This is part 2 of the sequence of blog posts on general theory of processes. For part 1 see here. I will be building upon the content presented there.

Introduction

“Mathematics exists solely for the honour of the human mind.” — Jacobi in a letter to Legendre after the death of Fourier. Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena.

We previously discussed Choquet's theory of capacities and its applications in measure theory. The goal of this blog post is to discuss the debut, section and projection theorems in stochastic processes. These theorems form the core of the “general theory of processes”, developed primarily by the Strasbourg school of probability under the tutelage of Paul-André Meyer. At the risk of being overly simplistic, general theory of processes is the study of filtrations and stopping times—martingales will be absent in our discussion.

Unlike part 1 where there were no prerequisites other than basic measure theory, this part assumes a good amount of familiarity with stochastic processes, at the level of (Karatzas and Shreve, 1998a), (Le Gall, 2016). Without that the material here will feel unmotivated and difficult. On the other hand, the theorems proved here are skipped even in advanced courses in stochastic processes as they aren't the most useful for applications, but rather are necessary to fill in gaps if a rigorous treatment is warranted. Nevertheless the theory presented is extremely profound and beautiful, and reading it will be an honour for the mind, if nothing else.

Setting The Stage

Completion of Measure Spaces and Filtrations

Thus the measure space $(E, \mathscr{E}, \mu)$ is complete if and only if every $\mu$-negligible subset of $E$ belongs to $\mathscr E.$

Recall that an outer measure on $E$ is a function $\mu^* \colon \mathfrak{P}(E) \to [0, \infty]$ such that

1. $\mu^*(\varnothing) = 0$,

2. if $A \subseteq B \subseteq E$, then $\mu^*(A) \le \mu^*(B)$, and

3. if $\{A_n\}_{n \in \mathbb N}$ is a sequence of subsets of $E$, then

   $$\begin{aligned} \mu^*\left( \bigcup_{n \in \mathbb N} A_n \right) \le \sum_{n \in \mathbb N} \mu^*(A_n).\end{aligned}$$

Also recall that a subset $B \subseteq E$ is called $\mu^*$-measurable if

It is a simple exercise to show that if $B \subseteq E$ is such that either $\mu^*(B) = 0$ or $\mu^*(B^\mathsf{c}) = 0$, then $B$ is $\mu^*$-measurable. Finally, recall that if $\mathscr{M}_{\mu^*}$ denotes the collection of all $\mu^*$-measurable subsets of $E$, then $\mathscr{M}_{\mu^*}$ is a $\sigma$-algebra, and the restriction of $\mu^*$ to $\mathscr{M}_{\mu^*}$ is a measure on $\mathscr{M}_{\mu^*}.$ Therefore, it follows that the measure space $(E, \mathscr{M}_{\mu^*}, \mu^*)$ is complete. In particular, the Lebesgue measure on the $\sigma$-algebra of Lebesgue subsets of $\mathbb R$ is complete. It can be shown that the restriction of Lebesgue measure to the $\sigma$-algebra of Borel subsets of $\mathbb R$ is not complete. We next discuss a result that allows us to complete any measure space.

If we denote by $\mathscr{N}$ the collection of all $\mu$-negligible sets for the measure space $(E, \mathscr{E}, \mu)$, then it is easy to see that $$\begin{aligned} \mathscr E_\mu = \{B \cup N \,:\, B \in \mathscr E \text{ and } N \in \mathscr N\},\end{aligned}$$ and $\overline \mu (B \cup N) = \mu(B).$ From this it follows that if the measure space $(E, \mathscr{E}, \mu)$ is complete, then it is its own completion.

A very nice property that holds if we assume that the measure space $(E, \mathscr{E}, \mu)$ is complete is that if $f$ and $g$ are real-valued functions defined $E$ such that $f$ is measurable and and $f = g$ holds $\mu$-a.e., then $g$ is also measurable. To see this, let $B \in \mathcal{B}(\mathbb R)$ be any Borel set. Then $\{f \in B\} \in \mathscr E$ and

by our assumptions. Thus, $\{f = g\} = \{f \neq g \}^\mathsf{c} \in \mathscr E$, and allows us to conclude that $g$ is measurable.

The above property can fail if the measure space is not complete. Similar problems can arise in the study of stochastic processes, and therefore completeness assumptions are made. For this whole blog post, we shall place ourselves on a complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$ endowed with a filtration $\mathbb{F} = \{\mathcal{F}_t\} _ {0 \le t < \infty}$, i.e., a family of sub-$\sigma$-algebras of $\mathcal{F}$ which is increasing in the sense that

We sometimes denote this setup conveniently as $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ and call it a filtered probability space. We shall assume that this filtration satisfies the usual conditions:

1. Right-continuity, i.e., $\mathcal{F}_t = \mathcal{F}_{t+} := \bigcap_{s > t} \mathcal{F}_s$ for all $t \in [0, \infty)$, and

2. $\mathcal{F}_0$ contains all the $\mathbb{P}$-negligible events in $\mathcal{F}$.

The structure of the filtration implies that we can write $\mathcal{F}_{t+}$ equivalently as $\bigcap_{n \in \mathbb N} \mathcal{F}_{t + 1/n}.$

Often we will need to start with the natural filtration $$\begin{aligned} \mathcal{F}_t^X := \sigma(X_s; \;0 \le s \le t), \quad t \ge 0,\end{aligned}$$ generated by the process $X = \{X_t\}_{t \ge 0},$ which is the smallest filtration with respect to which the process $X$ is adapted, i.e., $X_t$ is $\mathcal{F}_t^X$-measurable for each $t \ge 0.$ And then we will need to augment it to make it satisfy the usual conditions. To that end, we define the minimal augmented filtration generated by $X$ to be the smallest filtration that is right continuous and complete and with respect to which the process $X$ is adapted. This can be constructed in the following three steps:

1. First, let $\left\{\mathcal{F}_t^{X}\right\}_{t \ge 0}$ be the natural filtration as in (3).

2. Let $\mathscr{N}$ be the collection of all $\mathbb{P}$-negligible sets for the complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (we can always complete it using Theorem 11 if it isn’t). For each $t \ge 0,$ let $$\begin{aligned} \left(\mathcal{F}^X_t\right)_{\mathbb{P}} = \left\{F \cup N \,:\, F \in \mathcal{F}_t^X \text{ and } N \in \mathscr N\right\}\end{aligned}$$ be the completion of $\mathcal{F}_t^X$ just like in (2).

3. Finally, for each $t \ge 0,$ let $$\begin{aligned} \mathcal{F}_t := \left( \left(\mathcal{F}^X_t\right)_{\mathbb{P}} \right)_+ = \bigcap_{s > t} \left(\mathcal{F}^X_s\right)_{\mathbb{P}}\end{aligned}$$ define the right-continuous filtration. To show right-continuity of $\{\mathcal{F}_t\}_{t \ge 0},$ we need to show that

   $$\begin{aligned} \bigcap_{s > t} \left(\mathcal{F}^X_s\right)_{\mathbb{P}} = \mathcal{F}_t = \mathcal{F}_{t+} = \bigcap_{r > t} \bigcap_{s > r} \left(\mathcal{F}^X_s\right)_{\mathbb{P}}.\end{aligned}$$
   But this is obvious since $F \in \left(\mathcal{F}^X_s\right)_{\mathbb{P}}$ for every $s > r > t$ is same as saying $F \in \left(\mathcal{F}^X_s\right)_{\mathbb{P}}$ for every $s > t.$

These steps give a filtration $\{\mathcal{F}_t\}_{t \ge 0},$ which is right continuous and complete and with respect to which the process $X$ is adapted. It is clear that it is also the smallest such filtration. Does it matter if we do step 3 before step 2, i.e., is it true that

It would be very disappointing if they weren’t equal. Let us start by showing From (2), if $F \in \left( \left(\mathcal{F}^X_t\right)_{+} \right)_{\mathbb{P}},$ then $F = B \cup N$ for some $B \in \left(\mathcal{F}^X_t\right)_{+}$ and $N \in \mathscr N.$ Then $B \in \mathcal{F}^X_s$ for every $s > t,$ which means that $B \cup N \in \left(\mathcal{F}^X_s\right)_{\mathbb{P}},$ showing $F \in \left(\mathcal{F}^X_s\right)_{\mathbb{P}}$ for every $s > t.$

For the other side,

let $F \in \left( \left(\mathcal{F}^X_t\right)_{\mathbb{P}} \right)_+.$ We need to construct $B \in \left(\mathcal{F}^X_t\right)_{+}$ and $N \in \mathscr N$ such that $F = B \cup N.$ By the definition of right-continuity, $F \in \left(\mathcal{F}^X_{t + 1/n}\right)_{\mathbb{P}}$ for every $n \in \mathbb N,$ which means there exist sequences $\{B_n\}_{n \in \mathbb N}$ and $\{N_n\}_{n \in \mathbb N}$ satisfying $B_n \in \mathcal{F}^X_{t + 1/n},$ $N_n \in \mathscr N,$ and $F = B_n \cup N_n$ for each $n \in \mathbb N.$ Define Then note that $\bigcap_{n \ge k} B_n \in \mathcal{F}_{t + 1/k}^X,$ which implies that for each $m \in \mathbb N,$ This says that $B \in \left(\mathcal{F}^X_t\right)_{+}.$ Define $N := F \setminus B,$ and note This shows the desired inclusion, and we are done.

Stochastic Processes

For our purposes, a stochastic process is simply a collection of real-valued random variables $X = \{X_t\}_{t \ge 0}$ on $(\Omega, \mathcal{F}).$ The sample paths of $X$ are the mapping $[0, \infty) \ni t \mapsto X_t( \omega) \in \mathbb R$ obtained when fixing $\omega \in \Omega.$

We need terminology to talk about when two stochastic processes are the “same”.

We will need to make stronger measurability assumptions for the random variables $\{X_t\}_{t \ge 0}$ than just assuming that each $X_t$ is measurable.

Recall that every progressively measurable process is both measurable and adapted; and every adapted process with right-continuous or with left-continuous paths, is progressively measurable (Karatzas and Shreve, 1998a), (Le Gall, 2016). On the other hand, we have the following result due to Meyer (1966).

It can be verified that $\mathscr{P} _ \star$ is the collection of all sets $A \in \mathcal{F} \otimes \mathcal{B}([0, \infty))$ such that the process $X_t(\omega) = \mathbf{1}_A(\omega, t)$ is progressively measurable. This characterization gives the useful result that a subset $A$ of $[0, \infty) \times \Omega$ belongs to $\mathscr{P} _ \star$ if and only if, for every $t \in [0, \infty)$, $A \cap ([0,t] \times \Omega) \in \mathcal{B}([0,t]) \otimes \mathcal{F}_t.$

Stopping Times

Because the filtration $\mathbb{F}$ is right-continuous this is equivalent to $\{T < t\} \in \mathcal{F}_t$ for all $t \in [0, \infty)$ as can be seen by writing

It is easily checked that if $T$ and $S$ are stopping times, then so are

Likewise, if $\{T_n\}_{n \in \mathbb N}$ is a sequence of stopping times, then are also stopping times. Stopping times go hand in glove with progressive measurability. Suppose $X = \{X_t\}_{t \ge 0}$ is $\mathbb{F}$-progressively measurable and $T$ an $\mathbb{F}$-stopping . Then, the stopped process is $\mathbb{F}$-progressively measurable.

The following intuitive properties are useful and their proofs can be found in any standard textbook (Karatzas and Shreve, 1998a), (Le Gall, 2016).

Predictable Stopping Times

A very important result states that every stopping time is the decreasing limit of a sequence of discrete stopping time. More concretely, if $T$ is a stopping time, and we define

for $n,k \in \mathbb N,$ then each $T_n$ is a discrete stopping time and we have $T_n \downarrow T.$ When can we approximate a stopping with an increasing sequence of stopping times? Of course, being able to do that means that our stopping time is a very special one. We give a name to such stopping times.

Note $\{T \le t\} = \bigcap_{n \in \mathbb N} \{T_n \le t\} \in \mathcal{F}(t)$ and therefore if in the definition above we had not required $T$ to be a stopping time, it would still be a stopping time. A canonical example of a predictable stopping time is the first time Brownian motion $W$ hits or exceeds a certain level $b.$ It is announced by the sequence of first times $W$ hits or exceeds the levels $b - 1/n$, for ${n \in \mathbb N}.$ In fact, we have (taken from (Almost Sure blog)):

Of course, in the above proof we cheated by assuming the Debut Theorem for stopping times, proving which is our next agenda. In the same flavor as Lemma-6 we have

In particular, both the events $\{T \le S\}$ and $\{T = S\}$ belong to $\mathcal{F}(S-).$

Debut of a Progressive Set

Notice that we can write $\{Z_A \le Z\} = A \cup \left(A^\mathsf{c} \cap \{Z = \infty\}\right)$ as a disjoint union. If $T$ is a stopping time and $A \in \mathcal{F}(T)$, then since $\{T_A \le t\} = \{T \le t\} \cap A \in \mathcal{F}(t)$ for all $t \in [0, \infty)$, $T_A$ is also a stopping time.

The following important theorem now becomes an easy corollary.

Optional and Predictable Processes

Since all right-continuous or left-continuous adapted processes are progressively measurable, all optional or predictable processes are adapted. We need the concept of stochastic intervals to ease some notation and see some examples of optional and predictable sets.

Notice that $\llbracket U \rrbracket = \llbracket U,U \rrbracket = \llbracket 0, U \rrbracket \setminus \llbracket 0, U \llbracket$, and that $0_A$ is predictable with the announcing sequence $\{0_A \wedge n\} _ {n \in \mathbb N}.$ Denote by $\mathfrak{S}$ the collection of all stopping times and by $\mathfrak{S}^{(\mathscr{P})}$ the collection of all predictable stopping times ($\mathfrak{S}$ is S in fraktur font).

It is a simple exercise to show

Properties of Debuts

Recall the setting of section "Measurable Section" from part 1. We showed there that if $A \in \mathcal{B}([0, \infty)) \otimes \mathcal{F},$ then we can define a measurable mapping $T : \Omega \to [0, \infty]$ such that for all $\omega \in \pi(A)$ we have $(T(\omega), \omega) \in A$, or stated more succinctly, $\llbracket T \rrbracket \subseteq A.$ Now, what if we want something stronger, say, $T$ should be a stopping time? As we will see, we will have to make stronger assumptions on the set $A$ and even then if we want $T$ to be a stopping time we will have to let go of the nice property that $\{T < \infty\}= \pi(A).$

But first, we need to prove a certain property of debuts which we will need in proving the aforementioned section theorem, and that's the goal of this section.

It isn't difficult to see that $\mathscr{P}$ coincides with the mosaic generated by the paving on $[0, \infty) \times \Omega$ that consists of finite unions of stochastic intervals of the form $\llbracket S,T \rrbracket$, where $S,T \in \mathfrak{S}^{(\mathscr{P})}.$ Generalizing this, let us consider a collection $\mathfrak{A}$ ($\mathfrak{A}$ is A in fraktur font) of stopping times that contains $0$ and $\infty$, and which is closed under a.s. equality and under finitely many lattice operations ($\wedge$ and $\vee$). We denote by $\mathcal{J} := \{\llbracket U,V \llbracket\,:\,S,T \in \mathfrak{A}\}$, and by $\mathcal{T}$ the collection of finite unions of elements of $\mathcal{J}.$

Simple observations like $\llbracket S,T \llbracket^\mathsf{c} = \llbracket 0, S \llbracket \, \cup \, \llbracket T, \infty \llbracket$ or $\llbracket S,T \llbracket \, \cap \, \llbracket U,V \llbracket = \llbracket S \vee U, (S \vee U) \vee (T \wedge V) \llbracket$ can be used to show that $\mathcal{J}$ is a paving on $[0, \infty) \times \Omega$ and $\mathcal{T}$ is an algebra on $[0, \infty) \times \Omega.$

For more structure, let us impose the following properties on the collection $\mathfrak{A}$:

1. For any pair $S,T \in \mathfrak{A}$, the stopping time $S _ {\{S < T\}}$ (recall the notation of Definition 25 and the observation right after it) belongs to $\mathfrak{A}$;

2. For any increasing sequence $\{S_n\} _ {n \in \mathbb N} \subseteq \mathfrak{A}$, the limit $\lim _ {n \in \mathbb N} \uparrow S_n$ belongs to $\mathfrak{A}.$

Property 1 ensures that the debut of an element of $\mathcal{T}$ belongs to $\mathfrak{A}.$ This is because $S_{\{S < T\}} = D_{\llbracket S,T \llbracket}$ and because the debut of union of a finite collection of elements from $\mathcal{J}$ is equal to to the minimum of the debuts of those elements.

Property 2 ensures that the debut of an element of $\mathcal{T} _ \delta$ (which, recall, equals $\left\{\bigcup _ {n \in \mathbb N} A_n \,:\, A_n \in \mathcal{T} \text{ for all } {n \in \mathbb N}\right\}$) belongs to to $\mathfrak{A}.$ This is the main result of this section, and is proved below as Theorem 16.

Coming out of the abstraction for a bit, we note that the collection $\mathfrak{S}$ of all stopping times satisfies all the conditions imposed above on $\mathfrak{A}.$ Similarly for the collection $\mathfrak{S}^{(\mathscr{P})}$ of all predictable stopping times. The first claim is trivial to see. To show the second claim, we first prove the following lemma.

Coming back to the second claim above, if $S,T \in \mathfrak{S}^{(\mathscr{P})}$, then by Lemma 7, $\{S < T\} \in \mathcal{F}(S-)$, and thus by Lemma 10, $S _ {\{S < T\}} \in \mathfrak{S}^{(\mathscr{P})}.$ The rest of the conditions are trivial to check.

The next result is Proposition 1.2.26 from (Karatzas and Shreve, 1998a), and is left unproven there.

Section Theorems

We are now ready to come back to proving section theorems. We start by proving a general section theorem.

As mentioned in the beginning of the section “Properties of Debuts”, we didn't get the nice equality $$\begin{aligned} \llbracket T \rrbracket \subseteq A \text{ and } \{T < \infty\} = \pi(A)\end{aligned}$$ as we got in Measurable Section theorem, but instead obtained an approximation (7). The condition $\llbracket T_\varepsilon \rrbracket \subseteq A$ makes sure that $\{T_\varepsilon < \infty\} \subseteq \pi(A)$, and the $\mathbb{P}\left[\pi(A)\right] \le \mathbb{P}(T_\varepsilon < \infty) + \varepsilon$ part ensures that the measure of the difference $\mathbb{P}\left[ \pi(A) \setminus \{T_\varepsilon < \infty\} \right]$ of these two events can be made as small as desired.

To see that it is not always possible to choose a stopping time $T$ that satisfies (8) if $A \in \mathscr{O}$, we will construct a filtration $\{\mathcal{F}_t\}_{t \ge 0}$ and choose a set $A \in \mathscr{O}$ that forces $\pi(A) \setminus \{T < \infty\} \neq \varnothing$ for every stopping time $T$ (the argument is from (Almost Sure blog)).

To this end, let $\tau : \Omega \to (0, \infty)$ be a random variable such that $\mathbb{P}\{\tau < t\} > 0$ for all $t > 0.$ For example, let $\tau$ be such that its distribution is uniform on $(0,1).$ Let $\{\mathcal{F}_t\}_{t \ge 0}$ be the completed filtration such that $\mathcal{F}_t$ is generated by $\{\{\tau \le s\} \,:\, s \le t\}$, and thus $\tau$ becomes a stopping time. Let $A = \rrbracket 0, \tau \llbracket\,,$ which we know from Lemma 8 belongs to $\mathscr{O}.$ Note that $\mathbb{P}\{\pi(A)\} = 1$ by our construction.

It is easy to see that $\mathcal{F}_t$ is trivial when restricted to $\{\tau > t\}$, i.e., contains only sets of measure $0$ or $1.$ So, every $\mathcal{F}_t$-measurable random variable is a.s. constant on the event $\{\tau > t\}.$ Therefore, any stopping time $T$ is deterministic on the event $\{T < \tau\}.$ So, if $\llbracket T \rrbracket \subseteq A$, we have

a.s. for some fixed $s > 0.$ But then A similar argument can be used to show that it is not possible to choose a predictable stopping time $T$ that satisfies (8) if $A \in \mathscr{P}.$

Applications of Section Theorems

Recall the measurable graph theorem (Theorem 8 in part 1) which implies that a map $T : \Omega \to [0, \infty]$ is measurable if and only if its graph $\llbracket T \rrbracket$ is measurable. We also have the following neat result:

Suppose $X$ and $Y$ are measurable processes such that

for each $\mathcal{F}$-measurable $T : \Omega \to [0, \infty].$ Then what can we say about $X$ and $Y?$ We will show that $X$ and $Y$ are indistinguishable! To this end, let It is measurable since $X$ and $Y$ are measurable and due to our completeness assumption. We want to show that $\mathbb{P}\{\pi(F)\} = 0$, so suppose to the contrary that $\mathbb{P}\{\pi(F)\} > 0.$ Any $\mathcal{F}$-measurable $T : \Omega \to [0, \infty]$ satisfying $\llbracket T \rrbracket \subseteq F$ must also satisfy $X_T \mathbf{1}_{\{T < \infty\}} \neq Y_T \mathbf{1}_{\{T < \infty\}}.$ Now Measurable Section theorem (Theorem 10) says there exists a random variable $T : \Omega \to [0,\infty]$ such that $\llbracket T \rrbracket \subseteq F$ and $\{T < \infty\} = \pi(F)$, and therefore contradicting our initial assumption.

Remarkably, we have similar results for optional and predictable processes.

Projection Theorems

To motivate optional and predictable projections, let us start with a fundamental problem in filtering theory: Assume an underlying complete probability space $(\Omega, \mathcal{F}, \mathbb{P}).$ There is an underlying signal $X = \{X_t\} _ {t \ge 0}$ which is modelled as a stochastic process and which we are interested in studying. Our observation process has noise and therefore instead of observing $X$ we observe a process $Y = \{Y_t\} _ {t \ge 0}$ such that

for Wiener process $W = \{W_t\}_{t \ge 0}.$ Therefore, it makes sense to define $\mathbb{F} = \{\mathcal{F}_t\} _ {t \ge 0}$ to be the minimal augmented filtration generated by $Y.$ The problem, then, is to compute an estimate for $X$ based on the observable data at time $t$, viz. $\mathcal{F}_t.$

We could look at $$\begin{aligned} Z_t := \mathbb{E} \left( X_t \mid \mathcal{F}_t \right),\end{aligned}$$ as an estimate for $X_t$ at each time $t \ge 0.$ The process $Z = \{Z_t\}_{t \ge 0}$ is, of course, adapted. However, since conditional expectation is defined only up to $\mathbb{P}$-a.s., what version of $Z$ should we choose? (9) does not fix the paths of the process $Z$, which requires specifying its values at the uncountable set of time in $[0, \infty).$ We would be very lucky if it were possible for us to choose a version of $Z$ such that (9) holds not only for all $t \in [0, \infty)$ but also for all finite stopping times. And indeed this is possible! This is part of the statement of the optional projection theorem.

On the other hand, if we want an estimate of $X$ based on observable data before time $t$, then our estimate would be

where recall $\mathcal{F}_{t-} := \sigma\left(\bigcup _ {s < t} \mathcal{F}_s\right).$ Again it is possible to choose a version of $Z$ such that this equality holds not only for all $t \in [0, \infty)$ but also for all finite predictable stopping times, and this is part of the statement of the predictable projection theorem.

Remarks:

1. Taking expectations in (10) we get $$\begin{aligned} \mathbb{E} \left(X_T \mathbf{1}_{\{T < \infty\}}\right) = \mathbb{E} \left(X_T^o \mathbf{1}_{\{T < \infty\}}\right)\end{aligned}$$ for all stopping times $T \in \mathfrak{S}.$ Now suppose (12) holds for all stopping times $T \in \mathfrak{S}.$ Fix an arbitrary stopping time $S \in \mathfrak{S}$ and a set $A \in \mathcal{F}_S$, and write (12) for $T = S_A$ to get

   $$\begin{aligned} \mathbb{E} \left(X_S \mathbf{1}_{\{S < \infty\}} \mathbf{1}_A\right) = \mathbb{E} \left(X_S^o \mathbf{1}_{\{S < \infty\}} \mathbf{1}_A\right).\end{aligned}$$
   But this immediately implies (10) for stopping time $S.$ Therefore, requiring (10) to hold for all stopping times is equivalent to requiring (12) to hold for for all stopping times.

2. Similarly, requiring (11) to hold for all predictable stopping times is equivalent to requiring $$\begin{aligned} \mathbb{E}\left( X_T \mathbf{1}_{\{T < \infty\}} \right) = \mathbb{E}\left(X^p_T \mathbf{1}_{\{T < \infty\}}\right)\end{aligned}$$ to hold for all predictable stopping times.

3. Operators $^o$ and $^p$ are linear operators, i.e., for bounded, measurable processes $X,Y$ and $a,b \in \mathbb R$,

   $$\begin{aligned} (aX + bY)^o &= a X^o + bY^o, \text{ and} \\ (aX + bY)^p &= a X^p + bY^p,\end{aligned}$$
   as can be easily verified by using the properties of conditional expectation. The formulation of (12) and (13) makes it obvious that $\left(X^o\right)^o = X^o$ and $\left(X^p\right)^p = X^p.$ This explains the terminology of “projection”.

We end our discussion with a result on time change. We call it a “time change” because given an adapted increasing process $A$ with right-continuous paths and $A_0 = 0,$ we imagine a clock which runs according to $A$ in the sense that at time $t$ this clock shows time $A_t.$

References

• Dellacherie, Claude. Capacités et processus stochastiques, Springer-Verlag, 1972.

• Dellacherie, Claude and Meyer, Paul-André. Probabilities and Potential, North-Holland Publishing Company, 1978.

• He, Sheng-wu and Wang, Jia-gang and Yan, Jia-an. Semimartingale Theory and Stochastic Calculus, CRC Press, 1992.

• Karatzas, Ioannis and Shreve, Steven. Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics Volume 113, Springer-Verlag New York, 1998.

• Karatzas, Ioannis and Shreve, Steven. Methods of Mathematical Finance, Probability Theory and Stochastic Modelling Volume 39, Springer-Verlag New York, 1998.

• Le Gall, Jean-François. Brownian Motion, Martingales, and Stochastic Calculus, Graduate Texts in Mathematics Volume 274, Springer International Publishing, 2016.

• Lowther, George. Almost Sure blog.