Brownian bridges were described in a previous post, along with various different methods by which they can be constructed. Since a Brownian bridge on an interval is continuous and equal to zero at both endpoints, we can consider extending to the entire real line by partitioning the real numbers into intervals of length T and replicating the path of the process across each of these. This will result in continuous and periodic sample paths, suggesting another method of representing Brownian bridges. That is, by Fourier expansion. As we will see, the Fourier coefficients turn out to be independent normal random variables, giving a useful alternative method of constructing a Brownian bridge.

There are actually a couple of distinct Fourier expansions that can be used, which depends on precisely how we consider extending the sample paths to the real line. A particularly simple result is given by the sine series, which I describe first. This is shown for an example Brownian bridge sample path in figure 1 above, which plots the sequence of approximations formed by truncating the series after a small number of terms. This tends uniformly to the sample path, although it is quite slow to converge as should be expected when approximating such a rough path by smooth functions. Also plotted, is the series after the first 100 terms, by which time the approximation is quite close to the target. For simplicity, I only consider standard Brownian bridges, which are defined on the unit interval . This does not reduce the generality, since bridges on an interval can be expressed as scaled versions of standard Brownian bridges.

Theorem 1 A standard Brownian bridge B can be decomposed as

(1)

over , where is an IID sequence of standard normals. This series converges uniformly in t, both with probability one and in the norm for all .

A Brownian bridge can be defined as standard Brownian motion conditioned on hitting zero at a fixed future time T, or as any continuous process with the same distribution as this. Rather than conditioning, a slightly easier approach is to subtract a linear term from the Brownian motion, chosen such that the resulting process hits zero at the time T. This is equivalent, but has the added benefit of being independent of the original Brownian motion at all later times.

Lemma 1 Let X be a standard Brownian motion and be a fixed time. Then, the process

(1)

over is independent from .

Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times , we just need to show that is zero. Using the covariance structure we obtain,

as required. ⬜

This leads us to the definition of a Brownian bridge.

Definition 2 A continuous process is a Brownian bridge on the interval if and only it has the same distribution as for a standard Brownian motion X.

In case that , then B is called a standard Brownian bridge.

There are actually many different ways in which Brownian bridges can be defined, which all lead to the same result.

As a Brownian motion minus a linear term so that it hits zero at T. This is definition 2.

As a Brownian motion X scaled as . See lemma 9 below.

As a joint normal process with prescribed covariances. See lemma 7 below.

As a Brownian motion conditioned on hitting zero at T. See lemma 14 below.

As a Brownian motion restricted to the times before it last hits zero before a fixed positive time T, and rescaled to fit a fixed time interval. See lemma 15 below.

As a solution to a stochastic differential equation with drift term forcing it to hit zero at T. See lemma 18 below.

There are other constructions beyond these, such as in terms of limits of random walks, although I will not cover those in this post. Continue reading “Brownian Bridges”→

Lévy processes, which are defined as having stationary and independent increments, were introduced in the previous post. It was seen that the distribution of a d-dimensional Lévy process X is determined by the characteristics via the Lévy-Khintchine formula,

(1)

The positive semidefinite matrix describes the Brownian motion component of X, b is a drift term, and is a measure on such that is the rate at which jumps of X occur. Then, equation (1) gives us the characteristic function of the increments of the process.

In the current post, I will investigate some of the properties of such processes, and how they are related to the characteristics. In particular, we will be concerned with pathwise properties of X. It is known that Brownian motion and Cauchy processes have infinite variation in every nonempty time interval, whereas other Lévy processes — such as the Poisson process — are piecewise constant, only jumping at a discrete set of times. There are also purely discontinuous Lévy processes which have infinitely many discontinuities, yet are of finite variation, on every interval (e.g., the gamma process). Continue reading “Properties of Lévy Processes”→

Continuous-time stochastic processes with stationary independent increments are known as Lévy processes. In the previous post, it was seen that processes with independent increments are described by three terms — the covariance structure of the Brownian motion component, a drift term, and a measure describing the rate at which jumps occur. Being a special case of independent increments processes, the situation with Lévy processes is similar. However, stationarity of the increments does simplify things a bit. We start with the definition.

Definition 1 (Lévy process) A d-dimensional Lévy processX is a stochastic process taking values in such that

independent increments: is independent of for any .

stationary increments: has the same distribution as for any .

continuity in probability: in probability as s tends to t.

More generally, it is possible to define the notion of a Lévy process with respect to a given filtered probability space . In that case, we also require that X is adapted to the filtration and that is independent of for all . In particular, if X is a Lévy process according to definition 1 then it is also a Lévy process with respect to its natural filtration . Note that slightly different definitions are sometimes used by different authors. It is often required that is zero and that X has cadlag sample paths. These are minor points and, as will be shown, any process satisfying the definition above will admit a cadlag modification.

For example, the symmetric Cauchy distribution on the real numbers with scale parameter has probability density function p and characteristic function given by,

(1)

From the characteristic function it can be seen that if X and Y are independent Cauchy random variables with scale parameters and respectively then is Cauchy with parameter . We can therefore consistently define a stochastic process such that has the symmetric Cauchy distribution with parameter independent of , for any . This is called a Cauchy process, which is a purely discontinuous Lévy process. See Figure 1.

Lévy processes are determined by the triple , where describes the covariance structure of the Brownian motion component, b is the drift component, and describes the rate at which jumps occur. The distribution of the process is given by the Lévy-Khintchine formula, equation (3) below.

Theorem 2 (Lévy-Khintchine) Let X be a d-dimensional Lévy process. Then, there is a unique function such that

(2)

for all and . Also, can be written as

(3)

where , b and are uniquely determined and satisfy the following,

is a positive semidefinite matrix.

.

is a Borel measure on with and,

(4)

Furthermore, uniquely determine all finite distributions of the process .

Conversely, if is any triple satisfying the three conditions above, then there exists a Lévy process satisfying (2,3).

In a previous post, it was seen that all continuous processes with independent increments are Gaussian. We move on now to look at a much more general class of independent increments processes which need not have continuous sample paths. Such processes can be completely described by their jump intensities, a Brownian term, and a deterministic drift component. However, this class of processes is large enough to capture the kinds of behaviour that occur for more general jump-diffusion processes. An important subclass is that of Lévy processes, which have independent and stationary increments. Lévy processes will be looked at in more detail in the following post, and includes as special cases, the Cauchy process, gamma processes, the variance gamma process, Poisson processes, compound Poisson processes and Brownian motion.

Recall that a process has the independent increments property if is independent of for all times . More generally, we say that X has the independent increments property with respect to an underlying filtered probability space if it is adapted and is independent of for all . In particular, every process with independent increments also satisfies the independent increments property with respect to its natural filtration. Throughout this post, I will assume the existence of such a filtered probability space, and the independent increments property will be understood to be with regard to this space.

The process X is said to be continuous in probability if in probability as s tends to t. As we now state, a d-dimensional independent increments process X is uniquely specified by a triple where is a measure describing the jumps of X, determines the covariance structure of the Brownian motion component of X, and b is an additional deterministic drift term.

Theorem 1 Let X be an -valued process with independent increments and continuous in probability. Then, there is a unique continuous function , such that and

(1)

for all and . Also, can be written as

(2)

where , and are uniquely determined and satisfy the following,

is a continuous function from to such that and is positive semidefinite for all .

is a continuous function from to , with .

is a Borel measure on with , for all and,

(3)

Furthermore, uniquely determine all finite distributions of the process .

Conversely, if is any triple satisfying the three conditions above, then there exists a process with independent increments satisfying (1,2).

A random variable has the standard n-dimensional normal distribution if its components are independent normal with zero mean and unit variance. A well known fact of such distributions is that they are invariant under rotations, which has the following consequence. The distribution of is invariant under rotations of and, hence, is fully determined by the values of and . This is known as the noncentral chi-square distribution with n degrees of freedom and noncentrality parameter , and denoted by . The moment generating function can be computed,

(1)

which holds for all with real part bounded above by 1/2.

A consequence of this is that the norm of an n-dimensional Brownian motionB is Markov. More precisely, letting be its natural filtration, then has the following property. For times , conditional on , is distributed as . This is known as the `n-dimensional’ squared Bessel process, and denoted by .

Alternatively, the process X can be described by a stochastic differential equation (SDE). Applying integration by parts,

(2)

As the standard Brownian motions have quadratic variation , the final term on the right-hand-side is equal to . Also, the covarations are zero for from which it can be seen that

is a continuous local martingale with . By Lévy’s characterization, W is a Brownian motion and, substituting this back into (2), the squared Bessel process X solves the SDE

(3)

The standard existence and uniqueness results for stochastic differential equations do not apply here, since is not Lipschitz continuous. It is known that (3) does in fact have a unique solution, by the Yamada-Watanabe uniqueness theorem for 1-dimensional SDEs. However, I do not need and will not make use of this fact here. Actually, uniqueness in law follows from the explicit computation of the moment generating function in Theorem 5 below.

Although it is nonsensical to talk of an n-dimensional Brownian motion for non-integer n, Bessel processes can be extended to any real . This can be done either by specifying its distributions in terms of chi-square distributions or by the SDE (3). In this post I take the first approach, and then show that they are equivalent. Such processes appear in many situations in the theory of stochastic processes, and not just as the norm of Brownian motion. It also provides one of the relatively few interesting examples of stochastic differential equations whose distributions can be explicitly computed.

The distribution generalizes to all real , and can be defined as the unique distribution on with moment generating function given by equation (1). If and are independent, then has moment generating function and, therefore, has the distribution. That such distributions do indeed exist can be seen by constructing them. The distribution is a special case of the Gamma distribution and has probability density proportional to . If is a sequence of independent random variables with the standard normal distribution and T independently has the Poisson distribution of rate , then , which can be seen by computing its moment generating function. Adding an independent random variable Y to this produces the variable .

The definition of squared Bessel processes of any real dimension is as follows. We work with respect to a filtered probability space .

Definition 1 A process X is a squared Bessel process of dimension if it is continuous, adapted and, for any , conditional on , has the distribution.

In the previous post, the concept of Feller processes was introduced. These are Markov processes whose transition function satisfies certain continuity conditions. Many of the standard processes we study satisfy the Feller property, such as standard Brownian motion, Poisson processes, Bessel processes and Lévy processes as well as solutions to many stochastic differential equations. It was shown that all Feller processes admit a cadlag modification. In this post I state and prove some of the other useful properties satisfied by such processes, including the strong Markov property, quasi-left-continuity and right-continuity of the filtration. I also describe the basic properties of the infinitesimal generators. The results in this post are all fairly standard and can be found, for example, in Revuz and Yor (Continuous Martingales and Brownian Motion).

As always, we work with respect to a filtered probability space . Throughout this post we consider Feller processes X and transition functions defined on the lccb (locally compact with a countable base) space E which, taken together with its Borel sigma-algebra, defines a measurable space .

Recall that the law of a homogeneous Markov processX is described by a transition function on some measurable space . This specifies that the distribution of conditional on the history up until an earlier time is given by the measure . Equivalently,

for any bounded and measurable function . The strong Markov property generalizes this idea to arbitrary stopping times.

Definition 1 Let X be an adapted process and be a transition function.

Then, X satisfies the strong Markov property if, for each stopping time , conditioned on the process is Markov with the given transition function and with respect to the filtration .

As we see in a moment, Feller processes satisfy the strong Markov property. First, as an example, consider a standard Brownian motion B, and let be the first time at which it hits a fixed level . The reflection principle states that the process defined to be equal to B up until time and reflected about K afterwards, is also a standard Brownian motion. More precisely,

is a Brownian motion. This useful idea can be used to determine the distribution of the maximum . If then either the process itself ends up above K or it hits K and then drops below this level by time t, in which case . So, by the reflection principle,

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 . 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 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 , 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, denotes the continuous real-valued functions vanishing at infinity. That is, is in if it is continuous and, for any , the set is compact. Equivalently, its extension to the one-point compactification of E given by is continuous. The set is a Banach space under the uniform norm,

We can now state the general definition of Feller transition functions and processes. A topological space is also regarded as a measurable space by equipping it with its Borel sigma algebra , 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 is Feller if, for all ,

.

is continuous with respect to the norm topology on .

.

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

In these notes, the approach taken to stochastic calculus revolves around stochastic integration and the theory of semimartingales. An alternative starting point would be to consider Markov processes. Although I do not take the second approach, all of the special processes considered in the current section are Markov, so it seems like a good idea to introduce the basic definitions and properties now. In fact, all of the special processes considered (Brownian motion, Poisson processes, Lévy processes, Bessel processes) satisfy the much stronger property of being Feller processes, which I will define in the next post.

Intuitively speaking, a process X is Markov if, given its whole past up until some time s, the future behaviour depends only its state at time s. To make this precise, let us suppose that X takes values in a measurable space and, to denote the past, let be the sigma-algebra generated by . The Markov property then says that, for any times and bounded measurable function , the expected value of conditional on is a function of . Equivalently,

A Poisson process is a continuous-time stochastic process which counts the arrival of randomly occurring events. Commonly cited examples which can be modeled by a Poisson process include radioactive decay of atoms and telephone calls arriving at an exchange, in which the number of events occurring in each consecutive time interval are assumed to be independent. Being piecewise constant, Poisson processes have very simple pathwise properties. However, they are very important to the study of stochastic calculus and, together with Brownian motion, forms one of the building blocks for the much more general class of Lévy processes. I will describe some of their properties in this post.

A random variable N has the Poisson distribution with parameter , denoted by , if it takes values in the set of nonnegative integers and

(1)

for each . The mean and variance of N are both equal to , and the moment generating function can be calculated,

which is valid for all . From this, it can be seen that the sum of independent Poisson random variables with parameters and is again Poisson with parameter . The Poisson distribution occurs as a limit of binomial distributions. The binomial distribution with success probability p and m trials, denoted by , is the sum of m independent -valued random variables each with probability p of being 1. Explicitly, if then

In the limit as and such that , it can be verified that this tends to the Poisson distribution (1) with parameter .

Poisson processes are then defined as processes with independent increments and Poisson distributed marginals, as follows.

Definition 1 A Poisson process X of rate is a cadlag process with and independently of for all .

An immediate consequence of this definition is that, if X and Y are independent Poisson processes of rates and respectively, then their sum is also Poisson with rate . Continue reading “Poisson Processes”→