Almost Sure

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)

Equation (2) refers to {\mathcal{F}_{-1}} in the case {n=0}, and I take this to equal {\mathcal{F}_0} by convention. The optional projection is an important concept in filtering theory where X represents a process which is not directly observable. That is, it is not adapted, so that {X_n} is not {\mathcal{F}_n}-measurable. For example, we may only be able to estimate X through an observation process {Z_n=X_n+\epsilon_n} which includes some random noise {\epsilon_n}. If we only observe Z, so that it is used to generate the underlying filtration, the process X will not generally be adapted. The optional projection at any time is the expected value of the process conditional on the observable information up to and including that time, and the predictable projection is conditioned on the information observable strictly before that time.

A simple, but important, example is given by the optional projection of a constant process {X_n=U} for an integrable random variable U

\displaystyle  {}^{\rm o}\!X_n={\mathbb E}[U\,\vert\mathcal{F}_n]. (3)

This is the martingale with value converging to {{}^{\rm o}\!X_\infty={\mathbb E}[U\,\vert\mathcal{F}_\infty]}. In this example, the predictable projection is given simply by

\displaystyle  {}^{\rm p}\!X_n={}^{\rm o}\!X_{(n-1)_+}={\mathbb E}[U\,\vert\mathcal{F}_{n-1}].

In discrete time, a process X is optional if {X_n} is {\mathcal{F}_n}-measurable at each time n. So, `optional’ is just a synonym for `adapted’. The process is predictable if {X_n} is {\mathcal{F}_{n-1}}-measurable. Then, the optional projection of a process is clearly optional, and the predictable projection is predictable. Furthermore, a process is optional if and only if it is equal to its optional projection, {{}^{\rm o}\!X=X}, and is predictable if and only if it equals its predictable projection.

Optional and predictable projection are defined in terms of conditional expectations, and inherit their elementary properties. For example, linearity is straightforward,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {}^{\rm o}(\lambda X+\mu Y)&\displaystyle=\lambda\,^{\rm o}\!X+\mu\,^{\rm o}Y,\smallskip\\ \displaystyle {}^{\rm p}(\lambda X+\mu Y)&\displaystyle=\lambda\,^{\rm p}\!X+\mu\,^{\rm p}Y, \end{array} (4)

Here, {\lambda} and {\mu} are arbitrary real numbers or, more generally, {\mathcal{F}_0}-measurable random variables. Similarly,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {}^{\rm o}(ZX)&\displaystyle=Z\,^{\rm o}\!X,\smallskip\\ \displaystyle {}^{\rm p}(ZX)&\displaystyle=Z\,^{\rm p}\!X, \end{array} (5)

with the first equality holding whenever Z is optional and the second if it is predictable. As optional processes are equal to their own optional projection and predictable processes equal their predictable projection, projecting twice is the same as projecting once.

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {}^{\rm o}({}^{\rm o}\!X)&\displaystyle={}^{\rm o}\!X,\smallskip\\ \displaystyle {}^{\rm p}({}^{\rm p}\!X)&\displaystyle={}^{\rm p}\!X. \end{array} (6)

This justifies the terminology `projection’, as linear operators which, when applied twice give the same result as applying once. It is also straightforward to show that, whenever the predictable projection of a process X exists then,

\displaystyle  {}^{\rm p}({}^{\rm o}\!X)={}^{\rm o}({}^{\rm p}\!X) = {}^{\rm p}\!X. (7)

That {{}^{\rm p}\!X} equals {{}^{\rm p}({}^{\rm o}\!X)} is an immediate consequence of the tower law of iterated conditional expectations. That it also equals {{}^{\rm o}({}^{\rm p}\!X)} follows from the fact that predictable processes are also optional.

Equations (1) and (2) above can be generalised to arbitrary stopping times. A random time {\tau\colon\Omega\rightarrow{\mathbb Z}^+\cup\{\infty\}} is a stopping time if {\{\tau\le n\}\in\mathcal{F}_n} for each n, and is a predictable stopping time if {\{\tau\le n\}\in\mathcal{F}_{n-1}}. Note that, if {\tau} is a predictable stopping time then {(\tau-1)_+} is a stopping time, which will be denoted by {\tau-}. For any optional process X and stopping time {\tau}, the stopped process {X^\tau_n=X_{n\wedge\tau}} is again optional.

As in the continuous-time situation, the filtration can be extended to stopping times {\tau} by defining the sigma-algebra,

\displaystyle  \mathcal{F}_\tau=\left\{A\in\mathcal{F}_\infty\colon A\cap\{\tau\le n\}\in\mathcal{F}_n{\rm\ all\ }n=0,1,\ldots\right\}.

A straightforward application of the definition shows that for any stopping time {\tau} and optional process X then {1_{\{\tau < \infty\}}X_\tau} is {\mathcal{F}_\tau}-measurable. Similarly, if X is a predictable process and {\tau} a predictable stopping time, then {1_{\{\tau < \infty\}}X_\tau} is {\mathcal{F}_{\tau-}}-measurable. In important property of the definitions of optional and predictable projection, is that they extend naturally to sampling the process at a stopping time. Equation (8) below is similar to the optional sampling theorem for martingales and, in fact, is equivalent to it when {{}^{\rm o}\!X} is the martingale given by (3).

Theorem 2 Let X be a measurable process.

  1. If the optional projection exists then {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]} is almost-surely finite for each stopping time {\tau} and, almost surely,
    \displaystyle  1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau={\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_{\tau}]. (8)
  2. If the predictable projection exists then {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_{\tau-}]} is almost-surely finite for each predictable stopping time {\tau} and, almost surely,
    \displaystyle  1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau={\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_{\tau-}]. (9)

Proof: I make use of the fact that {\{\tau=n\}\in\mathcal{F}_\tau\cap\mathcal{F}_n}, and that the sigma-algebra {\mathcal{F}_\tau} coincides with {\mathcal{F}_n} on this event to derive,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle 1_{\{\tau=n\}}{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau] =&\displaystyle 1_{\{\tau=n\}}{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_n]\smallskip\\ =&\displaystyle{\mathbb E}[1_{\{\tau=n\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_n]\smallskip\\ =&1_{\{\tau=n\}}{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_n]. \end{array} (10)

If {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_n]} is almost surely finite then summing (10) over n shows that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_\tau]} is almost surely finite. Applying (10) with X in place of {\lvert X\rvert} gives (8).

Moving on to the second statement, if {\tau} is a predictable stopping time then (10) holds with {\mathcal{F}_{\tau-}} and {\mathcal{F}_{n-1}} in place of {\mathcal{F}_\tau} and {\mathcal{F}_n} respectively. If the predictable projection exists then, similarly as with the optional projection, we immediately get that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal{F}_{\tau-}]} is almost surely finite and (9) holds. ⬜

For processes which are dominated by an integrable random variable, the projections can be defined without using conditional expectations at all. Instead, it is only necessary to look at the expected values of the process at stopping times.

Theorem 3 If {\sup_{n\ge0}\lvert X_n\rvert} is integrable then the optional and predictable projections of X exist. The optional projection is the unique optional process such that {1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau} is integrable and

\displaystyle  {\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}X_\tau] (11)

for all stopping times {\tau}. The predictable projection is the unique predictable process such that {1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau} is integrable and

\displaystyle  {\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}X_\tau] (12)

for all predictable stopping times {\tau}.

Proof: As {\lvert X_m\rvert\le\sup_{n\ge 0}\lvert X_n\rvert} is integrable, the optional and predictable projections exist by definition. Letting {{}^{\rm o}\!X} and {{}^{\rm p}\!X} be the projections, then (11) and (12) follow from taking expectations of (8) and (9). Checking that the value of {{}^{\rm o}\!X_\tau} and {{}^{\rm p}\!X_\tau} are integrable is similar,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[1_{\{\tau < \infty\}}\lvert{}^{\rm o}\!X_\tau\rvert]&\displaystyle= {\mathbb E}[\lvert {\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_\tau]\rvert]\smallskip\\ &\displaystyle\le{\mathbb E}[\sup_n\lvert X_n\rvert] < \infty\smallskip\\ \displaystyle{\mathbb E}[1_{\{\tau < \infty\}}\lvert{}^{\rm p}\!X_\tau\rvert]&\displaystyle= {\mathbb E}[\lvert {\mathbb E}[1_{\{\tau < \infty\}}X_\tau\,\vert\mathcal{F}_{\tau-}]\rvert]\smallskip\\ &\displaystyle\le{\mathbb E}[\sup_n\lvert X_n\rvert] < \infty \end{array} (13)

The first inequality applies for any stopping time {\tau} and the second when {\tau} is predictable.

Conversely, suppose that {{}^{\rm o}\!X} and {{}^{\rm p}\!X} are, respectively, optional and predictable processes satisfying (11) and (12), and that integrability condition (13) is satisfied. For each {n=0,1,\ldots} and {A\in\mathcal{F}_n}, let {\tau} be the stopping time equal to {n} on event {A} and infinity otherwise.

\displaystyle  {\mathbb E}[1_A{}^{\rm o}\!X_n]={\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm o}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}\!X_\tau]={\mathbb E}[1_AX_n].

This implies (1) by the definition of conditional expectations. If {A\in\mathcal{F}_{n-1}} then {\tau} is predictable and

\displaystyle  {\mathbb E}[1_A{}^{\rm p}\!X_n]={\mathbb E}[1_{\{\tau < \infty\}}{}^{\rm p}\!X_\tau]={\mathbb E}[1_{\{\tau < \infty\}}\!X_\tau]={\mathbb E}[1_AX_n],

giving (2). ⬜

Dual projections

A dual projection can be constructed by applying the optional or predictable projection, as described above, to the increments of a process. The increments of a discrete time stochastic process, X, are defined here as the backward differences,

\displaystyle  \Delta X_n=X_n-X_{n-1}.

This is itself a stochastic process with time index n running over the non-negative integers. I will use the convention {X_{-1}=0} so that {\Delta X_0=X_0}. A process can be recovered from its increments,

\displaystyle  X_n=\sum_{k=0}^n\Delta X_k. (14)

It is easily seen that a process X is optional if and only {\Delta X} is optional and is predictable if and only if {\Delta X} is predictable.

The dual projections of X are defined by applying the optional and predictable projections to {\Delta X}.

Definition 4 Let A be a measurable process. Then,

  1. The dual optional projection, {A^{\rm o}}, exists if and only if the optional projection of {\Delta A} exists, in which case it is the unique process satisfying

    \displaystyle  \Delta A^{\rm o}={}^{\rm o}(\Delta A).

  2. The dual predictable projection, {A^{\rm p}}, exists if and only if the predictable projection of {\Delta A} exists, in which case it is the unique process satisfying

    \displaystyle  \Delta A^{\rm p}={}^{\rm p}(\Delta A).

Substituting in expression (14), the definitions of the dual projections can be expressed more directly as,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle A^{\rm o}_n=\sum_{k=0}^n{}^{\rm o}(\Delta A)_k=\sum_{k=0}^n{\mathbb E}[\Delta A_k\,\vert\mathcal{F}_k],\smallskip\\ &\displaystyle A^{\rm p}_n=\sum_{k=0}^n{}^{\rm p}(\Delta A)_k=\sum_{k=0}^n{\mathbb E}[\Delta A_k\,\vert\mathcal{F}_{k-1}]. \end{array} (15)

Identities (4,6,7) for the projections can be extended to the dual projections, simply by applying them to the increments.

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle (\lambda A+\mu B)^{\rm o}=\lambda A^{\rm o}+\mu B^{\rm o},\smallskip\\ &\displaystyle (\lambda A+\mu B)^{\rm p}=\lambda A^{\rm p}+\mu B^{\rm p},\smallskip\\ &\displaystyle (A^{\rm o})^{\rm o}= A^{\rm o},\smallskip\\ &\displaystyle (A^{\rm p})^{\rm p}= A^{\rm p},\smallskip\\ &\displaystyle (A^{\rm o})^{\rm p}=(A^{\rm p})^{\rm o}=A^{\rm p}. \end{array}

Here A and B are any raw IV processes for which the projections on the right hand side of the respective identities exist. The terms {\lambda} and {\mu} are real numbers or, more generally, can be {\mathcal{F}_0}-measurable random variables.

Identities (5) above do not extend directly to dual processes, as multiplying a process by another is not the same as multiplying its increments. Instead, we should look at the integral {X\cdot A} of a process X with respect to A. This is defined by multiplying the increments of A by X, {\Delta(X\cdot A)=X\Delta A}, or equivalently,

\displaystyle  (X\cdot A)_n\equiv\sum_{k=0}^n X_k\Delta A_k.

Applying (5) to {X\Delta A} gives the correct extension to the dual projections,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle (X\cdot A)^{\rm o}&\displaystyle=X\cdot A^{\rm o},\smallskip\\ \displaystyle (X\cdot A)^{\rm p}&\displaystyle=X\cdot A^{\rm p}, \end{array} (16)

for any optional X and raw IV process A, whenever the projections on the right hand side exist. Similarly, we can compute the dual projection of an integral {X\cdot A} whenever A is optional (resp., predictable) by taking the projection of the integral X,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle (X\cdot A)^{\rm o}&\displaystyle= {}^{\rm o}\!X\cdot A,\smallskip\\ \displaystyle (X\cdot A)^{\rm p}&\displaystyle={}^{\rm p}\!X\cdot A. \end{array} (17)

The first equation holds whenever the optional projection of X exists and A is optional, and the second holds when the predictable projection of X exists and A is predictable. These identities follow from looking at the increments. For the optional projection, this is,

\displaystyle  \Delta(X\cdot A)^{\rm o}={}^{\rm o}(X\Delta A)={}^{\rm o}\!X\cdot\Delta A=\Delta({}^{\rm o}\!X\cdot A).

As their names suggest, the dual projections can also be understood as a kind of adjoint or dual of the optional and predictable projections. A process A will be said to be a raw IV process process if its total variation {\sum_n\lvert\Delta A_n\rvert} is integrable. The `raw’ in the name is just to distinguish from the usage of `IV process’ specifically for adapted processes. We are not requiring A to be adapted.

There is a natural duality given by

\displaystyle  \langle X,A\rangle \equiv {\mathbb E}\left[\sum_n X_n\Delta A_n\right]={\mathbb E}\left[(X\cdot A)_\infty\right]

for any uniformly bounded X and raw IV process A. The definition of the dual projections as adjoint or dual to the optional and predictable projection is given by the following.

Theorem 5 Let A be a raw IV process.

  1. The optional projection {A^{\rm o}} is the unique raw IV process satisfying
    \displaystyle  \langle X,A^{\rm o}\rangle=\langle {}^{\rm o}\!X,A\rangle (18)

    for every bounded process X.

  2. The dual predictable projection {A^{\rm p}} is the unique raw IV process satisfying
    \displaystyle  \langle X,A^{\rm p}\rangle=\langle {}^{\rm p}\!X,A\rangle (19)

    for every bounded process X.

Proof: By countable additivity, (18) holds for all bounded process X if and only if it holds for processes of the for {X_n=1_{\{n=m\}}U} for all {m\in\mathbb{Z}^+} and bounded random variables U. However, for such processes, (18) reduces to

\displaystyle  {\mathbb E}\left[U\Delta A^{\rm o}_m\right]={\mathbb E}\left[{\mathbb E}[U\,\vert\mathcal{F}_m]\Delta A_m\right].

By standard properties of conditional expectations, the right hand side can be rearranged to give

\displaystyle  {\mathbb E}\left[U\Delta A^{\rm o}_m\right]={\mathbb E}\left[U{\mathbb E}[\Delta A_m\,\vert\mathcal{F}_m]\right].

As this is for arbitrary bounded random variables U and {m\in\mathbb{Z}^+}, it is equivalent to the statement that {\Delta A^{\rm o}_m={\mathbb E}[\Delta A_m\,\vert\mathcal{F}_m]}. That is, (18) is equivalent to {\Delta A^{\rm o}} being the optional projection of {\Delta A} which, by definition, is equivalent to {A^{\rm o}} being the dual optional projection of A.

The equivalence of (19) with {A^{\rm p}} being the dual predictable projection of A follows by the same argument, replacing optional projections with predictable projections and {\mathcal{F}_m} with {\mathcal{F}_{m-1}}. ⬜

The Doob decomposition

The Doob decomposition theorem gives a unique decomposition of an integrable adapted process X into the sum of a martingale M starting from zero ({M_0=0}) and a predictable process A,

\displaystyle  X = M + A. (20)

There are some variations in the precise statement of this theorem regarding whether M or A is required to start from zero. Due to the conventions adopted in this post, it is most convenient here to require M to start from zero. Then, we necessarily have {A_0=X_0} and looking at the conditional expectations of the increments of X. Using the notation {\Delta X_n\equiv X_n-X_{n-1}} as above,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle {\mathbb E}[\Delta X_n\,\vert\mathcal{F}_{n-1}] &\displaystyle={\mathbb E}[\Delta M_n\,\vert\mathcal{F}_{n-1}] +{\mathbb E}[\Delta A_n\,\vert\mathcal{F}_{n-1}],\smallskip\\ &\displaystyle=\Delta A_n. \end{array}

The final equality here is using the martingale property of M, {{\mathbb E}[\Delta M_n\,\vert\mathcal{F}_{n-1}] =0}, and that A is predictable so that {{\mathbb E}[\Delta A_n\,\vert\mathcal{F}_{n-1}]=\Delta A_n}. Hence we can solve for A,

\displaystyle  A_n=X_0+\sum_{k=1}^n{\mathbb E}[\Delta X_k\,\vert\mathcal{F}_{k-1}]. (21)

Conversely, we can plug solution (21) back into (20) to get {M=X-A} and,

\displaystyle  {\mathbb E}[\Delta M_n\,\vert\mathcal{F}_{n-1}] ={\mathbb E}[\Delta X_n\,\vert\mathcal{F}_{n-1}]-{\mathbb E}[\Delta A_n\,\vert\mathcal{F}_{n-1}] =0,

so M is a martingale.

The Doob decomposition is often applied in the case where X is a submartingale, in which case A is increasing. In particular, if X is a square integrable martingale, then {X^2} is a submartingale and the Doob decomposition can be applied. This defines an increasing predictable process, denoted by {\langle X\rangle}, such that {X^2-\langle X\rangle} is a martingale. The process {\langle X\rangle} is called the predictable quadratic variation of X. There is also a process known as the optional quadratic variation (or, just, the quadratic variation) defined by

\displaystyle  [ X ]_n=X_0^2+\sum_{k=1}^n (\Delta X_k)^2.

The optional quadratic variation is defined more generally, and does not require X to be either square integrable or a martingale. However, it is increasing and, in the case where X is a square integrable martingale, then {X^2-\langle X\rangle}, {X^2-[X]} and {\langle X\rangle-[X]} are all martingales. One reason for defining the quadratic variation processes is that they satisfy a discrete version of the Ito Isometry,

\displaystyle  {\mathbb E}\left[(H\cdot X)^2_n\right]={\mathbb E}\left[(H\cdot\langle X\rangle)_n\right]={\mathbb E}\left[(H\cdot[X])_n\right].

This holds for all bounded predictable H and square integrable martingales X.

The relevance of the Doob decomposition to this post is that it is closely related to dual predictable projection.

Lemma 6 Let X be an integrable adapted process. Then {X^{\rm p}} is the unique predictable process such that {X-X^{\rm p}} is a martingale starting from zero.

Furthermore, X is a submartingale if and only if {X^{\rm p}} is increasing.

Proof: We already showed above that the unique predictable process A such that {X-A} is a martingale starting from zero is given by (21). Comparing with (15), this is equivalent to {A=X^{\rm p}}.

Finally, using

\displaystyle  \Delta X^{\rm p}_n={\mathbb E}[\Delta X_n\,\vert\mathcal{F}_{n-1}],

which is positive everywhere if and only if X is a submartingale, we see that X is a submartingale if and only if {X^{\rm p}} is increasing. ⬜

Lemma 6 only applies when X is adapted. If we only know that it is integrable then the lemma cannot be applied directly. Instead, we can first take the dual optional projection {X^{\rm o}} and apply the result to this. As the dual predictable projection of {X^{\rm o}} is {(X^{\rm o})^{\rm p}=X^{\rm p}} we obtain the following.

Corollary 7 Let X be an integrable process. Then, {X^{\rm o}-X^{\rm p}} is a martingale.

The transition to continuous time

The constructions given above in the discrete-time situation are rather basic, with the optional and predictable projection defined directly as conditional expectations by (1,2). The properties of these projections follows from this, and the theory is not complicated. Constructing the projections of continuous-time processes is considerably more difficult, but also leads to a very powerful and interesting theory. Let us now consider how the ideas developed above can be carried over to continuous-time. This section is just to get a feel of how projections could be handled in continuous time, and is not intended to be rigorous.

An obvious, but naive, thing to do is to simply modify Definition 1 by allowing the time index to vary over the continuous index set {\mathbb{R}^+}. This leads to defining the optional projection {{}^{\rm o}\!X} and predictable projection {{}^{\rm p}\!X} by

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle {}^{\rm o}X_t={\mathbb E}[X_t\,\vert\mathcal{F}_t],\smallskip\\ &\displaystyle {}^{\rm p}X_t={\mathbb E}[X_t\,\vert\mathcal{F}_{t-}], \end{array} (22)

for each {t\in\mathbb{R}^+}. Note that this only defines the projections at any time up to a zero probability and, unfortunately, this is not enough to determine the sample paths of {{}^{\rm o}\!X} or {{}^{\rm p}\!X} up to evanescence, or to determine their values at random times. As was previously mentioned in these notes, when working in continuous time, it is important to choose a good version of any stochastic processes.

One approach to choosing good versions of the projection is to impose pathwise regularity conditions such as left or right continuity. Consider the case where X has constant paths equal to an integrable random variable, {X_t=U}. Then, (22) says that {{}^{\rm o}\!X_t=M_t} where M is the martingale {M_t={\mathbb E}[U\,\vert\mathcal{F}_t]}. As we know, assuming right-continuity of the underlying filtration, this will have a cadlag modification. Hence, the optional projection will have a cadlag modification, which is uniquely determined up to evanescence. Similarly, (22) gives {{}^{\rm p}\!X_t=M_{t-}} for the predictable projection, which has a unique left-continuous modification up to evanescence. I do investigate this approach in these notes, constructing a right-continuous optional projection and a left-continuous predictable projection, and show that this can be made rigorous.

A drawback of imposing left or right continuity requirements on the projection is that it imposes rather restrictive conditions on the process X. To stand a chance that the projection has a modification with the necessary pathwise properties requires imposing these same properties on the original process. There is a much more general approach to defining the projection of any jointly measurable process satisfying mild integrability conditions. We impose the conditions that {{}^{\rm o}\!X} and {{}^{\rm p}\!X} satisfy the continuous-time definition of optional and, respectively, predictable processes. This is, at least, consistent with imposing the pathwise left or right continuity condition, since right-continuous adapted processes are optional and left-continuous adapted processes are predictable. However, defining the projection at each deterministic time t by (22) is still not enough to uniquely determine the result up to evanescence. To get around this shortcoming we note that, in light of Theorem 2 above, the defining equation (22) of the optional projection {{}^{\rm o}\!X} should hold when t is replaced by a stopping time, and that for the predictable projection {{}^{\rm p}\!X} should hold when t is a predictable stopping time.

This leads us to the following definitions in continuous time. The optional projection {{}^{\rm o}\!X} is an optional process satisfying (22) at each stopping time t, and the predictable projection {{}^{\rm p}\!X} is a predictable process satisfying (22) at each predictable stopping time t. As it turns out, this is enough to uniquely define the projections up to evanescence, as a result of the section theorems. This is the standard approach to optional and predictable projection, which is a very useful and powerful theory. However, as it relies on easy to state but surprisingly difficult to prove section theorems for its consistency, it is often considered as one of the more advanced parts of stochastic calculus.

Finally, consider the dual projections. According to definition 4, they are defined so that the increments of the dual projection are the projection of the increments,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\Delta A^{\rm o}={}^{\rm o}\!\Delta A,\smallskip\\ &\displaystyle\Delta A^{\rm p}={}^{\rm p}\!\Delta A. \end{array} (23)

In the continuous time setting, the discrete increments considered above do not have any meaning. Instead, if A is cadlag then {\Delta A} can be interpreted as the instantaneous jumps, {\Delta A_t=A_t-A_{t-}}. While this makes sense, and the dual projections do satisfy (23), it is to weak a condition. We cannot reconstruct a process from its jumps — consider continuous processes, which have no jumps.

If A is an FV process then the infinitesimal increments dA can be understood in the sense of Lebesgue-Stieltjes integration.

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

This is still not a precise statement though, since dA is not represented as a real-valued process so does not have well-defined optional and predictable projections. It can be made precise though, by integrating over a bounded process {\xi}, taking expectations, and rearranging the optional/predictable projection slightly,

\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}

This is the continuous-time version of the definition of dual projections to that provided by theorem 5 above.

An alternative approach to the continuous-time dual projections, is to approximate by taking the discrete projections along partitions {0=t_0 < t_1 < t_2 <\cdots}, and then take the limit as the mesh of the partition goes to zero. This can also be shown to give well-defined dual projections.

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1 Comment »

  1. good notes! Please keep writing great stuff!!

    Comment by Huy Nguyễn — 23 December 18 @ 3:47 PM | Reply


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