In the previous post, the concept of Feller processes was introduced. These are Markov processes whose transition function satisfies certain continuity conditions. Many of the standard processes we study satisfy the Feller property, such as standard Brownian motion, Poisson processes, Bessel processes and Lévy processes as well as solutions to many stochastic differential equations. It was shown that all Feller processes admit a cadlag modification. In this post I state and prove some of the other useful properties satisfied by such processes, including the strong Markov property, quasi-left-continuity and right-continuity of the filtration. I also describe the basic properties of the infinitesimal generators. The results in this post are all fairly standard and can be found, for example, in Revuz and Yor (Continuous Martingales and Brownian Motion).
As always, we work with respect to a filtered probability space . Throughout this post we consider Feller processes X and transition functions defined on the lccb (locally compact with a countable base) space E which, taken together with its Borel sigma-algebra, defines a measurable space .
Recall that the law of a homogeneous Markov process X is described by a transition function on some measurable space . This specifies that the distribution of conditional on the history up until an earlier time is given by the measure . Equivalently,
for any bounded and measurable function . The strong Markov property generalizes this idea to arbitrary stopping times.
Definition 1 Let X be an adapted process and be a transition function.
Then, X satisfies the strong Markov property if, for each stopping time , conditioned on the process is Markov with the given transition function and with respect to the filtration .
As we see in a moment, Feller processes satisfy the strong Markov property. First, as an example, consider a standard Brownian motion B, and let be the first time at which it hits a fixed level . The reflection principle states that the process defined to be equal to B up until time and reflected about K afterwards, is also a standard Brownian motion. More precisely,
is a Brownian motion. This useful idea can be used to determine the distribution of the maximum . If then either the process itself ends up above K or it hits K and then drops below this level by time t, in which case . So, by the reflection principle,
In fact, the reflection principle is a consequence of the strong Markov property, as follows. First, by the strong Markov property, the process is a Brownian motion independently of the stopped process . As the law of standard Brownian motion is symmetric, this has the same distribution as . So, has the same joint distribution as .
Let us now write out explicitly what the strong Markov property means. If is a stopping time then, conditioning on , the Markov property applied to under the filtration is equivalent to
(almost surely) for all times . Simplifying this a bit gives the following.
Lemma 2 A process X is strong Markov with transition function on state space E if and only if, for every finite stopping time , every , and bounded measurable ,
Then, using the fact that and agree on the set ,
Letting T increase to infinity gives (1). ⬜
We now prove that Feller processes are strong Markov.
Theorem 3 Every cadlag Feller process satisfies the strong Markov property.
Proof: Suppose that X is a cadlag Feller process with transition function . We just need to show that (2) holds for a bounded stopping time and fixed time . First, the case where takes values in a countable set is easily handled, and does not even require the Feller property. Just the Markov property is needed. Letting Z be a bounded -measurable random variable, is -measurable for all fixed times s, giving
Next let be any stopping time and, for each nonnegative integer n, let be the first time after which is a multiple of 1/n. That is , which is a stopping time taking values in a countable subset of . So,
for bounded -measurable Z. From the definition of the Feller property, if then is jointly continuous in t and right-continuous in s. So, taking limits
In the study of continuous-time stochastic processes it is common to assume that the underlying filtration satisfies the `usual conditions’. That is, it is complete and right-continuous. On the other hand, in these notes, I have not done this. Completeness of the filtration has been used throughout, so that the debut theorem for right-continuous processes holds and so that we can take cadlag versions of processes. However, I have not been assuming right-continuity of the filtration. For Feller processes, as it turns out, right-continuity is automatic in any case. That is, the complete filtration generated by a Feller process is right-continuous.
Theorem 4 Let X be a Feller process and
be its natural filtration. Suppose that is the sigma algebra generated by together with the zero probability sets in .
Then, is a right-continuous filtration,
Proof: Suppose that X is Feller with state space E and transition function . The aim is to prove the identity
(almost surely) for every bounded -measurable random variable Z. As is the sigma-algebra generated by (up to zero probability sets), it is enough to prove the result for Z of the form
for times and functions . The functional monotone class theorem then extends this to all bounded and -measurable Z.
First, consider for some time and . As the Feller process X has a cadlag modification, will be right-continuous in probability w.r.t. t. So, the Markov property gives
Extending the result to all Z of the form (4) is just an application of induction on n. So, suppose that the (3) holds for n replaced by n-1. If then Z is -measurable, and the result is trivial. If then setting gives
Finally, if then setting and
the Markov property gives . Also, the induction hypothesis gives (3) with in place of Z. So,
as required. ⬜
One consequence of the right-continuity of the filtration is the following zero-one law. Consider, for example, a standard Brownian motion B. It can be shown hits zero infinitely often for t in any neighborhood of 0, with probability one. Events such as this are -measurable for each and, therefore, measurable. The following result states that all such events have probability zero or one. So, in fact, any Feller process with fixed initial state must either equal x infinitely often in any neighbourhood of 0 with probability one or, again with probability one, there must be a nonempty interval on which it does not equal x.
Corollary 5 (Blumenthal’s zero-one law) Let X be a Feller process with fixed initial value . Letting be its natural filtration, every satisfies or .
Proof: As is fixed at x, the sigma-algebra it generates contains only sets with probability zero or one. If is the sigma-algebra generated by together with the zero probability sets then, by Theorem 4, . However, is generated by sets of zero probability. ⬜
Recall that, for a Poisson process X, its jump times are totally inaccessible. That is, for any predictable stopping time , it is almost surely continuous at time , so . This property holds for all Feller processes, and is referred to as quasi-left-continuity. By definition, a stopping time is predictable if there exists a sequence of stopping times increasing to . Then, as . For arbitrary jointly measurable processes, quasi-left-continuity is defined as follows.
Definition 6 A process X taking values in a topological space E is quasi-left-continuous if and only if, for each finite stopping time and sequence of stopping times increasing to ,
almost surely as .
We now prove that Feller processes satisfy this property.
Theorem 7 Every cadlag Feller process is quasi-left-continuous.
Proof: Let X be a cadlag Feller process on state space E and with transition function . Then choose a finite predictable stopping time and a sequence of stopping times increasing to . The aim is to prove the identity
for bounded measurable functions . Then, taking gives as required.
As the Borel sigma-algebra on an lccb space E is generated by functions in , it is sufficient to prove (5) for for functions , as the functional monotone class theorem then extends (5) to all bounded and measurable u. Given any such functions f, g, the continuity of in both x and t together with the strong Markov property (2) applied at the stopping times gives,
for any . Taking the limit as gives
as required. ⬜
Another property of homogeneous Poisson processes is that the times between jumps are exponentially distributed. This also carries through to arbitrary Feller processes in the following form; the time spent at any fixed level x, if it is not zero or infinite, has the exponential distribution. In fact, the following result almost holds for all right-continuous Markov processes. The Feller property is only used to show that is a stopping time (using right-continuity of the filtration) and that X is discontinuous at whenever (using the strong Markov property).
Theorem 8 Let X be a cadlag Feller process with fixed initial value , and set
If the underlying filtration is complete then is a stopping time. Furthermore, one of the following properties holds.
- almost surely.
- almost surely.
- There is a such that has the distribution. In this case, with probability one, so X is discontinuous at .
Proof: Without loss of generality we can assume that the underlying filtration is the one generated by X together with the zero probability sets, since the filtration can always be replaced by this without altering the conclusion of the theorem. By right-continuity of X,
Theorem 4 says that the filtration is right continuous and, by the following, is a stopping time.
Now define the function by . For any , conditioning on the set gives and, by the Markov property, the process has the same distribution as . Therefore, conditioned on has the same distribution as ,
The only solutions to this functional equation for a right-continuous and decreasing function are,
- , in which case almost surely.
- , in which case almost surely.
- for some constant . So, has the exponential distribution of rate . Now define the process . By definition of , is almost surely zero. However, by the strong Markov property, conditioned on the event the random variable has the same distribution as and is therefore almost surely positive. So,
Except in certain simple cases it is often not possible to explicitly write out the transition function describing a Feller process. Instead, the infinitesimal generator is used. This approximately describes the transition kernel for small times t, and can be viewed as the derivative of at time 0, . As the transition function is likely not to be differentiable in any strong sense, the generator is only defined on some subset of .
Definition 9 Let be a Feller transition function on the lccb space E. Then, is said to be in the domain of the infinitesimal generator if the limit
exists under the uniform topology on .
The operator is called the infinitesimal generator of the semigroup .
Equation (6) can alternatively be written as
where denotes a term vanishing faster than t as . So, the generator A gives the first-order approximation to for small t.
Theorem 10 Let . Then, for all . Furthermore, is differentiable (w.r.t. the norm topology on ) and
Proof: By definition, tends to in as , over . Applying the bounded linear operator to this gives
as over . This shows that has the right-hand derivative . We need to extend this to show that it is actually differentiable. By the Feller property, is continuous in s. In fact, all functions with a continuous right-hand derivative are differentiable. We can integrate to get
as . Therefore, is indeed differentiable. Next,
as , over . By definition this means that and . ⬜
Recall from the previous post that, associated with the transition function , there is a resolvent defined for any as
This is a transition kernel on E and, if is Feller, then for all . Restricted to , is a continuous linear map satisfying and as . Resolvents and infinitesimal generators are closely related.
Theorem 11 Let be a Feller transition function and . Then, is equal to the image of the resolvent on and, considering as a map from to ,
Proof: The second of equations (9) is just a rearrangement of the first. We can apply to for any and ,
By the Feller property, as , giving
So, by equation (7), and . Rearranging gives
Now, choosing any , equation (8) and integration by parts gives,
Using the resolvent simplifies the proofs of some of the properties of the generator. In particular, its domain is dense in and A is closed. In the following lemma, the topology given by the uniform norm on is used.
Lemma 12 The domain is a dense subset of , and A is a closed operator. That is, if converge to such that , then and .
A Feller transition function is uniquely determined by its generator. However, this result not nearly as useful as it sounds. This is because, in all but a small number of special cases, we do not know what the domain is. Usually, we just have A defined on some dense subspace of , such as the twice continuously differentiable functions, and this is not enough to apply the following result.
Lemma 13 Let and be Feller transition functions on an lccb space E with generators A and B respectively.
Suppose that the domains of the generators satisfy and, restricted to , . Then, for all t.
Proof: Denote the resolvents of and by and respectively. Theorem 11 gives
for all . So, the resolvents are identical. However, resolvents are just Laplace transforms of the transition function and, by invertibility of Laplace transforms, for almost every t. By continuity in t, as required. ⬜
Finally, the generator of a Feller process can be rewritten in terms of a martingale problem. The martingale approach, introduced by Stroock and Varadhan, is a very useful method in the theory of stochastic differential equations and Markov processes. For example, it is instrumental in the famous Stroock-Varadhan uniqueness theorem which shows that stochastic differential equations of the form
satisfy uniqueness in law for the n-dimensional process X. This is under the condition that are continuous functions such that is a nonsingular nxn matrix for any x, and are bounded and measurable.
The martingale description of the generator is as follows.
Lemma 14 Let X be a cadlag Feller process with transition function and generator A. Then, for , the process
is a martingale.
Proof: Choose times and let Z be a bounded -measurable random variable. Taking expectations of and applying the Markov property gives,
as required. Here, (8) has been used to express as a derivative. ⬜
Lemma 14 almost gives a complete description of the generator associated with a Feller process X, but not quite. To strengthen this result to give an alternative characterization of the generator, we must consider running the Markov process X from each possible starting position .
In the following theorem, we let be the set of cadlag functions with coordinate process . Then, is the sigma-algebra generated by and . For each , Corollary 4 of the previous post implies that there is a unique probability measure on under which X is a Feller process with the given transition function and . Then, is a filtered probability space.
Theorem 15 Let be a Feller transition function and . Then, and if and only the process
is a martingale under the measure for all .
Proof: If then Lemma 14 says that M is a martingale. Conversely, suppose that M is a martingale under the measure . Then,
However, the Feller property says that as , giving
By equation (7), this says that and . ⬜
For example, suppose that is an n-dimensional process satisfying the stochastic differential equation
for an m-dimensional Brownian motion B and measurable functions . Defining the function by , consider the differential operator
For any twice-continuously differentiable function , Ito’s lemma gives
So, is a local martingale. If it is known that solutions to the SDE are in fact Feller processes, such as is the case when the coefficients are Lipschitz continuous, then Theorem 15 says that any twice continuously differentiable such that vanishes at infinity is in the domain of the generator, and the generator agrees with L on such functions.