Brownian Excursions

A normalized Brownian excursion is a nonnegative real-valued process with time ranging over the unit interval, and is equal to zero at the start and end time points. It can be constructed from a standard Brownian motion by conditioning on being nonnegative and equal to zero at the end time. We do have to be careful with this definition, since it involves conditioning on a zero probability event. Alternatively, as the name suggests, Brownian excursions can be understood as the excursions of a Brownian motion X away from zero. By continuity, the set of times at which X is nonzero will be open and, hence, can be written as the union of a collection of disjoint (and stochastic) intervals (σ, τ).

In fact, Brownian motion can be reconstructed by simply joining all of its excursions back together. These are independent processes and identically distributed up to scaling. Because of this, understanding the Brownian excursion process can be very useful in the study of Brownian motion. However, there will by infinitely many excursions over finite time periods, so the procedure of joining them together requires some work. This falls under the umbrella of ‘excursion theory’, which is outside the scope of the current post. Here, I will concentrate on the properties of individual excursions.

In order to select a single interval, start by fixing a time T > 0. As X_T is almost surely nonzero, T will be contained inside one such interval (σ, τ). Explicitly,

$\displaystyle \begin{aligned} &\sigma=\sup\left\{t\le T\colon X_t=0\right\},\\ &\tau=\inf\left\{t\ge T\colon X_t=0\right\}, \end{aligned}$

(1)

so that σ < T < τ < ∞ almost surely. The path of X across such an interval is t ↦ X_σ + t for time t in the range [0, τ - σ]. As it can be either nonnegative or nonpositive, we restrict to the nonnegative case by taking the absolute value. By invariance, S^-1/2X_tS is also a standard Brownian motion, for each fixed S > 0. Using a stochastic factor S = τ – σ, the width of the excursion is normalised to obtain a continuous process {B_t}_{t ∈ [0, 1]} given by

$\displaystyle B_t=(\tau-\sigma)^{-1/2}\lvert X_{\sigma+t(\tau-\sigma)}\rvert.$

(2)

By construction, this is strictly positive over 0 < t < 1 and equal to zero at the endpoints t ∈ {0, 1}.

Figure 2: Constructing a Brownian excursion

The only remaining ambiguity is in the choice of the fixed time T.

Lemma 1 The distribution of B defined by (2) does not depend on the choice of the time T > 0.

Proof: This follows from scaling invariance of Brownian motion. Consider any other fixed positive time T̃, and use the construction above with T̃, σ̃, τ̃, B̃ in place of T, σ, τ, B respectively. We need to show that B̃ and B have the same distribution. Using the scaling factor S = T̃/T, then X′_t = S^-1/2X_tS is a standard Brownian motion. Also, σ′= σ̃/S and τ′= τ̃/S are random times given in the same way as σ and τ, but with the Brownian motion X′ in place of X in (1). So,

$\displaystyle \tilde B_t=(\tau^\prime-\sigma^\prime)^{-1/2}\lvert X^\prime_{\sigma^\prime+t(\tau^\prime-\sigma^\prime)}\rvert$

has the same distribution as B. ⬜

This leads to the definition used here for Brownian excursions.

Definition 2 A continuous process {B_t}_{t ∈ [0, 1]} is a Brownian excursion if and only it has the same distribution as (2) for a standard Brownian motion X and time T > 0.

In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.

As a normalized excursion away from zero of a Brownian motion. This is definition 2.
As a normalized excursion away from zero of a Brownian bridge. This is theorem 6.
As a Brownian bridge conditioned on being nonnegative. See theorem 9 below.
As the sample path of a Brownian bridge, translated so that it has minimum value zero at time 0. This is a very interesting and useful method of directly computing excursion sample paths from those of a Brownian bridge. See theorem 12 below, sometimes known as the Vervaat transform.
As a Markov process with specified transition probabilities. See theorem 15 below.
As a transformation of Bessel process paths, see theorem 16 below.
As a Bessel bridge of order 3. This can be represented either as a Bessel process conditioned on hitting zero at time 1., or as the vector norm of a 3-dimensional Brownian bridge. See lemma 17 below.
As a solution to a stochastic differential equation. See theorem 18 below.

Continue reading “Brownian Excursions” →

Brownian Bridges

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 ${T > 0}$ be a fixed time. Then, the process

$\displaystyle B_t = X_t - \frac tTX_T$ (1)

over ${0\le t\le T}$ is independent from ${\{X_t\}_{t\ge T}}$ .

Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times ${s\le T\le t}$ , we just need to show that ${{\mathbb E}[B_sX_t]}$ is zero. Using the covariance structure ${{\mathbb E}[X_sX_t]=s\wedge t}$ we obtain,

$\displaystyle {\mathbb E}[B_sX_t]={\mathbb E}[X_sX_t]-\frac sT{\mathbb E}[X_TX_t]=s-\frac sTT=0$

as required. ⬜

This leads us to the definition of a Brownian bridge.

Definition 2 A continuous process ${\{B_t\}_{t\in[0,T]}}$ is a Brownian bridge on the interval ${[0,T]}$ if and only it has the same distribution as ${X_t-\frac tTX_T}$ for a standard Brownian motion X.

In case that ${T=1}$ , 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 ${tT^{-1/2}X_{T/t-1}}$ . 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 Markov process. See lemma 13 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” →

Bessel Processes

A random variable ${N=(N^1,\ldots,N^n)}$ has the standard n-dimensional normal distribution if its components ${N^i}$ 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 ${Z\equiv\Vert N+\boldsymbol{\mu}\Vert^2}$ is invariant under rotations of ${\boldsymbol{\mu}\in{\mathbb R}^n}$ and, hence, is fully determined by the values of ${n\in{\mathbb N}}$ and ${\mu=\Vert\boldsymbol{\mu}\Vert^2\in{\mathbb R}_+}$ . This is known as the noncentral chi-square distribution with n degrees of freedom and noncentrality parameter ${\mu}$ , and denoted by ${\chi^2_n(\mu)}$ . The moment generating function can be computed,

$\displaystyle M_Z(\lambda)\equiv{\mathbb E}\left[e^{\lambda Z}\right]=\left(1-2\lambda\right)^{-\frac{n}{2}}\exp\left(\frac{\lambda\mu}{1-2\lambda}\right),$

(1)

which holds for all ${\lambda\in{\mathbb C}}$ with real part bounded above by 1/2.

A consequence of this is that the norm ${\Vert B_t\Vert}$ of an n-dimensional Brownian motion B is Markov. More precisely, letting ${\mathcal{F}_t=\sigma(B_s\colon s\le t)}$ be its natural filtration, then ${X\equiv\Vert B\Vert^2}$ has the following property. For times ${s<t}$ , conditional on ${\mathcal{F}_s}$ , ${X_t/(t-s)}$ is distributed as ${\chi^2_n(X_s/(t-s))}$ . This is known as the `n-dimensional’ squared Bessel process, and denoted by ${{\rm BES}^2_n}$ .

A Poisson process sample path — Figure 1: Squared Bessel processes of dimensions n=1,2,3

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

$\displaystyle dX = 2\sum_iB^i\,dB^i+\sum_id[B^i].$

(2)

As the standard Brownian motions have quadratic variation ${[B^i]_t=t}$ , the final term on the right-hand-side is equal to ${n\,dt}$ . Also, the covarations ${[B^i,B^j]}$ are zero for ${i\not=j}$ from which it can be seen that

$\displaystyle W_t = \sum_i\int_0^t1_{\{B\not=0\}}\frac{B^i}{\Vert B\Vert}\,dB^i$

is a continuous local martingale with ${[W]_t=t}$ . By Lévy’s characterization, W is a Brownian motion and, substituting this back into (2), the squared Bessel process X solves the SDE

$\displaystyle dX=2\sqrt{X}\,dW+ndt.$

(3)

The standard existence and uniqueness results for stochastic differential equations do not apply here, since ${x\mapsto2\sqrt{x}}$ 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 ${n\ge0}$ . 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 ${\chi^2_n(\mu)}$ distribution generalizes to all real ${n\ge0}$ , and can be defined as the unique distribution on ${{\mathbb R}_+}$ with moment generating function given by equation (1). If ${Z_1\sim\chi_m(\mu)}$ and ${Z_2\sim\chi_n(\nu)}$ are independent, then ${Z_1+Z_2}$ has moment generating function ${M_{Z_1}(\lambda)M_{Z_2}(\lambda)}$ and, therefore, has the ${\chi^2_{m+n}(\mu+\nu)}$ distribution. That such distributions do indeed exist can be seen by constructing them. The ${\chi^2_n(0)}$ distribution is a special case of the Gamma distribution and has probability density proportional to ${x^{n/2-1}e^{-x/2}}$ . If ${Z_1,Z_2,\ldots}$ is a sequence of independent random variables with the standard normal distribution and T independently has the Poisson distribution of rate ${\mu/2}$ , then ${\sum_{i=1}^{2T}Z_i^2\sim\chi_0^2(\mu)}$ , which can be seen by computing its moment generating function. Adding an independent ${\chi^2_n(0)}$ random variable Y to this produces the ${\chi^2_n(\mu)}$ variable ${Z\equiv Y+\sum_{i=1}^{2T}Z_i^2}$ .

The definition of squared Bessel processes of any real dimension ${n\ge0}$ is as follows. We work with respect to a filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$ .

Definition 1 A process X is a squared Bessel process of dimension ${n\ge0}$ if it is continuous, adapted and, for any ${s<t}$ , conditional on ${\mathcal{F}_s}$ , ${X_t/(t-s)}$ has the ${\chi^2_n\left(X_s/(t-s)\right)}$ distribution.

Continue reading “Bessel Processes” →

Properties of Feller Processes

In the previous post, the concept of Feller processes was introduced. These are Markov processes whose transition function ${\{P_t\}_{t\ge0}}$ 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 ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$ . Throughout this post we consider Feller processes X and transition functions ${\{P_t\}_{t\ge0}}$ defined on the lccb (locally compact with a countable base) space E which, taken together with its Borel sigma-algebra, defines a measurable space ${(E,\mathcal{E})}$ .

Recall that the law of a homogeneous Markov process X is described by a transition function ${\{P_t\}_{t\ge0}}$ on some measurable space ${(E,\mathcal{E})}$ . This specifies that the distribution of ${X_t}$ conditional on the history up until an earlier time ${s<t}$ is given by the measure ${P_{t-s}(X_s,\cdot)}$ . Equivalently,

$\displaystyle {\mathbb E}[f(X_t)\mid\mathcal{F}_s]=P_{t-s}f(X_s)$

for any bounded and measurable function ${f\colon E\rightarrow{\mathbb R}}$ . The strong Markov property generalizes this idea to arbitrary stopping times.

Definition 1 Let X be an adapted process and ${\{P_t\}_{t\ge 0}}$ be a transition function.

Then, X satisfies the strong Markov property if, for each stopping time ${\tau}$ , conditioned on ${\tau<\infty}$ the process ${\{X_{\tau+t}\}_{t\ge0}}$ is Markov with the given transition function and with respect to the filtration ${\{\mathcal{F}_{\tau+t}\}_{t\ge0}}$ .

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 ${\tau}$ be the first time at which it hits a fixed level ${K>0}$ . The reflection principle states that the process ${\tilde B}$ defined to be equal to B up until time ${\tau}$ and reflected about K afterwards, is also a standard Brownian motion. More precisely,

$\displaystyle \tilde B_t=\begin{cases} B_t,&\textrm{if }t\le\tau,\\ 2K-B_t,&\textrm{if }t\ge\tau, \end{cases}$

is a Brownian motion. This useful idea can be used to determine the distribution of the maximum ${B^*_t=\max_{s\le t}B_s}$ . If ${B^*_t\ge K}$ 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 ${\tilde B_t>K}$ . So, by the reflection principle,

$\displaystyle {\mathbb P}(B^*_t\ge K)={\mathbb P}(B_t\ge K)+{\mathbb P}(\tilde B_t> K)=2{\mathbb P}(B_t\ge K).$

Continue reading “Properties of Feller Processes” →

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 ${f\in C_0(E)}$ ,

${P_tf\in C_0(E)}$ .

${t\mapsto P_tf}$ is continuous with respect to the norm topology on ${C_0(E)}$ .

${P_0f=f}$ .

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

Continue reading “Feller Processes” →

Markov Processes

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 ${(E,\mathcal{E})}$ and, to denote the past, let ${\mathcal{F}_t}$ be the sigma-algebra generated by ${\{X_s\colon s\le t\}}$ . The Markov property then says that, for any times ${s\le t}$ and bounded measurable function ${f\colon E\rightarrow{\mathbb R}}$ , the expected value of ${f(X_t)}$ conditional on ${\mathcal{F}_s}$ is a function of ${X_s}$ . Equivalently,

$\displaystyle {\mathbb E}\left[f(X_t)\mid\mathcal{F}_s\right]={\mathbb E}\left[f(X_t)\mid X_s\right]$

(1)

(almost surely). More generally, this idea makes sense with respect to any filtered probability space ${\mathbb{F}=(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$ . A process X is Markov with respect to ${\mathbb{F}}$ if it is adapted and (1) holds for times ${s\le t}$ . Continue reading “Markov Processes” →

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