Almost Sure

15 November 16

Optional Processes

The optional sigma-algebra, ${\mathcal{O}}$, was defined earlier in these notes as the sigma-algebra generated by the adapted and right-continuous processes. Then, a stochastic process is optional if it is ${\mathcal{O}}$-measurable. However, beyond the definition, very little use was made of this concept. While right-continuous adapted processes are optional by construction, and were used throughout the development of stochastic calculus, there was no need to make use of the general definition. On the other hand, optional processes are central to the theory of optional section and projection. So, I will now look at such processes in more detail, starting with the following alternative, but equivalent, ways of defining the optional sigma-algebra. Throughout this post we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})}$, and all stochastic processes will be assumed to be either real-valued or to take values in the extended reals ${\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}$.

Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on ${{\mathbb R}_+\times\Omega}$.

{${[\tau,\infty)}$: ${\tau}$ is a stopping time}.

• ${Z1_{[\tau,\infty)}}$ as ${\tau}$ ranges over the stopping times and Z over the ${\mathcal{F}_\tau}$-measurable random variables.
• The right-continuous adapted processes.
• The optional-sigma algebra was previously defined to be generated by the right-continuous adapted processes. However, any of the four collections of sets and processes stated in Theorem 1 can equivalently be used, and the definitions given in the literature do vary. So, I will restate the definition making use of this equivalence.

Definition 2 The optional sigma-algebra, ${\mathcal{O}}$, is the sigma-algebra on ${{\mathbb R}_+\times\Omega}$ generated by any of the collections of sets/processes in Theorem 1.

A stochastic process is optional iff it is ${\mathcal{O}}$-measurable.

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.

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