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}}.

(more…)

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21 January 19

Pathwise Properties of Optional and Predictable Projections

Recall that the the optional and predictable projections of a process are defined, firstly, by a measurability property and, secondly, by their values at stopping times. Namely, the optional projection is measurable with respect to the optional sigma-algebra, and its value is defined at each stopping time by a conditional expectation of the original process. Similarly, the predictable projection is measurable with respect to the predictable sigma-algebra and its value at each predictable stopping time is given by a conditional expectation. While these definitions can be powerful, and many properties of the projections follow immediately, they say very little about the sample paths. Given a stochastic process X defined on a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})} with optional projection {{}^{\rm o}\!X} then, for each {\omega\in\Omega}, we may be interested in the sample path {t\mapsto{}^{\rm o}\!X_t(\omega)}. For example, is it continuous, right-continuous, cadlag, etc? Answering these questions requires looking at {{}^{\rm o}\!X_t(\omega)} simultaneously at the uncountable set of times {t\in{\mathbb R}^+}, so the definition of the projection given by specifying its values at each individual stopping time, up to almost-sure equivalence, is not easy to work with. I did establish some of the basic properties of the projections in the previous post, but these do not say much regarding sample paths.

I will now establish the basic properties of the sample paths of the projections. Although these results are quite advanced, most of the work has already been done in these notes when we established some pathwise properties of optional and predictable processes in terms of their behaviour along sequences of stopping times, and of predictable stopping times. So, the proofs in this post are relatively simple and will consist of applications of these earlier results.

Before proceeding, let us consider what kind of properties it is reasonable to expect of the projections. Firstly, it does not seem reasonable to expect the optional projection {{}^{\rm o}\!X} or the predictable projection {{}^{\rm p}\!X} to satisfy properties not held by the original process X. Therefore, in this post, we will be concerned with the sample path properties which are preserved by the projections. Consider a process with constant paths. That is, {X_t=U} at all times t, for some bounded random variable U. This has about as simple sample paths as possible, so any properties preserved by the projections should hold for the optional and predictable projections of X. However, we know what the projections of this process are. Letting M be the martingale defined by {M_t={\mathbb E}[U\,\vert\mathcal F_t]} then, assuming that the underlying filtration is right-continuous, M has a cadlag modification and, furthermore, this modification is the optional projection of X. So, assuming that the filtration is right-continuous, the optional projection of X is cadlag, meaning that it is right-continuous and has left limits everywhere. So, we can hope that the optional projection preserves these properties. If the filtration is not right-continuous, then M need not have a cadlag modification, so we cannot expect optional projection to preserve right-continuity in this case. However, M does still have a version with left and right limits everywhere, which is the optional projection of X. So, without assuming right-continuity of the filtration, we may still hope that the optional projection preserves the existence of left and right limits of a process. Next, the predictable projection is equal to the left limits, {{}^{\rm p}\!X_t=M_{t-}}, which is left-continuous with left and right limits everywhere. Therefore, we can hope that predictable projections preserve left-continuity and the existence of left and right limits. The existence of cadlag martingales which are not continuous, such as the compensated Poisson process, imply that optional projections do not generally preserve left-continuity and the predictable projection does not preserve right-continuity.

Recall that I previously constructed a version of the optional projection and the predictable projection for processes which are, respectively, right-continuous and left-continuous. This was done by defining the projection at each deterministic time and, then, enforcing the respective properties of the sample paths. We can use the results in those posts to infer that the projections do indeed preserve these properties, although I will now more direct proofs in greater generality, and using the more general definition of the optional and predictable projections.

We work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}. As usual, we say that the sample paths of a process satisfy any stated property if they satisfy it up to evanescence. Since integrability conditions will be required, I mention those now. Recall that a process X is of class (D) if the set of random variables {X_\tau}, over stopping times {\tau}, is uniformly integrable. It will be said to be locally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{\{\tau_n > 0\}}1_{[0,\tau_n]}X} is of class (D) for each n. Similarly, it will be said to be prelocally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{[0,\tau_n)}X} is of class (D) for each n.

Theorem 1 Let X be pre-locally of class (D), with optional projection {{}^{\rm o}\!X}. Then,

  • if X has left limits, so does {{}^{\rm o}\!X}.
  • if X has right limits, so does {{}^{\rm o}\!X}.

Furthermore, if the underlying filtration is right-continuous then,

  • if X is right-continuous, so is {{}^{\rm o}\!X}.
  • if X is cadlag, so is {{}^{\rm o}\!X}.

(more…)

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}.

(more…)

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)

(more…)

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}.

(more…)

25 October 16

Optional Projection For Right-Continuous Processes

In filtering theory, we have a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})} and a signal process {\{X_t\}_{t\in{\mathbb R}_+}}. The sigma-algebra {\mathcal{F}_t} represents the collection of events which are observable up to and including time t. The process X is not assumed to be adapted, so need not be directly observable. For example, we may only be able to measure an observation process {Z_t=X_t+\epsilon_t}, which incorporates some noise {\epsilon_t}, and generates the filtration {\mathcal{F}_t}, so is adapted. The problem, then, is to compute an estimate for {X_t} based on the observable data at time t. Looking at the expected value of X conditional on the observable data, we obtain the following estimate for X at each time {t\in{\mathbb R}_+},

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

The process Y is adapted. However, as (1) only defines Y up to a zero probability set, it does not give us the paths of Y, which requires specifying its values simultaneously at the uncountable set of times in {{\mathbb R}_+}. Consequently, (1) does not tell us the distribution of Y at random times. So, it is necessary to specify a good version for Y.

Optional projection gives a uniquely defined process which satisfies (1), not just at every time t in {{\mathbb R}_+}, but also at all stopping times. The full theory of optional projection for jointly measurable processes requires the optional section theorem. As I will demonstrate, in the case where X is right-continuous, optional projection can be done by more elementary methods.

Throughout this post, it will be assumed that the underlying filtered probability space satisfies the usual conditions, meaning that it is complete and right-continuous, {\mathcal{F}_{t+}=\mathcal{F}_t}. Stochastic processes are considered to be defined up to evanescence. That is, 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 on X. Often, to avoid such issues, optional projection is defined for uniformly bounded processes. For a bit more generality, I will relax this requirement a bit and use prelocal integrability. Recall that, in these notes, a process X is prelocally integrable if there exists a sequence of stopping times {\tau_n} increasing to infinity and such that

\displaystyle  1_{\{\tau_n > 0\}}\sup_{t < \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 (Optional Projection) Let X be a right-continuous and prelocally integrable process. Then, there exists a unique right-continuous process Y satisfying (1).

Uniqueness is immediate, as (1) determines Y, almost-surely, at each fixed time, and this is enough to uniquely determine right-continuous processes up to evanescence. Existence of Y is the important part of the statement, and the proof will be left until further down in this post.

The process defined by Theorem 1 is called the optional projection of X, and is denoted by {{}^{\rm o}\!X}. That is, {{}^{\rm o}\!X} is the unique right-continuous process satisfying

\displaystyle  {}^{\rm o}\!X_t={\mathbb E}[X_t\;\vert\mathcal{F}_t]{\rm\ \ (a.s.)} (3)

for all times t. In practise, the process X will usually not just be right-continuous, but will also have left limits everywhere. That is, it is cadlag.

Theorem 2 Let X be a cadlag and prelocally integrable process. Then, its optional projection is cadlag.

A simple example of optional projection is where {X_t} is constant in t and equal to an integrable random variable U. Then, {{}^{\rm o}\!X_t} is the cadlag version of the martingale {{\mathbb E}[U\;\vert\mathcal{F}_t]}. (more…)

21 October 16

The Projection Theorems

Back when I first started this series of posts on stochastic calculus, the aim was to write up the notes which I began writing while learning the subject myself. The idea behind these notes was to give a more intuitive and natural, yet fully rigorous, approach to stochastic integration and semimartingales than the traditional method. The stochastic integral and related concepts were developed without requiring advanced results such as optional and predictable projection or the Doob-Meyer decomposition which are often used in traditional approaches. Then, the more advanced theory of semimartingales was developed after stochastic integration had already been established. This now complete! The list of subjects from my original post have now all been posted. Of course, there are still many important areas of stochastic calculus which are not adequately covered in these notes, such as local times, stochastic differential equations, excursion theory, etc. I will now focus on the projection theorems and related results. Although these are not required for the development of the stochastic integral and the theory of semimartingales, as demonstrated by these notes, they are still very important and powerful results invaluable to much of the more advanced theory of continuous-time stochastic processes. Optional and predictable projection are often regarded as quite advanced topics beyond the scope of many textbooks on stochastic calculus. This is because they require some descriptive set theory and, in particular, some understanding of analytic sets. The level of knowledge required for applications to stochastic calculus is not too great though, and I aim to give complete proofs of the projection theorems in these notes. However, the proofs of these theorems do require ideas which are not particularly intuitive from the viewpoint of stochastic calculus, and hence the desire to avoid them in the initial development of the stochastic integral. The theory of semimartingales and stochastic integration will not used at all in the series of posts on the projection theorems, and all that will be required from these stochastic calculus notes are the initial posts on filtrations and processes. I will also mention quasimartingales, although only the definition and very basic properties will be required.

The subjects related to the projection theorems which I will cover are,

  • The Debut Theorem. I have already covered the debut theorem for right-continuous processes. This is a special case of the more general result which applies to arbitrary progressively measurable processes.
  • The Optional and Predictable Section Theorems. These very powerful results state that optional processes are determined, up to evanescence, by their values at stopping times and, similarly, predictable processes are determined by their values at predictable stopping times.
  • Optional and Predictable Projection. This forms the core of these sequence of posts, and follows in a straightforward way from the section theorems. As the section theorems are required to prove them, the projection theorems are also regarded as an advanced topic. However, for right-continuous and left-continuous processes it is possible to construct respectively the optional and predictable projections in a more elementary and natural way, without involving the section theorems.
  • Dual Optional and Predictable Projection. The dual projections are, as the name suggests, dual to the optional and predictable projections mentioned above. These apply to increasing integrable processes or, more generally, to processes with integrable variation. For a process X, the dual projections can be thought of as the optional and predictable projections applied to the differential {dX}.
  • The Doléans Measure. The Doléans measure can be defined for class (D) submartingales and, applied to the square of a martingale, can be used to construct the stochastic integral for square integrable martingales. Although this does not involve the projection theorems, the Doléans measure in conjunction with dual predictable projection gives a slick proof of the Doob-Meyer decomposition. The Doléans measure also exists for quasimartingales and, similarly, the Doob-Meyer decomposition can be extended to such processes.

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