Almost Sure

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}.

When the integrability criterion of Theorem 1 or Theorem 2 is satisfied then I will simply say that the optional projection {{}^{\rm o}\!X} or, respectively, the predictable projection {{}^{\rm p}\!X} exists. I will give the proofs of these theorems in a moment but, first, state some equivalent forms.

In Theorems 1 and 2, we can restrict the stopping times to be bounded. This avoids having to multiply by the term {1_{\{\tau < \infty\}}}, which was required to avoid sampling the processes at time {\tau=\infty}.

Theorem 3 The optional projection of a measurable process X exists if and only if {{\mathbb E}[\lvert X_\tau\rvert\,\vert\mathcal{F}_{\tau}]} is almost surely finite for each bounded stopping time {\tau}. Then, {{}^{\rm o}\!X} is the unique optional process satisfying

\displaystyle  {}^{\rm o}\!X_\tau={\mathbb E}[X_\tau\,\vert\mathcal{F}_\tau] (3)

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

Proof: From the definition, if the optional projection exists then {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]} is almost surely finite for every stopping time {\tau} and, in particular, for every bounded stopping time. Conversely, for any fixed time T, {\mathcal{F}_\tau} and {\mathcal{F}_{\tau\wedge T}} agree on the event {\{\tau \le T\}}. So,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle 1_{\{\tau\le T\}}{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau] &\displaystyle = 1_{\{\tau\le T\}}{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_{\tau\wedge T}]\smallskip\\ &\displaystyle= {\mathbb E}[1_{\{\tau\le T\}}\lvert X_{\tau\wedge T}\rvert\,\vert\mathcal{F}_{\tau\wedge T}] \end{array}

is almost surely finite. Letting T increase to infinity, {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]} is almost surely finite as required.

If the optional projection exists, (3) is the same as (1) for bounded stopping times. So, (3) holds and, by optional section, uniquely defines {{}^{\rm o}\!X} up to evanescence. ⬜

Predictable projection can be defined similarly.

Theorem 4 The predictable projection of X exists if and only if {{\mathbb E}[\lvert X_\tau\rvert\,\vert\mathcal{F}_{\tau-}]} is almost surely finite for each bounded predictable stopping time {\tau}. Then, {{}^{\rm p}\!X} is the unique predictable process satisfying

\displaystyle  {}^{\rm p}\!X_\tau={\mathbb E}[X_\tau\,\vert\mathcal{F}_{\tau-}] (4)

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

Proof: The proof is exactly the same as that given for Theorem 3, only with `stopping time’ replaced by `predictable stopping time’ and {\mathcal{F}_\tau} replaced by {\mathcal{F}_{\tau-}}. ⬜

Next, restricting to processes whose values at stopping times are integrable, optional projection can be defined by taking expectations rather than conditional expectations.

Theorem 5 Let X be a measurable process such that {1_{\{\tau < \infty\}}X_\tau} is integrable for each stopping time {\tau}. Then, the optional projection, {{}^{\rm o}\!X}, exists and is the unique optional process such that, for each stopping time {\tau}, {1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau} is integrable and satisfies

\displaystyle  {\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}X_\tau]. (5)

Proof: If {1_{\{\tau < \infty\}}X_\tau} is integrable, then so is {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]}, so the optional projection exists and, taking expectations of (1) gives (5).

Conversely, suppose that (5) holds for every stopping time {\tau}. For any {A\in\mathcal{F}_\tau}, let {\tau_A} denote the stopping time equal to {\tau} on the event A and {\infty} otherwise. Using (5),

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {\mathbb E}[1_A1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau] &\displaystyle = {\mathbb E}[1_{\{\tau_A < \infty\}}{}^{\rm o}\!X_{\tau_A}]\smallskip\\ &=\displaystyle {\mathbb E}[1_{\{\tau_A < \infty\}}X_{\tau_A}]\smallskip\\ &=\displaystyle {\mathbb E}[1_A1_{\{\tau < \infty\}}X_\tau]. \end{array}

As {1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau} is {\mathcal{F}_\tau}-measurable, by definition of conditional expectations this is equivalent to (1). ⬜

Similarly, for predictable projection.

Theorem 6 Let X be a measurable process such that {1_{\{\tau < \infty\}}X_\tau} is integrable for each predictable stopping time {\tau}. Then, the predictable projection, {{}^{\rm p}\!X}, exists and is the unique predictable process such that, for each predictable stopping time {\tau}, {1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau} is integrable and satisfies

\displaystyle  {\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}X_\tau]. (6)

Proof: The proof is exactly the same as that given for Theorem 5, with `stopping time’ replaced by `predictable stopping time’ and {\mathcal{F}_\tau} replaced by {\mathcal{F}_{\tau-}}. The fact that {\tau_A} is a predictable stopping time when {A\in\mathcal{F}_{\tau-}} is also required. ⬜

Existence of optional and predictable projections

If they exist, then the uniqueness of optional and predictable projections is an immediate consequence of the section theorems. This is because (1) and (2) almost surely determine the values of the optional process {{}^{\rm o}\!X} and predictable process {{}^{\rm p}\!X} at every stopping time and, respectively, at every predictable stopping time, uniquely identifying them up to evanescence. In contrast, proving existence requires a bit more work, but only involves the basic theory of continuous martingales. I start with the projections of constant processes. Under the usual conditions, this makes use of the existence of cadlag versions of martingales. As I am not assuming right-continuity of the filtration, we need to relax the cadlag condition a bit to include martingales which are right-continuous outside of a countable set of times.

Lemma 7 Let X be the constant process {X_t=U}, for an integrable random variable U. Let M be a version of the martingale

\displaystyle  M_t={\mathbb E}[U\,\vert\mathcal{F}_t]

which has right and left limits everywhere, and is right-continuous outside of a countable set {S\subseteq{\mathbb R}_+}. Such versions always exist.

Then, the optional and predictable projections of X exist, and are given by

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle {}^{\rm o}\!X_t=M_t,\smallskip\\ &\displaystyle {}^{\rm p}\!X_t=M_{t-}. \end{array}

Proof: Start by considering a simple stopping time, {\tau}, taking values in {A\cup\{\infty\}} for a finite set {A\subset{\mathbb R}_+}. Then,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {\mathbb E}[1_{\{\tau < \infty\}}M_\tau] &\displaystyle= {\mathbb E}\left[\sum_{t\in A}M_t\right]={\mathbb E}\left[\sum_{t\in A}U\right]\smallskip\\ &\displaystyle={\mathbb E}[1_{\{\tau < \infty\}}U]. \end{array}

As in the proof of Theorem 5 above, this proves that (1) or, equivalently,

\displaystyle  1_{\{\tau < \infty\}}M_\tau={\mathbb E}\left[1_{\{\tau < \infty\}}U\,\vert\mathcal{F}_\tau\right] (7)

is satisfied at simple stopping times. Now, for an arbitrary stopping time {\tau}, let {\tau_n} be a sequence of simple stopping times decreasing to {\tau} and with {\tau_n=\tau} eventually whenever {\tau\in S}. For example, let {S_n} be a sequence of finite subsets of {{\mathbb R}_+} increasing to a dense subset of {{\mathbb R}_+} containing S, and set

\displaystyle  \tau_n=\inf\left\{t\in S_n\colon t\ge\tau\right\}.

As the conditional expectations given by (7) are uniformly integrable, and {M_{\tau_n}\rightarrow M_\tau}, we can take limits,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {\mathbb E}[1_{\{\tau < \infty\}} M_\tau] &\displaystyle= \lim_{n\rightarrow\infty}{\mathbb E}[1_{\{\tau_n < \infty\}}M_{\tau_n}]=\lim_{n\rightarrow\infty}{\mathbb E}[1_{\{\tau_n < \infty\}}U]\smallskip\\ &\displaystyle={\mathbb E}[1_{\{\tau < \infty\}}U]. \end{array}

So, as in the proof of Theorem 5, {{}^{\rm o}\!X=M} as required.

Next, if {\tau} is a predictable stopping time then choose a sequence of stopping times {\tau_n} announcing {\tau}. So, {M_{\tau_n}\rightarrow M_{\tau-}} and, for each fixed time T,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {\mathbb E}[1_{\{\tau\le T\}}M_{\tau-}] &\displaystyle=\lim_{n\rightarrow\infty}{\mathbb E}[1_{\{\tau_n\le T\}}M_{\tau_n}] =\lim_{n\rightarrow\infty}{\mathbb E}[1_{\{\tau_n\le T\}}U]\smallskip\\ &\displaystyle = {\mathbb E}[1_{\{\tau\le T\}}U]. \end{array}

Letting T increase to infinity gives {{\mathbb E}[1_{\{\tau < \infty\}}M_{\tau-}]={\mathbb E}[1_{\{\tau < \infty\}}U]}, so {{}^{\rm p}\!X=M_-} as required. ⬜

The completion of the proof of existence of optional and predictable projections is now just an application of the monotone class theorem.

Proof of Theorem 1: Let {\mathcal{H}} denote the space of measurable processes, X, such that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]} is almost surely finite for every stopping time {\tau}, and let {\mathcal{H}_0} denote the set of {X\in\mathcal{H}} such that there exists an optional process, {{}^{\rm o}\!X}, satisfying (1). We just need to show that {\mathcal{H}_0=\mathcal{H}}. The monotone class theorem implies this, so long as we can verify the following.

  1. There exists a pi-system {\mathcal{A}} generating the sigma-algebra {\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}} such that {1_A\in\mathcal{H}_0} for all {A\in\mathcal{A}}.
  2. {\mathcal{H}_0} is closed under finite linear combinations.
  3. For any sequence of nonnegative {X^n\in\mathcal{H}_0} increasing to a limit {X\in\mathcal{H}}, then {X\in\mathcal{H}_0}.

For the first statement above, let {\mathcal{A}} denote the products {A\times B} for {A\in\mathcal{B}({\mathbb R}_+)} and {B\in\mathcal{F}}. By Lemma 7, the constant process {X=1_B} has an optional projection. So, {1_{A\times B}} has optional projection {1_A{}^{\rm o}\!X}, and is in {\mathcal{H}_0}.

For the second statement above, it is clear that the optional projection of a linear combination of processes is equal to the linear combination of their optional projections.

For the third statement, consider a sequence of nonnegative {X^n\in\mathcal{H}_0} increasing to a limit {X\in\mathcal{H}}. Consider setting {{}^{\rm o}\!X=\sup_n{}^{\rm o}\!X^n}, which is optional. For a stopping time {\tau}, monotone convergence gives

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle 1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau &\displaystyle =\sup_n{\mathbb E}[1_{\{\tau < \infty\}}X^n_\tau\,\vert\mathcal{F}_\tau]\smallskip\\ &\displaystyle ={\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_\tau] \end{array}

almost surely. So, (1) is satisfied. As {X\in\mathcal{H}}, this also shows that {{}^{\rm o}\!X_\tau} is almost surely finite on {\{\tau < \infty\}} and, therefore, (1) will still be satisfied if {{}^{\rm o}\!X} is replaced by the real-valued optional process {1_{\{\lvert{}^{\rm o}\!X\rvert < \infty\}}{}^{\rm o}\!X}. ⬜

Proof of Theorem 2: This follows by an identical proof to the one just given for Theorem 1, with `optional’ replaced by `predictable’, `stopping time’ replaced by `predictable stopping time’, and {\mathcal{F}_\tau} replaced by {\mathcal{F}_{\tau-}}. ⬜

Notes

The approach to the projection theorems in this post is fairly standard, although there are some slight variations within the literature. For one thing, unlike many texts, I am not assuming that the usual conditions are satisfied. In particular, the filtration need not be right-continuous. The only complication that this leads to is that, in Lemma 7 above, restricting to cadlag modifications of the martingale M is not sufficient, and it is necessary to allow it to be non-right-continuous at a countable set of times.

Some texts on the subject restrict attention to bounded processes. I do not do this. Instead, for more generality, I use weak integrability requirements which are just sufficient to ensure that a real-valued projection exists. This approach is also used, for example, in the book Semimartingale Theory and Stochastic Calculus, 1992, by Shang-wu He, Jia-gang Wang and Jia-an Yan. In the other direction, for non-negative processes, the projections are often defined without any integrability requirements at all. This is because conditional expectations can be defined for all non-negative random variables, so equations (1) and (2) are well-defined for non-negative processes X. However, in this case, the projections can be infinite.

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