Almost Sure

8 November 16

Measurable Projection and the Debut Theorem

I will discuss some of the immediate consequences of the following deceptively simple looking result.

Theorem 1 (Measurable Projection) If {(\Omega,\mathcal{F},{\mathbb P})} is a complete probability space and {A\in\mathcal{B}({\mathbb R})\otimes\mathcal{F}} then {\pi_\Omega(A)\in\mathcal{F}}.

The notation {\pi_B} is used to denote the projection from the cartesian product {A\times B} of sets A and B onto B. That is, {\pi_B((a,b)) = b}. As is standard, {\mathcal{B}({\mathbb R})} is the Borel sigma-algebra on the reals, and {\mathcal{A}\otimes\mathcal{B}} denotes the product of sigma-algebras.

Theorem 1 seems almost obvious. Projection is a very simple map and we may well expect the projection of, say, a Borel subset of {{\mathbb R}^2} onto {{\mathbb R}} to be Borel. In order to formalise this, we could start by noting that sets of the form {A\times B} for Borel A and B have an easily described, and measurable, projection, and the Borel sigma-algebra is the closure of the collection such sets under countable unions and under intersections of decreasing sequences of sets. Furthermore, the projection operator commutes with taking the union of sequences of sets. Unfortunately, this method of proof falls down when looking at the limit of decreasing sequences of sets, which does not commute with projection. For example, the decreasing sequence of sets {S_n=(0,1/n)\times{\mathbb R}\subseteq{\mathbb R}^2} all project onto the whole of {{\mathbb R}}, but their limit is empty and has empty projection.

There is an interesting history behind Theorem 1, as mentioned by Gerald Edgar on MathOverflow (1) in answer to The most interesting mathematics mistake? In a 1905 paper, Henri Lebesgue asserted that the projection of a Borel subset of the plane onto the line is again a Borel set (Lebesgue, (3), pp 191–192). This was based on the erroneous assumption that projection commutes with the limit of a decreasing sequence of sets. The mistake was spotted, in 1916, by Mikhail Suslin, and led to his investigation of analytic sets and to begin the study of what is now known as descriptive set theory. See Kanamori, (2), for more details. In fact, as was shown by Suslin, projections of Borel sets need not be Borel. So, by considering the case where {\Omega={\mathbb R}} and {\mathcal{F}=\mathcal{B}({\mathbb R})}, Theorem 1 is false if the completeness assumption is dropped. I will give a proof of Theorem 1 but, as it is a bit involved, this is left for a later post.

For now, I will state some consequences of the measurable projection theorem which are important to the theory of continuous-time stochastic processes, starting with the following. Throughout this post, the underlying probability space {(\Omega,\mathcal{F})} is assumed to be complete, and stochastic processes are taken to be real-valued, or take values in the extended reals {\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}, with time index ranging over {{\mathbb R}_+}. For a first application of measurable projection, it allows us to show that the supremum of a jointly measurable processes is measurable.

Lemma 2 If X is a jointly measurable process and {S\in\mathcal{B}(\mathbb{R}_+)} then {\sup_{s\in S}X_s} is measurable.

Proof: Setting {U=\sup_{s\in S}X_s} then, for each real K, {U > K} if and only if {X_s > K} for some {s\in S}. Hence,

\displaystyle  U^{-1}\left((K,\infty]\right)=\pi_\Omega\left((S\times\Omega)\cap X^{-1}\left((K,\infty]\right)\right).

By the measurable projection theorem, this is in {\mathcal{F}} and, as sets of the form {(K,\infty]} generate the Borel sigma-algebra on {\mathbb{\bar R}}, U is {\mathcal{F}}-measurable. ⬜

Next, the running maximum of a jointly measurable process is again jointly measurable.

Lemma 3 If X is a jointly measurable process then {X^*_t\equiv\sup_{s\le t}X_s} is also jointly measurable.


21 October 16

The Projection Theorems

Back when I first started this series of posts on stochastic calculus, the aim was to write up the notes which I began writing while learning the subject myself. The idea behind these notes was to give a more intuitive and natural, yet fully rigorous, approach to stochastic integration and semimartingales than the traditional method. The stochastic integral and related concepts were developed without requiring advanced results such as optional and predictable projection or the Doob-Meyer decomposition which are often used in traditional approaches. Then, the more advanced theory of semimartingales was developed after stochastic integration had already been established. This now complete! The list of subjects from my original post have now all been posted. Of course, there are still many important areas of stochastic calculus which are not adequately covered in these notes, such as local times, stochastic differential equations, excursion theory, etc. I will now focus on the projection theorems and related results. Although these are not required for the development of the stochastic integral and the theory of semimartingales, as demonstrated by these notes, they are still very important and powerful results invaluable to much of the more advanced theory of continuous-time stochastic processes. Optional and predictable projection are often regarded as quite advanced topics beyond the scope of many textbooks on stochastic calculus. This is because they require some descriptive set theory and, in particular, some understanding of analytic sets. The level of knowledge required for applications to stochastic calculus is not too great though, and I aim to give complete proofs of the projection theorems in these notes. However, the proofs of these theorems do require ideas which are not particularly intuitive from the viewpoint of stochastic calculus, and hence the desire to avoid them in the initial development of the stochastic integral. The theory of semimartingales and stochastic integration will not used at all in the series of posts on the projection theorems, and all that will be required from these stochastic calculus notes are the initial posts on filtrations and processes. I will also mention quasimartingales, although only the definition and very basic properties will be required.

The subjects related to the projection theorems which I will cover are,

  • The Debut Theorem. I have already covered the debut theorem for right-continuous processes. This is a special case of the more general result which applies to arbitrary progressively measurable processes.
  • The Optional and Predictable Section Theorems. These very powerful results state that optional processes are determined, up to evanescence, by their values at stopping times and, similarly, predictable processes are determined by their values at predictable stopping times.
  • Optional and Predictable Projection. This forms the core of these sequence of posts, and follows in a straightforward way from the section theorems. As the section theorems are required to prove them, the projection theorems are also regarded as an advanced topic. However, for right-continuous and left-continuous processes it is possible to construct respectively the optional and predictable projections in a more elementary and natural way, without involving the section theorems.
  • Dual Optional and Predictable Projection. The dual projections are, as the name suggests, dual to the optional and predictable projections mentioned above. These apply to increasing integrable processes or, more generally, to processes with integrable variation. For a process X, the dual projections can be thought of as the optional and predictable projections applied to the differential {dX}.
  • The Doléans Measure. The Doléans measure can be defined for class (D) submartingales and, applied to the square of a martingale, can be used to construct the stochastic integral for square integrable martingales. Although this does not involve the projection theorems, the Doléans measure in conjunction with dual predictable projection gives a slick proof of the Doob-Meyer decomposition. The Doléans measure also exists for quasimartingales and, similarly, the Doob-Meyer decomposition can be extended to such processes.

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