# 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.

## 6 March 17

### The Projection Theorems

In this post, I introduce the concept of optional and predictable projections of jointly measurable processes. Optional projections of right-continuous processes and predictable projections of left-continuous processes were constructed in earlier posts, with the respective continuity conditions used to define the projection. These are, however, just special cases of the general theory. For arbitrary measurable processes, the projections cannot be expected to satisfy any such pathwise regularity conditions. Instead, we use the measurability criteria that the projections should be, respectively, optional and predictable.

The projection theorems are a relatively straightforward consequence of optional and predictable section. However, due to the difficulty of proving the section theorems, optional and predictable projection is generally considered to be an advanced or hard part of stochastic calculus. Here, I will make use of the section theorems as stated in an earlier post, but leave the proof of those until after developing the theory of projection.

As usual, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\ge0},{\mathbb P})}$, and only consider real-valued processes. Any two processes are considered to be the same if they are equal up to evanescence. The optional projection is then defined (up to evanescence) by the following.

Theorem 1 (Optional Projection) Let X be a measurable process such that ${{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\;\vert\mathcal{F}_\tau]}$ is almost surely finite for each stopping time ${\tau}$. Then, there exists a unique optional process ${{}^{\rm o}\!X}$, referred to as the optional projection of X, satisfying

 $\displaystyle 1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau={\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_\tau]$ (1)

almost surely, for each stopping time ${\tau}$.

Predictable projection is defined similarly.

Theorem 2 (Predictable Projection) Let X be a measurable process such that ${{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\;\vert\mathcal{F}_{\tau-}]}$ is almost surely finite for each predictable stopping time ${\tau}$. Then, there exists a unique predictable process ${{}^{\rm p}\!X}$, referred to as the predictable projection of X, satisfying

 $\displaystyle 1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau={\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_{\tau-}]$ (2)

almost surely, for each predictable stopping time ${\tau}$.

## 28 February 17

### Pathwise Regularity of Optional and Predictable Processes

As I have mentioned before in these notes, when working with processes in continuous time, it is important to select a good modification. Typically, this means that we work with processes which are left or right continuous. However, in general, it can be difficult to show that the paths of a process satisfy such pathwise regularity. In this post I show that for optional and predictable processes, the section theorems introduced in the previous post can be used to considerably simplify the situation. Although they are interesting results in their own right, the main application in these notes will be to optional and predictable projection. Once the projections are defined, the results from this post will imply that they preserve certain continuity properties of the process paths.

Suppose, for example, that we have a continuous-time process X which we want to show to be right-continuous. It is certainly necessary that, for any sequence of times ${t_n\in{\mathbb R}_+}$ decreasing to a limit ${t}$, ${X_{t_n}}$ almost-surely tends to ${X_t}$. However, even if we can prove this for every possible decreasing sequence ${t_n}$, it does not follow that X is right-continuous. As a counterexample, if ${\tau\colon\Omega\rightarrow{\mathbb R}}$ is any continuously distributed random time, then the process ${X_t=1_{\{t\le \tau\}}}$ is not right-continuous. However, so long as the distribution of ${\tau}$ has no atoms, X is almost-surely continuous at each fixed time t. It is remarkable, then, that if we generalise to look at sequences of stopping times, then convergence in probability along decreasing sequences of stopping times is enough to guarantee everywhere right-continuity of the process. At least, it is enough so long as we restrict consideration to optional processes.

As usual, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$. Two processes are considered to be the same if they are equal up to evanescence, and any pathwise property is said to hold if it holds up to evanescence. That is, a process is right-continuous if and only is it is everywhere right-continuous on a set of probability 1. All processes will be taken to be real-valued, and a process is said to have left (or right) limits if its left (or right) limits exist everywhere, up to evanescence, and are finite.

Theorem 1 Let X be an optional process. Then,

1. X is right-continuous if and only if ${X_{\tau_n}\rightarrow X_\tau}$ in probability, for each uniformly bounded sequence ${\tau_n}$ of stopping times decreasing to a limit ${\tau}$.
2. X has right limits if and only if ${X_{\tau_n}}$ converges in probability, for each uniformly bounded decreasing sequence ${\tau_n}$ of stopping times.
3. X has left limits if and only if ${X_{\tau_n}}$ converges in probability, for each uniformly bounded increasing sequence ${\tau_n}$ of stopping times.

The only if’ parts of these statements is immediate, since convergence everywhere trivially implies convergence in probability. The importance of this theorem is in the if’ directions. That is, it gives sufficient conditions to guarantee that the sample paths satisfy the respective regularity properties.

Note that conditions for left-continuity are absent from the statements of Theorem 1. In fact, left-continuity does not follow from the corresponding property along sequences of stopping times. Consider, for example, a Poisson process, X. This is right-continuous but not left-continuous. However, its jumps occur at totally inaccessible times. This implies that, for any sequence ${\tau_n}$ of stopping times increasing to a finite limit ${\tau}$, it is true that ${X_{\tau_n}}$ converges almost surely to ${X_\tau}$. In light of such examples, it is even more remarkable that right-continuity and the existence of left and right limits can be determined by just looking at convergence in probability along monotonic sequences of stopping times. Theorem 1 will be proven below, using the optional section theorem.

For predictable processes, we can restrict attention to predictable stopping times. In this case, we obtain a condition for left-continuity as well as for right-continuity.

Theorem 2 Let X be a predictable process. Then,

1. X is right-continuous if and only if ${X_{\tau_n}\rightarrow X_\tau}$ in probability, for each uniformly bounded sequence ${\tau_n}$ of predictable stopping times decreasing to a limit ${\tau}$.
2. X is left-continuous if and only if ${X_{\tau_n}\rightarrow X_\tau}$ in probability, for each uniformly bounded sequence ${\tau_n}$ of predictable stopping times increasing to a limit ${\tau}$.
3. X has right limits if and only if ${X_{\tau_n}}$ converges in probability, for each uniformly bounded decreasing sequence ${\tau_n}$ of predictable stopping times.
4. X has left limits if and only if ${X_{\tau_n}}$ converges in probability, for each uniformly bounded increasing sequence ${\tau_n}$ of predictable stopping times.

Again, the proof is given below, and relies on the predictable section theorem. (more…)

## 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}${\cup}${${A\times\{0\}\colon A\in\mathcal{F}_0}$}.

• 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.

## 1 November 16

### Predictable Projection For Left-Continuous Processes

In the previous post, I looked at optional projection. Given a non-adapted process X we construct a new, adapted, process Y by taking the expected value of ${X_t}$ conditional on the information available up until time t. I will now concentrate on predictable projection. This is a very similar concept, except that we now condition on the information available strictly before time t.

It will be assumed, throughout this post, that the underlying filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})}$ satisfies the usual conditions, meaning that it is complete and right-continuous. This is just for convenience, as most of the results stated here extend easily to non-right-continuous filtrations. The sigma-algebra

$\displaystyle \mathcal{F}_{t-} = \sigma\left(\mathcal{F}_s\colon s < t\right)$

represents the collection of events which are observable before time t and, by convention, we take ${\mathcal{F}_{0-}=\mathcal{F}_0}$. Then, the conditional expectation of X is written as,

 $\displaystyle Y_t={\mathbb E}[X_t\;\vert\mathcal{F}_{t-}]{\rm\ \ (a.s.)}$ (1)

By definition, Y is adapted. However, at each time, (1) only defines Y up to a zero probability set. It does not determine the paths of Y, which requires specifying its values simultaneously at the uncountable set of times in ${{\mathbb R}_+}$. So, (1) does not tell us the distribution of Y at random times, and it is necessary to specify an appropriate version for Y. Predictable projection gives a uniquely defined modification satisfying (1). The full theory of predictable projection for jointly measurable processes requires the predictable section theorem. However, as I demonstrate here, in the case where X is left-continuous, predictable projection can be done by more elementary methods. The statements and most of the proofs in this post will follow very closely those given previously for optional projection. The main difference is that left and right limits are exchanged, predictable stopping times are used in place of general stopping times, and the sigma algebra ${\mathcal{F}_{t-}}$ is used in place of ${\mathcal{F}_t}$.

Stochastic processes will be defined up to evanescence, so two processes are considered to be the same if they are equal up to evanescence. In order to apply (1), some integrability requirements need to imposed. I will use local integrability. Recall that, in these notes, a process X is locally integrable if there exists a sequence of stopping times ${\tau_n}$ increasing to infinity and such that

 $\displaystyle 1_{\{\tau_n > 0\}}\sup_{t \le \tau_n}\lvert X_t\rvert$ (2)

is integrable. This is a strong enough condition for the conditional expectation (1) to exist, not just at each fixed time, but also whenever t is a stopping time. The main result of this post can now be stated.

Theorem 1 (Predictable Projection) Let X be a left-continuous and locally integrable process. Then, there exists a unique left-continuous process Y satisfying (1).

As it is left-continuous, the fact that Y is specified, almost surely, at any time t by (1) means that it is uniquely determined up to evanescence. The main content of Theorem 1 is the existence of Y, and the proof of this is left until later in this post.

The process defined by Theorem 1 is called the predictable projection of X, and is denoted by ${{}^{\rm p}\!X}$. So, ${{}^{\rm p}\!X}$ is the unique left-continuous process satisfying

 $\displaystyle {}^{\rm p}\!X_t={\mathbb E}[X_t\;\vert\mathcal{F}_{t-}]{\rm\ \ (a.s.)}$ (3)

for all times t. In practice, X will usually not just be left-continuous, but will also have right limits everywhere. That is, it is caglad (“continu à gauche, limites à droite”).

Theorem 2 Let X be a caglad and locally integrable process. Then, its predictable projection is caglad.

The simplest non-trivial example of predictable projection is where ${X_t}$ is constant in t and equal to an integrable random variable U. Then, ${{}^{\rm p}\!X_t=M_{t-}}$ is the left-limits of the cadlag martingale ${M_t={\mathbb E}[U\;\vert\mathcal{F}_t]}$, so ${{}^{\rm p}\!X}$ is easily seen to be a caglad process. (more…)

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