Month: October 2016

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]}. Continue reading “Optional Projection For Right-Continuous Processes” →

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.

Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample

f(t,x)
Figure 1: The function f, convex in x and decreasing in t

Here, I attempt to construct a counterexample to the hypotheses of the earlier post, Do convex and decreasing functions preserve the semimartingale property? There, it was asked, for any semimartingale X and function {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} such that {f(t,x)} is convex in x and right-continuous and decreasing in t, is {f(t,X_t)} necessarily a semimartingale? It was explained how this is equivalent to the hypothesis: for any function {f\colon[0,1]^2\rightarrow{\mathbb R}} such that {f(t,x)} is convex and Lipschitz continuous in x and decreasing in t, does it decompose as {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t. This is the form of the hypothesis which this post will be concerned with, so the example will only involve simple real analysis and no stochastic calculus. I will give some numerical calculations suggesting that the construction below is a counterexample, but do not have any proof of this. So, the hypothesis is still open.

Although the construction given here will be self-contained, it is worth noting that it is connected to the example of a martingale which moves along a deterministic path. If {\{M_t\}_{t\in[0,1]}} is the martingale constructed there, then

\displaystyle C(t,x)={\mathbb E}[(M_t-x)_+]

defines a function from {[0,1]\times[-1,1]} to {{\mathbb R}} which is convex in x and increasing in t. The question is then whether C can be expressed as the difference of functions which are convex in x and decreasing in t. The example constructed in this post will be the same as C with the time direction reversed, and with a linear function of x added so that it is zero at {x=\pm1}. Continue reading “Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample” →

A Martingale Which Moves Along a Deterministic Path

Sample Paths
Figure 1: Sample paths

In this post I will construct a continuous and non-constant martingale M which only varies on the path of a deterministic function {f\colon{\mathbb R}_+\rightarrow{\mathbb R}}. That is, {M_t=f(t)} at all times outside of the set of nontrivial intervals on which M is constant. Expressed in terms of the stochastic integral, {dM_t=0} on the set {\{t\colon M_t\not=f(t)\}} and,

\displaystyle M_t = \int_0^t 1_{\{M_s=f(s)\}}\,dM_s. (1)

In the example given here, f will be right-continuous. Examples with continuous f do exist, although the constructions I know of are considerably more complicated. At first sight, these properties appear to contradict what we know about continuous martingales. They vary unpredictably, behaving completely unlike any deterministic function. It is certainly the case that we cannot have {M_t=f(t)} across any interval on which M is not constant.

By a stochastic time-change, any Brownian motion B can be transformed to have the same distribution as M. This means that there exists an increasing and right-continuous process A adapted to the same filtration as B and such that {B_t=M_{A_t}} where M is a martingale as above. From this, we can infer that

\displaystyle B_t=f(A_t),

expressing Brownian motion as a function of an increasing process. Continue reading “A Martingale Which Moves Along a Deterministic Path” →