with adaptedness. This can be done in either of the following three ways, in order of increasing generality (see Lemma 4 below).
- The predictable sigma-algebra on , denoted by , is generated by the left-continuous and adapted processes. A stochastic process is said to be predictable if it is -measurable. Alternatively, the predictable processes are sometimes called previsible.
- The optional sigma-algebra on , denoted by , is generated by the right-continuous and adapted processes. A stochastic process is said to be optional if it is -measurable.
- A process is progressively measurable, or just progressive, if for each , the map
The most important of these definitions, at least in these notes, is that of predictable processes. While adapted right-continuous processes will be used extensively, there is not much need to generalize to optional processes. Similarly, progressive measurability isn’t used a lot except in the context of adapted right-continuous (and therefore optional) processes. On the other hand, predictable processes are extensively used as integrands for stochastic integrals and in the Doob-Meyer decomposition, and are often not restricted to the adapted and left-continuous case.
Given any set of real-valued functions on a set, which is closed under multiplication, the set of functions measurable with respect to the generated sigma-algebra can be identitified as follows. They form the smallest set of real-valued functions containing the generating set and which is closed under taking linear combinations and increasing limits. So, for example, the predictable processes form the smallest set containing the adapted left-continuous processes which is closed under linear combinations and such that the limit of an increasing sequence of predictable processes is predictable.
Another way of defining predictable processes is in terms of continuous adapted processes.
Lemma 2 The predictable sigma-algebra is generated by the continuous and adapted processes.
Proof: Clearly every continuous adapted process is left-continuous and, therefore, is predictable. Conversely, if is an adapted left-continuous process then it can be written as a limit of the continuous processes
Continuity of follows from the fact that is continuous for bounded and measurable . In fact, as is bounded by , it has Lipschitz constant . The limit follows from left-continuity of , which is therefore in the sigma-algebra generated by the continuous adapted processes . ⬜
A further method of defining the predictable sigma-algebra is in terms of simple sets generating it. The following is sometimes used.
Lemma 3 The predictable sigma-algebra is generated by the sets of the form
Proof: If is any of the sets in the collection (1) then the process defined by is adapted and left-continuous, and therefore predictable. So, .
Conversely, let X be left-continuous and adapted. Then it is the limit of the piecewise constant functions
as n goes to infinity. Each of the summands on the right hand side is easily seen to be measurable with respect to the sigma-algebra generated by the collection (1). So, is also measurable. ⬜
In these notes, I refer to the collection of finite unions of sets in the collection (1) as the elementary or elementary predictable sets. Writing these as then .
Finally, the different forms of measurability can be listed in order of generality, starting with the predictable processes, up to the much larger class of jointly measurable adapted processes.
Lemma 4 Each of the following properties of a stochastic process implies the next
- adapted and jointly measurable.
Proof: As the predictable sigma-algebra is generated by the continuous adapted processes, which are also optional by definition, it follows that all predictable processes are optional.
Now, if is a right-continuous and adapted process and , then the process is right-continuous and -measurable at all times. By the joint measurability of right-continuous processes, is -measurable. As this holds for all times , is progressively measurable.
Finally, consider a progressively measurable process . From the definitions, it is jointly measurable. Furthermore, for any time , restricted to is -measurable. Therefore, is -measurable for all and, in particular, is adapted. ⬜