Paul Lévy’s construction of Brownian motion

Contents

In 1923, Norbert Wiener demonstrated the existence of Standard Brownian motion in his paper "Differential Space." Subsequently, Paul Lévy has given the beautiful prove of this fact using dyadic numbers and properties of Gaussian variables. This article elaborates on Lévy's proof, supporting it by visually explaining key concepts.

Definition (Brownian Motion)
A real-valued stochastic process

\{B(t): t \geq 0\}

is called a standard Brownian Motion on

[0, 1]

with

B_0 = 0

if the following properties holds

the process has independent increments, i.e., for all times $0 \leq t_1 < t_2 < \cdots < t_n$ , the increments $B(t_n) - B(t_{n-1}), B(t_{n-1}) - B(t_{n-2}), \ldots, B(t_2) - B(t_1)$ are independent random variables,
for all $t \geq 0$ and $h > 0$ , the increments $B(t + h) - B(t)$ are normally distributed with expectation zero and variance $h$ ,
almost surely, the function $t \mapsto B(t)$ is continuous.

Theorem (Wiener 1923)
Standard Brownian motion exists.

Proof of Wiener Theorem

The foolowing proof follows [MörtersPeres] pages 9-12

Proof of Wiener Theorem

The foolowing proof follows [MörtersPeres] pages 9-12

\quad

Proof. We construct Brownian motion as a uniform limit of continuous functions, to ensure that it automatically has continuous paths.

Building Brownian Motion on Dyadic Set

The idea is to construct the Brownian Motion on the set of dyadic numbers

\begin{equation*}\mathcal{D} = \bigcup_{n \in \mathbb{N}} \mathcal{D}_n \quad \textrm{and} \quad \mathcal{D}_n = \left\{ \frac{k}{2^n} : 0 \leq k \leq 2^n \right\}\end{equation*}

and to interpolate the values on

D

linearly and check that the uniform limit of these continuous functions exists is a Brownian motion.

levyConstruction dyadic.png — The sequence of dyadic sets

To do this let

(\Omega, \mathcal{A}, P)

be a probability space on which a collection

\{Z_t : t \in \mathcal{D}\}

of independent, standard normally distributed random variables can be defined.
Let

B(0) := 0

and

B(1) := Z_1

. For each

n \in \mathbb{N}

we define the random variables

B(d), \, d \in \mathcal{D}_n

, such that

for all $r < s < t$ in $\mathcal{D}_n$ , the random variable $B(t) - B(s)$ is normally distributed with mean zero and variance $t - s$ , and is independent of $B(s) - B(r)$ ,
the vectors $(B(d) : d \in \mathcal{D}_n)$ and $(Z_t : t \in \mathcal{D} \setminus \mathcal{D}_n)$ are independent.

Note that we have already done this for

n=0, \, \mathcal{D}_0 = \{0,1\}

. Proceeding inductively we may assume that we have succeeded in doing it for some

n - 1

. We then define

B(d)

for

d \in \mathcal{D}_n \setminus \mathcal{D}_{n-1}

\begin{equation*}B(d) = \frac{B(d - 2^{-n}) + B(d + 2^{-n})}{2} + \frac{Z_d}{2^{(n+1)/2}}.\end{equation*}

Since

d \, \pm \, 2^{-n} \, \in \, \mathcal{D}_{n-1}

, then the first summand is the linear interpolation of the values of

B

at the neighbouring points of

d

\mathcal{D}_{n-1}

. Therefore

(B(d) : d \in \mathcal{D}_n)

is independent of

\{Z_t : t \in \mathcal{D} \setminus \mathcal{D}_n\}

and the second property is fulfilled.

levyConstruction induction_step.png — The one step of induction to construct $B(d)$

Now we need to verify that successive increments

B(d) - B(d - 2^{-n})

and

B(d + 2^{-n}) - B(d)

for

d \in \mathcal{D}_n \setminus \mathcal{D}_{n-1}

are independent.

\begin{align*}B(d) - B(d - 2^{-n}) &= \frac{B(d + 2^{-n}) - B(d - 2^{-n})}{2} + \frac{Z_d}{2^{(n+1)/2}}. \\B(d + 2^{-n}) - B(d) &= \frac{B(d + 2^{-n}) - B(d - 2^{-n})}{2} - \frac{Z_d}{2^{(n+1)/2}}.\end{align*}

By our induction assumptions

\frac{1}{2}[B(d+2^{-n}) - B(d-2^{-n})] \sim \mathcal{N}(0,2^{-n+1})

and depends only on

(Z_t : t \in \mathcal{D}_{n-1})

, it is independent of

Z_d/2^{(n+1)/2}

. Hence their sum

B(d) - B(d - 2^{-n})

and their difference

B(d + 2^{-n}) - B(d)

are independent and normally distributed with mean zero and variance

2^{-n}

. The same are true for any

d \in \mathcal{D}_{n}

. Two any increments over disjoint intervals are thus independent by summing independent increments over intervals of the form

(k 2^{-n}, (k+1) 2^{-n})

Formally, we can define the functions

\begin{equation*}F_0(t) = \begin{cases}Z_1 & \text{for } t = 1, \\0 & \text{for } t = 0, \\\text{linear in between},\end{cases}\end{equation*}

and, for each

n \geq 1

\begin{equation*}F_n(t) = \begin{cases}2^{-(n+1)/2} Z_t & \text{for } t \in D_n \setminus D_{n-1} \\0 & \text{for } t \in D_{n-1} \\\text{linear between consecutive points in } D_n.\end{cases}\end{equation*}

We can define the partial sum, which for given

m

incorporates all induction steps up to

m

\begin{equation*}B_t^m = \sum_{i=0}^{m} F_i(t)\end{equation*}

Uniform convergence $B_t^m$

We need to prove that with probability one

(B_t^m)_{m \in \mathbb{N}}

converges uniformly on

[0, 1]

. To do it, we can start with evaluation of the successive difference

\begin{equation*}\sup_{t \in [0,1]} | B_t^{m} - B_t^{m-1} | = \sup_{t \in [0,1]} | F_m(t) | = \| F_m \| \leq 2^{-(m+1)/2} \max \{ |Z_d|, \, d \in \mathcal{D}_m \}\end{equation*}

On the other hand, we can evaluate

\mathbb{P}(|Z_d| \geq c\sqrt{m})

using the following estimation of the standard normal distribution:

\begin{equation*}\mathbb{P}(|Z_d| \geq \lambda) \leq \sqrt{\frac{2}{\pi}} \frac{\sigma}{\lambda} \exp\left(-\frac{\lambda^2}{2\sigma^2}\right),\end{equation*}

so that the series

\begin{equation*}\mathbb{P} \left( 2^{-(m+1)/2} \max \{ |Z_d|, \, d \in \mathcal{D}_m \} > \frac{1}{m^2}\right) \leq 2^m \mathbb{P} \left( |Z_1| > \frac{2^{(m+1)/2}}{m^2} \right) \leq \frac{1}{\sqrt{2 \pi}} m^2 2^{\frac{m-1}{2}} \exp\left(-\frac{2^{m}}{m^4}\right)\end{equation*}

The right-hand side is summable, for example, by ratio test. Then using the Borel-Cantelli lemma we can conclude that

\exists n_0 \in \mathbb{N}, \forall m \geq n_0

\begin{equation*}\sup_{t \in [0,1]} | B_t^{m} - B_t^{m-1} | \leq \frac{1}{m^2} \quad \textrm{for large enough} \, m \, \textrm{a.s.}\end{equation*}

and the uniform convergence follows from:

\begin{equation*}\sup_{t \in [0,1]} | B_t^{m+n} - B_t^{m-1} | \leq \sum^n_{i=0} \sup_{t \in [0,1]} | B_t^{m+i} - B_t^{m+i-1} | \leq \sum^n_{i=0} \frac{1}{(m+i)^2}\end{equation*}

Finally, the series

\begin{equation*}B(t) = \sum_{n=0}^{\infty} F_n(t)\end{equation*}

is uniformly convergent on

[0,1]

. We denote the continuous limit by

\{B(t) : t \in [0,1]\}

Continuous extension $B_t$ to the whole $[0,1]$

It remains to check that the increments of this process have the right finite-dimensional distributions. This follows directly from the properties of

B

on the dense set

\mathcal{D} \cap [0,1]

and the continuity of the paths. Indeed, suppose that

t_1 < t_2 < \ldots < t_n

are in

[0,1]

. We find

t_{k_i} \in \mathcal{D}

such that

\lim_{k \to \infty} t_{k_i} = t_i

and infer from the continuity of

B

that, for

1 \leq i \leq n-1

\begin{equation*}B(t_{i+1}) - B(t_i) = \lim_{k \to \infty} (B(t_{k_{i+1}}) - B(t_{k_i})).\end{equation*}

\lim_{k \to \infty} \mathbb{E}[B(t_{k_{i+1}}) - B(t_{k_i})] = 0

and

\begin{equation*}\lim_{k \to \infty} \text{Cov}(B(t_{k_{i+1}}) - B(t_{k_i}), B(t_{k_j}) - B(t_{k_{j-1}})) = \lim_{k \to \infty} 1_{\{j=i+1\}}(t_{k_{i+1}} - t_{k_i}) = 1_{\{j=i+1\}}(t_{i+1} - t_i),\end{equation*}

the increments

B(t_{i+1}) - B(t_i)

are independent Gaussian random variables with mean 0 and variance

t_{i+1} - t_i

, as required.

Extension $B_t$ to $\mathbb{R}$

We have thus constructed a continuous process

B: [0,1] \rightarrow \mathbb{R}

with the same finite-dimensional distributions as Brownian motion. Take a sequence

B_0, B_1, \ldots

of independent

C[0,1]

-valued random variables with the distribution of this process, and define

\{B(t) : t \geq 0\}

by gluing together the parts, more precisely by

\begin{equation*}B(t) = B_{\lfloor t \rfloor}(t - \lfloor t \rfloor) + \sum_{\ell=0}^{\lfloor t \rfloor - 1} B_{\ell}(1), \text{ for all } t \geq 0.\end{equation*}

This defines a continuous random function

B: [0, \infty) \rightarrow \mathbb{R}

and one can see easily from what we have shown so far that it is a standard Brownian motion.

\Box

References

[MörtersPeres]
Peter Mörters and Yuval Peres. Brownian Motion. Cambridge University Press, 2010.

Paul Lévy’s construction of Brownian motion

Proof of Wiener Theorem

Proof of Wiener Theorem

Proof of Wiener Theorem

Building Brownian Motion on Dyadic Set

Uniform convergence BtmB_t^mBtm​

Continuous extension BtB_tBt​ to the whole [0,1][0,1][0,1]

Extension BtB_tBt​ to R\mathbb{R}R

References

References

Uniform convergence $B_t^m$

Continuous extension $B_t$ to the whole $[0,1]$

Extension $B_t$ to $\mathbb{R}$