Almost Sure

7 January 19

Proof of Optional and Predictable Section

In this post I give a proof of the theorems of optional and predictable section. These are often considered among the more advanced results in stochastic calculus, and many texts on the subject skip their proofs entirely. The approach here makes use of the measurable section theorem but, other than that, is relatively self-contained and will not require any knowledge of advanced topics beyond basic properties of probability measures.

Given a probability space {(\Omega,\mathcal F,{\mathbb P})} we denote the projection map from {\Omega\times{\mathbb R}^+} to {\Omega} by

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\pi_\Omega\colon \Omega\times{\mathbb R}^+\rightarrow\Omega,\smallskip\\ &\displaystyle\pi_\Omega(\omega,t)=\omega. \end{array}

For a set {S\subseteq\Omega\times{\mathbb R}^+} then, by construction, for every {\omega\in\pi_\Omega(S)} there exists a {t\in{\mathbb R}^+} with {(\omega,t)\in S}. Measurable section states that this choice can be made in a measurable way. That is, assuming that the probability space is complete, {\pi_\Omega(S)} is measurable and there is a measurable section {\tau\colon\Omega\rightarrow{\mathbb R}^+} satisfying {\tau\in S}. I use the shorthand {\tau\in S} to mean {(\omega,\tau(\omega))\in S}, and it is convenient to extend the domain of {\tau} to all of {\Omega} by setting {\tau=\infty} outside of {\pi_\Omega(S)}. So, we consider random times taking values in the extended nonnegative real numbers {\bar{\mathbb R}^+={\mathbb R}^+\cup\{\infty\}}. The property that {\tau\in S} whenever {\tau < \infty} can be expressed by stating that the graph of {\tau} is contained in S, where the graph is defined as

\displaystyle  [\tau]\equiv\left\{(\omega,t)\in\Omega\times{\mathbb R}^+\colon t=\tau(\omega)\right\}.

The optional section theorem is a significant extension of measurable section which is very important to the general theory of stochastic processes. It starts with the concept of stopping times and with the optional sigma-algebra on {\Omega\times{\mathbb R}^+}. Then, it says that if S is optional its section {\tau} can be chosen to be a stopping time. However, there is a slight restriction. It might not be possible to define such {\tau} everywhere on {\pi_\Omega(S)}, but instead only up to a set of positive probability {\epsilon}, where {\epsilon} can be made as small as we like. There is also a corresponding predictable section theorem, which says that if S is in the predictable sigma-algebra, its section {\tau} can be chosen to be a predictable stopping time.

I give precise statements and proofs of optional and predictable section further below, and also prove a much more general section theorem which applies to any collection of random times satisfying a small number of required properties. Optional and predictable section will follow as consequences of this generalised section theorem.

Both the optional and predictable sigma-algebras, as well as the sigma-algebra used in the generalised section theorem, can be generated by collections of stochastic intervals. Any pair of random times {\sigma,\tau\colon\Omega\rightarrow\bar{\mathbb R}^+} defines a stochastic interval,

\displaystyle  [\sigma,\tau)\equiv\left\{(\omega,t)\in\Omega\times{\mathbb R}^+\colon\sigma(\omega)\le t < \tau(\omega)\right\}.

The debut of a set {S\subseteq\Omega\times{\mathbb R}^+} is defined to be the random time

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle D(S)\colon\Omega\rightarrow\bar{\mathbb R}^+,\smallskip\\ &\displaystyle D(S)(\omega)=\inf\left\{t\in{\mathbb R}^+\colon(\omega,t)\in S\right\}. \end{array}

In general, even if S is measurable, its debut need not be, although it can be shown to be measurable in the case that the probability space is complete. For a random time {\tau} and a measurable set {A\subseteq\Omega}, we use {\tau_A} to denote the restriction of {\tau} to A defined by

\displaystyle  \tau_A(\omega)=\begin{cases} \tau(\omega),&{\rm\ if\ }\omega\in A,\\ \infty,&{\rm\ if\ }\omega\not\in A. \end{cases}

We start with the general situation of a collection of random times {\mathcal T} satisfying a few required properties and show that, for sufficiently simple subsets of {\Omega\times{\mathbb R}^+}, the section can be chosen to be almost surely equal to the debut. It is straightforward that the collection of all stopping times defined with respect to some filtration do indeed satisfy the required properties for {\mathcal T}, but I also give a proof of this further below. A nonempty collection {\mathcal A} of subsets of a set X is called an algebra, Boolean algebra or, alternatively, a ring, if it is closed under finite unions, finite intersections, and under taking the complement {A^c=X\setminus A} of sets {A\in\mathcal A}. Recall, also, that {\mathcal A_\delta} represents the countable intersections of A, which is the collection of sets of the form {\bigcap_nA_n} for sequences {A_1,A_2,\ldots} in {\mathcal A}.

Lemma 1 Let {(\Omega,\mathcal F,{\mathbb P})} be a probability space and {\mathcal T} be a collection of measurable times {\tau\colon\Omega\rightarrow\bar{\mathbb R}^+} satisfying,

  • the constant function {\tau=0} is in {\mathcal T}.
  • {\sigma\wedge\tau} and {\sigma_{\{\sigma < \tau\}}} are in {\mathcal T}, for all {\sigma,\tau\in\mathcal T}.
  • {\sup_n\mathcal\tau_n\in\mathcal T} for all sequences {\tau_1,\tau_2,\cdots} in {\mathcal T}.

Then, letting {\mathcal A} be the collection of finite unions of stochastic intervals {[\sigma,\tau)} over {\sigma,\tau\in\mathcal T}, we have the following,

  • {\mathcal A} is an algebra on {\Omega\times{\mathbb R}^+}.
  • for all {S\in\mathcal A_\delta}, its debut satisfies

    \displaystyle  [D(S)]\subseteq S,\ \{D(S) < \infty\}=\pi_\Omega(S),

    and there is a {\tau\in\mathcal T} with {[\tau]\subseteq[D(S)]} and {\tau = D(S)} almost surely.

(more…)

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29 November 16

The Section Theorems

Consider a probability space {(\Omega,\mathcal{F},{\mathbb P})} and a subset S of {{\mathbb R}_+\times\Omega}. The projection {\pi_\Omega(S)} is the set of {\omega\in\Omega} such that there exists a {t\in{\mathbb R}_+} with {(t,\omega)\in S}. We can ask whether there exists a map

\displaystyle  \tau\colon\pi_\Omega(S)\rightarrow{\mathbb R}_+

such that {(\tau(\omega),\omega)\in S}. From the definition of the projection, values of {\tau(\omega)} satisfying this exist for each individual {\omega}. By invoking the axiom of choice, then, we see that functions {\tau} with the required property do exist. However, to be of use for probability theory, it is important that {\tau} should be measurable. Whether or not there are measurable functions with the required properties is a much more difficult problem, and is answered affirmatively by the measurable selection theorem. For the question to have any hope of having a positive answer, we require S to be measurable, so that it lies in the product sigma-algebra {\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}}, with {\mathcal{B}({\mathbb R}_+)} denoting the Borel sigma-algebra on {{\mathbb R}_+}. Also, less obviously, the underlying probability space should be complete. Throughout this post, {(\Omega,\mathcal{F},{\mathbb P})} will be assumed to be a complete probability space.

It is convenient to extend {\tau} to the whole of {\Omega} by setting {\tau(\omega)=\infty} for {\omega} outside of {\pi_\Omega(S)}. Then, {\tau} is a map to the extended nonnegative reals {\bar{\mathbb R}_+={\mathbb R}_+\cup\{\infty\}} for which {\tau(\omega) < \infty} precisely when {\omega} is in {\pi_\Omega(S)}. Next, the graph of {\tau}, denoted by {[\tau]}, is defined to be the set of {(t,\omega)\in{\mathbb R}_+\times\Omega} with {t=\tau(\omega)}. The property that {(\tau(\omega),\omega)\in S} whenever {\tau(\omega) < \infty} is expressed succinctly by the inclusion {[\tau]\subseteq S}. With this notation, the measurable selection theorem is as follows.

Theorem 1 (Measurable Selection) For any {S\in\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}}, there exists a measurable {\tau\colon\Omega\rightarrow\bar{\mathbb R}_+} such that {[\tau]\subseteq S} and

\displaystyle  \left\{\tau < \infty\right\}=\pi_\Omega(S). (1)

As noted above, if it wasn’t for the measurability requirement then this theorem would just be a simple application of the axiom of choice. Requiring {\tau} to be measurable, on the other hand, makes the theorem much more difficult to prove. For instance, it would not hold if the underlying probability space was not required to be complete. Note also that, stated as above, measurable selection implies that the projection of S is equal to a measurable set {\{\tau < \infty\}}, so the measurable projection theorem is an immediate corollary. I will leave the proof of Theorem 1 for a later post, together with the proofs of the section theorems stated below.

A closely related problem is the following. Given a measurable space {(X,\mathcal{E})} and a measurable function, {f\colon X\rightarrow\Omega}, does there exist a measurable right-inverse on the image of {f}? This is asking for a measurable function, {g}, from {f(X)} to {X} such that {f(g(\omega))=\omega}. In the case where {(X,\mathcal{E})} is the Borel space {({\mathbb R}_+,\mathcal{B}({\mathbb R}_+))}, Theorem 1 says that it does exist. If S is the graph {\{(t,f(t))\colon t\in{\mathbb R}_+\}} then {\tau} will be the required right-inverse. In fact, as all uncountable Polish spaces are Borel-isomorphic to each other and, hence, to {{\mathbb R}_+}, this result applies whenever {(X,\mathcal{E})} is a Polish space together with its Borel sigma-algebra. (more…)

22 November 16

Predictable Processes

In contrast to optional processes, the class of predictable processes was used extensively in the development of stochastic integration in these notes. They appeared as integrands in stochastic integrals then, later on, as compensators and in the Doob-Meyer decomposition. Since they are also central to the theory of predictable section and projection, I will revisit the basic properties of predictable processes now. In particular, any of the collections of sets and processes in the following theorem can equivalently be used to define the predictable sigma-algebra. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})}. However, completeness is not actually required for the following result. All processes are assumed to be real valued, or 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 predictable stopping time}.

  • {Z1_{[\tau,\infty)}} as {\tau} ranges over the predictable stopping times and Z over the {\mathcal{F}_{\tau-}}-measurable random variables.
  • {\{A\times(t,\infty)\colon t\in{\mathbb R}_+,A\in\mathcal{F}_t\}\cup\{A\times\{0\}\colon A\in\mathcal{F}_0\}}.
  • The elementary predictable processes.
  • {{(\tau,\infty)}: {\tau} is a stopping time}{\cup}{{A\times\{0\}\colon A\in\mathcal{F}_0}}.

  • The left-continuous adapted processes.
  • The continuous adapted processes.
  • Compare this with the analogous result for sets/processes generating the optional sigma-algebra given in the previous post. The proof of Theorem 1 is given further below. First, recall that the predictable sigma-algebra was previously defined to be generated by the left-continuous adapted processes. However, it can equivalently be defined by any of the collections stated in Theorem 1. To make this clear, I now restate the definition making use if this equivalence.

    Definition 2 The predictable sigma-algebra, {\mathcal{P}}, 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 predictable iff it is {\mathcal{P}}-measurable.

    (more…)

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