Almost Sure

8 February 19

Dual Projections

The optional and predictable projections of stochastic processes have corresponding dual projections, which are the subject of this post. I will be concerned with their initial construction here, and show that they are well-defined. The study of their properties will be left until later. In the discrete time setting, the dual projections are relatively straightforward, and can be constructed by applying the optional and predictable projection to the increments of the process. In continuous time, we no longer have discrete time increments along which we can define the dual projections. In some sense, they can still be thought of as projections of the infinitesimal increments so that, for a process A, the increments of the dual projections {A^{\rm o}} and {A^{\rm p}} are determined from the increments {dA} of A as

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dA^{\rm o}={}^{\rm o}(dA),\smallskip\\ &\displaystyle dA^{\rm p}={}^{\rm p}(dA). \end{array} (1)

Unfortunately, these expressions are difficult to make sense of in general. In specific cases, (1) can be interpreted in a simple way. For example, when A is differentiable with derivative {\xi}, so that {dA=\xi dt}, then the dual projections are given by {dA^{\rm o}={}^{\rm o}\xi dt} and {dA^{\rm p}={}^{\rm p}\xi dt}. More generally, if A is right-continuous with finite variation, then the infinitesimal increments {dA} can be interpreted in terms of Lebesgue-Stieltjes integrals. However, as the optional and predictable projections are defined for real valued processes, and {dA} is viewed as a stochastic measure, the right-hand-side of (1) is still problematic. This can be rectified by multiplying by an arbitrary process {\xi}, and making use of the transitivity property {{\mathbb E}[\xi\,{}^{\rm o}(dA)]={\mathbb E}[({}^{\rm o}\xi)dA]}. Integrating over time gives the more meaningful expressions

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle {\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm o}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm o}\xi\,dA\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm p}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm p}\xi\,dA\right]. \end{array}

In contrast to (1), these equalities can be used to give mathematically rigorous definitions of the dual projections. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and processes are identified whenever they are equal up to evanescence. The terminology `raw IV process‘ will be used to refer to any right-continuous integrable process whose variation on the whole of {{\mathbb R}^+} has finite expectation. The use of the word `raw’ here is just to signify that we are not requiring the process to be adapted. Next, to simplify the expressions, I will use the notation {\xi\cdot A} for the integral of a process {\xi} with respect to another process A,

\displaystyle  \xi\cdot A_t\equiv\xi_0A_0+\int_0^t\xi\,dA.

Note that, whereas the integral {\int_0^t\xi\,dA} is implicitly taken over the range {(0,t]} and does not involve the time-zero value of {\xi}, I have included the time-zero values of the processes in the definition of {\xi\cdot A}. This is not essential, and could be excluded, so long as we were to restrict to processes starting from zero. The existence and uniqueness (up to evanescence) of the dual projections is given by the following result.

Theorem 1 (Dual Projections) Let A be a raw IV process. Then,

  • There exists a unique raw IV process {A^{\rm o}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm o}_\infty\right]={\mathbb E}\left[{}^{\rm o}\xi\cdot A_\infty\right] (2)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm o}} as the dual optional projection of A.

  • There exists a unique raw IV process {A^{\rm p}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm p}_\infty\right]={\mathbb E}\left[{}^{\rm p}\xi\cdot A_\infty\right] (3)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm p}} as the dual predictable projection of A.

Furthermore, if A is nonnegative and increasing then so are {A^{\rm o}} and {A^{\rm p}}.

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20 January 19

Properties of Optional and Predictable Projections

Having defined optional and predictable projections in an earlier post, I now look at their basic properties. The first nontrivial property is that they are well-defined in the first place. Recall that existence of the projections made use of the existence of cadlag modifications of martingales, and uniqueness relied on the section theorems. By contrast, once we accept that optional and predictable projections are well-defined, everything in this post follows easily. Nothing here requires any further advanced results of stochastic process theory.

Optional and predictable projections are similar in nature to conditional expectations. Given a probability space {(\Omega,\mathcal F,{\mathbb P})} and a sub-sigma-algebra {\mathcal G\subseteq\mathcal F}, the conditional expectation of an ({\mathcal F}-measurable) random variable X is a {\mathcal G}-measurable random variable {Y={\mathbb E}[X\,\vert\mathcal G]}. This is defined whenever the integrability condition {{\mathbb E}[\lvert X\rvert\,\vert\mathcal G] < \infty} (a.s.) is satisfied, only depends on X up to almost-sure equivalence, and Y is defined up to almost-sure equivalence. That is, a random variable {X^\prime} almost surely equal to X has the same conditional expectation as X. Similarly, a random variable {Y^\prime} almost-surely equal to Y is also a version of the conditional expectation {{\mathbb E}[X\,\vert\mathcal G]}.

The setup with projections of stochastic processes is similar. We start with a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and a (real-valued) stochastic process is a map

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

which we assume to be jointly-measurable. That is, it is measurable with respect to the Borel sigma-algebra {\mathcal B({\mathbb R})} on the image, and the product sigma-algebra {\mathcal B({\mathbb R})\otimes\mathcal F} on the domain. The optional and predictable sigma-algebras are contained in the product,

\displaystyle  \mathcal P\subseteq\mathcal O\subseteq \mathcal B({\mathbb R})\otimes\mathcal F.

We do not have a reference measure on {({\mathbb R}^+\times\Omega,\mathcal B({\mathbb R})\otimes\mathcal F)} in order to define conditional expectations with respect to {\mathcal O} and {\mathcal P}. However, the optional projection {{}^{\rm o}\!X} and predictable projection {{}^{\rm p}\!X} play similar roles. Assuming that the necessary integrability properties are satisfied, then the projections exist. Furthermore, the projection only depends on the process X up to evanescence (i.e., up to a zero probability set), and {{}^{\rm o}\!X} and {{}^{\rm p}\!X} are uniquely defined up to evanescence.

In what follows, we work with respect to a complete filtered probability space. Processes are always only considered up to evanescence, so statements involving equalities, inequalities, and limits of processes are only required to hold outside of a zero probability set. When we say that the optional projection of a process exists, we mean that the integrability condition in the definition of the projection is satisfied. Specifically, that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal F_\tau]} is almost surely finite. Similarly for the predictable projection.

The following lemma gives a list of initial properties of the optional projection. Other than the statement involving stopping times, they all correspond to properties of conditional expectations.

Lemma 1

  1. X is optional if and only if {{}^{\rm o}\!X} exists and is equal to X.
  2. If the optional projection of X exists then,
    \displaystyle  {}^{\rm o}({}^{\rm o}\!X)={}^{\rm o}\!X. (1)
  3. If the optional projections of X and Y exist, and {\lambda,\mu} are {\mathcal{F}_0}-measurable random variables, then,
    \displaystyle  {}^{\rm o}(\lambda X+\mu Y) = \lambda\,^{\rm o}\!X + \mu\,^{\rm o}Y. (2)
  4. If the optional projection of X exists and U is an optional process then,
    \displaystyle  {}^{\rm o}(UX) = U\,^{\rm o}\!X (3)
  5. If the optional projection of X exists and {\tau} is a stopping time then, the optional projection of the stopped process {X^\tau} exists and,
    \displaystyle  1_{[0,\tau]}{}^{\rm o}(X^\tau)=1_{[0,\tau]}{}^{\rm o}\!X. (4)
  6. If {X\le Y} and the optional projections of X and Y exist then, {{}^{\rm o}\!X\le{}^{\rm o}Y}.

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23 December 18

Projection in Discrete Time

It has been some time since my last post, but I am continuing now with the stochastic calculus notes on optional and predictable projection. In this post, I will go through the ideas in the discrete-time situation. All of the main concepts involved in optional and predictable projection are still present in discrete time, but the theory is much simpler. It is only really in continuous time that the projection theorems really show their power, so the aim of this post is to motivate the concepts in a simple setting before generalising to the full, continuous-time situation. Ideally, this would have been published before the posts on optional and predictable projection in continuous time, so it is a bit out of sequence.

We consider time running through the discrete index set {{\mathbb Z}^+=\{0,1,2,\ldots\}}, and work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_n\}_{n=0,1,\ldots},{\mathbb P})}. Then, {\mathcal{F}_n} is used to represent the collection of events observable up to and including time n. Stochastic processes will all be real-valued and defined up to almost-sure equivalence. That is, processes X and Y are considered to be the same if {X_n=Y_n} almost surely for each {n\in{\mathbb Z}^+}. The projections of a process X are defined as follows.

Definition 1 Let X be a measurable process. Then,

  1. the optional projection, {{}^{\rm o}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_n]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm o}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_n]. (1)
  2. the predictable projection, {{}^{\rm p}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_{n-1}]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm p}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_{n-1}]. (2)

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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}.

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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…)

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 cadlag adapted processes.
  • 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.

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