Brownian Bridge Fourier Expansions

Sine series
Figure 1: Sine series approximations to a Brownian bridge

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 {[0,T]} 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 {[0,1]}. This does not reduce the generality, since bridges on an interval {[0,T]} can be expressed as scaled versions of standard Brownian bridges.

Theorem 1 A standard Brownian bridge B can be decomposed as

\displaystyle  B_t=\sum_{n=1}^\infty\frac{\sqrt2Z_n}{\pi n}\sin(\pi nt) (1)

over {0\le t\le1}, where {Z_1,Z_2,\ldots} is an IID sequence of standard normals. This series converges uniformly in t, both with probability one and in the {L^p} norm for all {1\le p < \infty}.

Continue reading “Brownian Bridge Fourier Expansions”

Brownian Bridges

Brownian bridges
Figure 1: Brownian bridges on subintervals of Brownian motion

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”