# Almost Sure

## 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 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.

## 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…)

## 14 July 10

### Feller Processes

The definition of Markov processes, as given in the previous post, is much too general for many applications. However, many of the processes which we study also satisfy the much stronger Feller property. This includes Brownian motion, Poisson processes, Lévy processes and Bessel processes, all of which are considered in these notes. Once it is known that a process is Feller, many useful properties follow such as, the existence of cadlag modifications, the strong Markov property, quasi-left-continuity and right-continuity of the filtration. In this post I give the definition of Feller processes and prove the existence of cadlag modifications, leaving the further properties until the next post.

The definition of Feller processes involves putting continuity constraints on the transition function, for which it is necessary to restrict attention to processes lying in a topological space ${(E,\mathcal{T}_E)}$. It will be assumed that E is locally compact, Hausdorff, and has a countable base (lccb, for short). Such spaces always possess a countable collection of nonvanishing continuous functions ${f\colon E\rightarrow{\mathbb R}}$ which separate the points of E and which, by Lemma 6 below, helps us construct cadlag modifications. Lccb spaces include many of the topological spaces which we may want to consider, such as ${{\mathbb R}^n}$, topological manifolds and, indeed, any open or closed subset of another lccb space. Such spaces are always Polish spaces, although the converse does not hold (a Polish space need not be locally compact).

Given a topological space E, ${C_0(E)}$ denotes the continuous real-valued functions vanishing at infinity. That is, ${f\colon E\rightarrow{\mathbb R}}$ is in ${C_0(E)}$ if it is continuous and, for any ${\epsilon>0}$, the set ${\{x\colon \vert f(x)\vert\ge\epsilon\}}$ is compact. Equivalently, its extension to the one-point compactification ${E^*=E\cup\{\infty\}}$ of E given by ${f(\infty)=0}$ is continuous. The set ${C_0(E)}$ is a Banach space under the uniform norm,

$\displaystyle \Vert f\Vert\equiv\sup_{x\in E}\vert f(x)\vert.$

We can now state the general definition of Feller transition functions and processes. A topological space ${(E,\mathcal{T}_E)}$ is also regarded as a measurable space by equipping it with its Borel sigma algebra ${\mathcal{B}(E)=\sigma(\mathcal{T})}$, so it makes sense to talk of transition probabilities and functions on E.

Definition 1 Let E be an lccb space. Then, a transition function ${\{P_t\}_{t\ge 0}}$ is Feller if, for all ${f\in C_0(E)}$,

1. ${P_tf\in C_0(E)}$.
2. ${t\mapsto P_tf}$ is continuous with respect to the norm topology on ${C_0(E)}$.
3. ${P_0f=f}$.

A Markov process X whose transition function is Feller is a Feller process.

## 3 March 10

### Existence of the Stochastic Integral

The principal reason for introducing the concept of semimartingales in stochastic calculus is that they are precisely those processes with respect to which stochastic integration is well defined. Often, semimartingales are defined in terms of decompositions into martingale and finite variation components. Here, I have taken a different approach, and simply defined semimartingales to be processes with respect to which a stochastic integral exists satisfying some necessary properties. That is, integration must agree with the explicit form for piecewise constant elementary integrands, and must satisfy a bounded convergence condition. If it exists, then such an integral is uniquely defined. Furthermore, whatever method is used to actually construct the integral is unimportant to many applications. Only its elementary properties are required to develop a theory of stochastic calculus, as demonstrated in the previous posts on integration by parts, Ito’s lemma and stochastic differential equations.

The purpose of this post is to give an alternative characterization of semimartingales in terms of a simple and seemingly rather weak condition, stated in Theorem 1 below. The necessity of this condition follows from the requirement of integration to satisfy a bounded convergence property, as was commented on in the original post on stochastic integration. That it is also a sufficient condition is the main focus of this post. The aim is to show that the existence of the stochastic integral follows in a relatively direct way, requiring mainly just standard measure theory and no deep results on stochastic processes.

Recall that throughout these notes, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$. To recap, elementary predictable processes are of the form

 $\displaystyle \xi_t=Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_{k} (1)

for an ${\mathcal{F}_0}$-measurable random variable ${Z_0}$, real numbers ${s_k,t_k\ge 0}$ and ${\mathcal{F}_{s_k}}$-measurable random variables ${Z_k}$. The integral with respect to any other process X up to time t can be written out explicitly as,

 $\displaystyle \int_0^t\xi\,dX = \sum_{k=1}^n Z_k(X_{t_k\wedge t}-X_{s_k\wedge t}).$ (2)

The predictable sigma algebra, ${\mathcal{P}}$, on ${{\mathbb R}_+\times\Omega}$ is generated by the set of left-continuous and adapted processes or, equivalently, by the elementary predictable process. The idea behind stochastic integration is to extend this to all bounded and predictable integrands ${\xi\in{\rm b}\mathcal{P}}$. Other than agreeing with (2) for elementary integrands, the only other property required is bounded convergence in probability. That is, if ${\xi^n\in{\rm b}\mathcal{P}}$ is a sequence uniformly bounded by some constant K, so that ${\vert\xi^n\vert\le K}$, and converging to a limit ${\xi}$ then, ${\int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX}$ in probability. Nothing else is required. Other properties, such as linearity of the integral with respect to the integrand follow from this, as was previously noted. Note that we are considering two random variables to be the same if they are almost surely equal. Similarly, uniqueness of the stochastic integral means that, for each integrand, the integral is uniquely defined up to probability one.

Using the definition of a semimartingale as a cadlag adapted process with respect to which the stochastic integral is well defined for bounded and predictable integrands, the main result is as follows. To be clear, in this post all stochastic processes are real-valued.

Theorem 1 A cadlag adapted process X is a semimartingale if and only if, for each ${t\ge 0}$, the set

 $\displaystyle \left\{\int_0^t\xi\,dX\colon \xi{\rm\ is\ elementary}, \vert\xi\vert\le 1\right\}$ (3)

is bounded in probability.

## 4 January 10

### Extending the Stochastic Integral

In the previous post, I used the property of bounded convergence in probability to define stochastic integration for bounded predictable integrands. For most applications, this is rather too restrictive, and in this post the integral will be extended to unbounded integrands. As bounded convergence is not much use in this case, the dominated convergence theorem will be used instead.

The first thing to do is to define a class of integrable processes for which the integral with respect to ${X}$ is well-defined. Suppose that ${\xi^n}$ is a sequence of predictable processes dominated by any such ${X}$-integrable process ${\alpha}$, so that ${\vert\xi^n\vert\le\vert\alpha\vert}$ for each ${n}$. If this sequence converges to a limit ${\xi}$, then dominated convergence in probability states that the integrals converge in probability,

 $\displaystyle \int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX\ \ \text{(in probability)}$ (1)

as ${n\rightarrow\infty}$. (more…)

## 3 January 10

### The Stochastic Integral

Having covered the basics of continuous-time processes and filtrations in the previous posts, I now move on to stochastic integration. In standard calculus and ordinary differential equations, a central object of study is the derivative ${df/dt}$ of a function ${f(t)}$. This does, however, require restricting attention to differentiable functions. By integrating, it is possible to generalize to bounded variation functions. If ${f}$ is such a function and ${g}$ is continuous, then the Riemann-Stieltjes integral ${\int_0^tg\,df}$ is well defined. The Lebesgue-Stieltjes integral further generalizes this to measurable integrands.

However, the kinds of processes studied in stochastic calculus are much less well behaved. For example, with probability one, the sample paths of standard Brownian motion are nowhere differentiable. Furthermore, they have infinite variation over bounded time intervals. Consequently, if ${X}$ is such a process, then the integral ${\int_0^t\xi\,dX}$ is not defined using standard methods.

Stochastic integration with respect to standard Brownian motion was developed by Kiyoshi Ito. This required restricting the class of possible integrands to be adapted processes, and the integral can then be constructed using the Ito isometry. This method was later extended to more general square integrable martingales and, then, to the class of semimartingales. It can then be shown that, as with Lebesgue integration, a version of the bounded and dominated convergence theorems are satisfied.

In these notes, a more direct approach is taken. The idea is that we simply define the stochastic integral such that the required elementary properties are satisfied. That is, it should agree with the explicit expressions for certain simple integrands, and should satisfy the bounded and dominated convergence theorems. Much of the theory of stochastic calculus follows directly from these properties, and detailed constructions of the integral are not required for many practical applications. (more…)

## 24 December 09

### Local Martingales

Recall from the previous post that a cadlag adapted process ${X}$ is a local martingale if there is a sequence ${\tau_n}$ of stopping times increasing to infinity such that the stopped processes ${1_{\{\tau_n>0\}}X^{\tau_n}}$ are martingales. Local submartingales and local supermartingales are defined similarly.

An example of a local martingale which is not a martingale is given by the `double-loss’ gambling strategy. Interestingly, in 18th century France, such strategies were known as martingales and is the origin of the mathematical term. Suppose that a gambler is betting sums of money, with even odds, on a simple win/lose game. For example, betting that a coin toss comes up heads. He could bet one dollar on the first toss and, if he loses, double his stake to two dollars for the second toss. If he loses again, then he is down three dollars and doubles the stake again to four dollars. If he keeps on doubling the stake after each loss in this way, then he is always gambling one more dollar than the total losses so far. He only needs to continue in this way until the coin eventually does come up heads, and he walks away with net winnings of one dollar. This therefore describes a fair game where, eventually, the gambler is guaranteed to win.

Of course, this is not an effective strategy in practise. The losses grow exponentially and, if he doesn’t win quickly, the gambler must hit his credit limit in which case he loses everything. All that the strategy achieves is to trade a large probability of winning a dollar against a small chance of losing everything. It does, however, give a simple example of a local martingale which is not a martingale.

The gamblers winnings can be defined by a stochastic process ${\{Z_n\}_{n=1,\ldots}}$ representing his net gain (or loss) just before the n’th toss. Let ${\epsilon_1,\epsilon_2,\ldots}$ be a sequence of independent random variables with ${{\mathbb P}(\epsilon_n=1)={\mathbb P}(\epsilon_n=-1)=1/2}$. Here, ${\epsilon_n}$ represents the outcome of the n’th toss, with 1 referring to a head and -1 referring to a tail. Set ${Z_1=0}$ and

$\displaystyle Z_{n}=\begin{cases} 1,&\text{if }Z_{n-1}=1,\\ Z_{n-1}+\epsilon_n(1-Z_{n-1}),&\text{otherwise}. \end{cases}$

This is a martingale with respect to its natural filtration, starting at zero and, eventually, ending up equal to one. It can be converted into a local martingale by speeding up the time scale to fit infinitely many tosses into a unit time interval

$\displaystyle X_t=\begin{cases} Z_n,&\text{if }1-1/n\le t<1-1/(n+1),\\ 1,&\text{if }t\ge 1. \end{cases}$

This is a martingale with respect to its natural filtration on the time interval ${[0,1)}$. Letting ${\tau_n=\inf\{t\colon\vert X_t\vert\ge n\}}$ then the optional stopping theorem shows that ${X^{\tau_n}_t}$ is a uniformly bounded martingale on ${t<1}$, continuous at ${t=1}$, and constant on ${t\ge 1}$. This is therefore a martingale, showing that ${X}$ is a local martingale. However, ${{\mathbb E}[X_1]=1\not={\mathbb E}[X_0]=0}$, so it is not a martingale. (more…)

## 22 December 09

### U.C.P. and Semimartingale Convergence

A mode of convergence on the space of processes which occurs often in the study of stochastic calculus, is that of uniform convergence on compacts in probability or ucp convergence for short.

First, a sequence of (non-random) functions ${f_n\colon{\mathbb R}_+\rightarrow{\mathbb R}}$ converges uniformly on compacts to a limit ${f}$ if it converges uniformly on each bounded interval ${[0,t]}$. That is,

 $\displaystyle \sup_{s\le t}\vert f_n(s)-f(s)\vert\rightarrow 0$ (1)

as ${n\rightarrow\infty}$.

If stochastic processes are used rather than deterministic functions, then convergence in probability can be used to arrive at the following definition.

Definition 1 A sequence of jointly measurable stochastic processes ${X^n}$ converges to the limit ${X}$ uniformly on compacts in probability if

$\displaystyle {\mathbb P}\left(\sup_{s\le t}\vert X^n_s-X_s\vert>K\right)\rightarrow 0$

as ${n\rightarrow\infty}$ for each ${t,K>0}$.

## 18 December 09

As was mentioned in the initial post of these stochastic calculus notes, it is important to choose good versions of stochastic processes. In some cases, such as with Brownian motion, it is possible to explicitly construct the process to be continuous. However, in many more cases, it is necessary to appeal to more general results to assure the existence of such modifications.

The theorem below guarantees that many of the processes studied in stochastic calculus have a right-continuous version and, furthermore, these versions necessarily have left limits everywhere. Such processes are known as càdlàg from the French for “continu à droite, limites à gauche” (I often drop the accents, as seems common). Alternative terms used to refer to a cadlag process are rcll (right-continuous with left limits), R-process and right process. For a cadlag process ${X}$, the left limit at any time ${t>0}$ is denoted by ${X_{t-}}$ (and ${X_{0-}\equiv X_0}$). The jump at time ${t}$ is denoted by ${\Delta X_t=X_t-X_{t-}}$.

We work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$.

Theorem 1 below provides us with cadlag versions under the condition that elementary integrals of the processes cannot, in a sense, get too large. Recall that elementary predictable processes are of the form

$\displaystyle \xi=Z_01_{\{t=0\}}+\sum_{k=1}^nZ_k1_{\{s_k

for times ${s_k, ${\mathcal{F}_0}$-measurable random variable ${Z_0}$ and ${\mathcal{F}_{s_k}}$-measurable random variables ${Z_k}$. Its integral with respect to a stochastic process ${X}$ is

$\displaystyle \int_0^t \xi\,dX=\sum_{k=1}^nZ_k(X_{t_k\wedge t}-X_{s_{k}\wedge t}).$

An elementary predictable set is a subset of ${{\mathbb R}_+\times\Omega}$ which is a finite union of sets of the form ${\{0\}\times F}$ for ${F\in\mathcal{F}_0}$ and ${(s,t]\times F}$ for nonnegative reals ${s and ${F\in\mathcal{F}_s}$. Then, a process is an indicator function ${1_A}$ of some elementary predictable set ${A}$ if and only if it is elementary predictable and takes values in ${\{0,1\}}$.

The following theorem guarantees the existence of cadlag versions for many types of processes. The first statement applies in particular to martingales, submartingales and supermartingales, whereas the second statement is important for the study of general semimartingales.

Theorem 1 Let X be an adapted stochastic process which is right-continuous in probability and such that either of the following conditions holds. Then, it has a cadlag version.

• X is integrable and, for every ${t\in{\mathbb R}_+}$,

$\displaystyle \left\{{\mathbb E}\left[\int_0^t1_A\,dX\right]\colon A\textrm{ is elementary}\right\}$

is bounded.

• For every ${t\in{\mathbb R}_+}$ the set

$\displaystyle \left\{\int_0^t1_A\,dX\colon A\textrm{ is elementary}\right\}$

is bounded in probability.

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