Brownian Motion 1

Table of Contents

Second post in the series. Mainly about the construction of Brownian motion and its properties. I have moved the construction of Gaussian white noise to this post.

Gaussian White Noise

A Gaussan white noise with intensity $\mu$ is an isometry:

\[G: L^2(E, \mathcal{E}, \mu) \to K\]

where $K$ is a centered Gaussian space.

The isometry allows us to compute variance and covariance easily:

\[\mathbb{E}[G(f)^2] = \int f^2 d\mu\]

for $f \in L^2$ and $G(f)$ a Gaussian random variable.

Note covariance can be computed by inner product, since both spaces are Hilbert spaces.

Existence of Gaussian white noise: If $\mu$ is a $\sigma$-finite measure on $(E, \mathcal{E})$, then there exists a Gaussian white noise with intensity $\mu$ on some appropriate probability space $(\Omega, \mathcal{F}, P)$.

The proof follows from the existence of a Gaussian process given the covariance function (in the previous post). It suffices to set the covariance function $\Gamma(f, g) = \langle f, g \rangle_{L^2(\mu)}$ to construct a Gaussian process $(X_f)_{f\in L^2}$ and then we set $G(f)=X_f$.

When $L^2$ is separable, we can explicitly take an orthonormal basis $(\varphi_n)$ and set \(G(f) = \sum_{n=1}^\infty \xi_n \langle f, \varphi_n \rangle_{L^2(\mu)}\)

The Haar system is used in Levy’s construction of Brownian motion.

The following is useful when discussing the variation of Brownian motion.

And continuous local martingales in general, which is discussed in chapter 4.

(Lemma) Quadratic variation: Let $A$ be a set with finite measure and a sequence of partitions indexed by $n$, $$ A=A_1^n \cup \cdots A_{k_n}^n $$ whose mesh tends to zero (i.e. $\lim_{n\to \infty} \sup_{1\leq i \leq k_n} \mu (A_i^n) = 0$). Then $$ \lim_{n\to \infty} \sum_{i=1}^{k_n} G(\mathbf{1}_{A_i^n})^2 = \mu(A) $$ where the convergence is in $L^2$.

The proof is to write $\mathbb{E}[\left(\sum G(A_j)^2 - \mu(A) \right)]=var(\sum G(A_j)^2)$ and uses $var(X^2) = \mathbb{E}[X^4] - \mathbb{E}[X^2]^2=3\sigma^4 - \sigma^4 = 2\sigma^4$ for $X\sim N(0, \sigma^2)$.

Example:
This is actually an example from later sections, but it is a good illustration.
Let $B$ be a Brownian motion and $0=t_1^n < \cdots < t_{k_n}^n$ be a sequence of partitions of $[0, t]$ with mesh tending to zero. Then we have

\[\lim_{n\to \infty} \sum_{i=1}^{k_n} (B_{t_{i+1}^n} - B_{t_i^n})^2 = t\]

in $L^2$. Combined with the fact that $B_t$ is a.s. continuous, we see $B$ has infinite variation on any nontrivial interval, almost surely. Thus, it is NOT possible to define a Riemann-Stieltjes integral with respect to $B$, motivating the construction of stochastic integrals.

Pre-Brownian Motion

We first construct the pre-Brownian motion using Gaussian white noise, which has most of the properties of Brownian motion except for the continuity.

Let $G$ be a Gaussian white nosie on $(\mathbb{R}_{+}, \mathcal{B}(\mathbb{R}_{+}), \lambda)$, where $\lambda$ is the Lebesgue measure, then we define the pre-Brownian motion as the process $(B_t)_{t\geq 0}$ with

\[B_t = G(\mathbf{1}_{[0, t]})\]

From isometry, it is immediate that the covariance $\mathbb{E}[B_sB_t] = \min(s, t)$.

Equivalent characterization

Let $(X_t)_{t\geq 0}$ be a real-valued random process. The following are equivalent:

• $(X_t)_{t\geq 0}$ is a pre-Brownian motion.

• $(X_t)_{t\geq 0}$ is a centered Gaussian process with covariance $\mathbb{E}[X_sX_t] = \min(s, t)$.

• (Independent increments) $X_0=0$ almost surely. For any $0=t_0 < t_1 < \cdots < t_n$, the random variables $X_{t_0}, X_{t_1}-X_{t_0}, \cdots, X_{t_n}-X_{t_{n-1}}$ are independent and follows $\mathcal{N}(0, t_i-t_{i-1})$ distribution.

• (Memoryless) $X_0=0$ almost surely and for every $0 \leq s < t$, $X_t-X_s$ is independent of $\sigma(X_r, 0\leq r \leq s)$ and distributed as $\mathcal{N}(0, t-s)$.

To go from independent increments to the definition by white noise, we can construct from a step function

\[f = \sum \lambda_i \mathbf{1}_{(t_{i-1}, t_i]}\]

and set $G(f)=\sum_{i=1}^n \lambda_i (X_{t_i}-X_{t_{i-1}})$, we can verify this is an isometry on the space of step functions and then use approximation to extend to $L^2$.

It follows that $G$ is determined by $B$ and this motivates the definition of a Wiener integral.

\[f \mapsto G(f\mathbf{1}_{[0, t]}) = \int_0^t f(s) dB_s\]

So to show a process is a Brownian motion, we can show it is a pre-Brownian motion and has continuous sample paths. Even simpler, we can show it’s a centered Gaussian process with the covariance function specified and continuous sample paths.

Finite dimensional distribution: A random process is a pre-Brownian motion if and only if for every $0=t_0 < t_1 < \cdots < t_n$, the vector $(B_{t_1}, \ldots, B_{t_n})$ has the density: $$ p(x_1, \ldots, x_n) = \frac{1}{(2\pi)^{n/2}}\sqrt{t_1(t_2-t_1)\cdots (t_n-t_{n-1})} \exp \left( -\frac{1}{2} \sum_{i=1}^n \frac{(x_i-x_{i-1})^2}{t_i-t_{i-1}} \right) $$

Transformation of pre-Brownian motion

• (Scaling) For any $c>0$, $B^c_t := cB_{t/c^2}$ is a pre-Brownian motion.

• (Symmetry) $-B_t$ is a pre-Brownian motion.

• (Simple Markov) For any $s\geq 0$, $B_t^{(s)}=B_{s+t}-B_s$ is a pre-Brownian motion. (indexed by $t$)

• (Time inversion) The process $W_t$ defined by $W_0=0$ and $W_t = tB_{1/t}$ for $t>0$ is a pre-Brownian motion.

• (Time reversal) The process $B_t’:=B_1 - B_{1-t}$ is a pre-Brownian motion.

Continuous Sample Paths

By sample paths of a stochastic process, we mean the collection of mappings: \(T \ni t \mapsto X_t(\omega) \in \mathbb{R}\) The collection is indexed by $\omega \in \Omega$.

Modification and indistinguishability

Let $(X_t)_{t\in T}$ and $(\bar{X}_t)_{t \in T}$ be two stochastic processes. We say that $(\bar{X}_t)_{t \in T}$ is a modification of $(X_t)_{t\in T}$ if for every $t\geq 0$

\[\mathbb{P}(X_t = \bar{X}_t)=1\]

We say $(\bar{X}_{t})_{t\in T}$ is indistinguishable from $(X_t)_{t\in T}$ if for every $t\geq 0$ if there exists a negligible set $N$ such that:

\[\forall \omega \in \Omega \setminus N, \forall t \in T, \quad X_{t}(\omega) = \bar{X}_{t}(\omega)\]

Informally, this is $P(X_t = \bar{X}_{t}, \forall t\in T)=1$ (informally as this might not be a measurable set).

Example:

Let $X_t$ be a stochastic process s.t.

\[X(t)(\omega) = \begin{cases} 1 & t=\omega \\ 0 & t \neq \omega \end{cases}\]

Then $X_t$ is a modification of $0$ but not indistinguishable from $0$.

Remark: If $(X_t)_{t\in T}$ and $(\bar{X}_t)_{t\in T}$ are indistinguishable, then they are modifications of each other.

In fact, the converse of the remark above is also true under some conditions.

Proposition: Fix an interval $I \subset \mathbb{R}$. If $(X_t)_{t\in T}$ and $(\bar{X}_t)_{t\in T}$ are modifications of each other and both processes right/left continuous, then they are indistinguishable.

A proof can be found at this answer.

A process $X$ is said to be Hölder continuous with exponent $\alpha$ if for every $\omega \in \Omega$, there exists finite constant $C_{\alpha}(\omega)$ such that for all $s, t \in I$ a bounded interval,

\[d(X_s(\omega), X_t(\omega)) \leq C_{\alpha}(\omega) |t-s|^{\alpha}\]

where $d$ is a metric on $(E, d)$.

Now we present the main tool for constructing a continuous modification of a stochastic process.

Kolmogorov's lemma: Let $X = (X_{t})_{t\in I}$ be stochastic process indexed by a bounded interval $I$ and taking values in a complete metric space $(E, d)$. Assume that there exists three reals $q, \varepsilon, \alpha$ such that for all $s, t \in I$, $$ \mathbb{E}[d(X_s, X_t)^q] \leq C|t-s|^{1+\varepsilon} $$ Then $X$ has a modification which is Hölder continuous with exponent $\alpha$ for every $\alpha \in (0, \varepsilon/q)$. In particular, this modification is unique up to indistinguishability as we have remarked above.

The proof invovles first using a Markov inequality to get a bound on $P(d(X_{s}, X_{t}) \geq a)\leq Ca^{-q}\vert t-s\vert^{1+\varepsilon}$ and divide the bounded interval into small pieces of length $2^{-n}$. Then choosing $a=2^{-n\alpha}$ and using the Borel-Cantelli lemma to show that $\limsup \bigcup_{i=1}^{2^n} {d(X_{(i-1)2^{-n}}, X_{i2^{-n}}) \geq 2^{-n\alpha}}$ has probability zero.

Then another technical lemma from analysis is used. Let $D$ be all the reals in $I$ that can be written as a $i2^{-n}$ for some $i, n \in \mathbb{N}$, which is dense.

Lemma: Assume $\exists \alpha>0, K<\infty$, s.t. $\forall n\geq 1$, $\forall i \in {1, 2, \ldots, 2^n-1}$, let $f$ be a mapping defined on $D \subseteq E$, s.t.

\[d(f((i-1)2^{-n}), f(i2^{-n})) \leq K2^{-n\alpha}\]

Then we can extend this to any $s,t\in D$,

\[d(f(s), f(t)) \leq \frac{2K}{1-2^{-\alpha}}|t-s|^{\alpha}\]

By completeness assumption of $E$, we can extend $f$ to all of $I$ and the extension is Hölder continuous with exponent $\alpha$. More explicitly, the extension is defined by

\[\tilde{X}_t = \lim_{s\to t, s\in D} X_s(\omega)\]

when the constant $K$ is finite (which is a event of probability one).

The final step is to check this limit is equal to the $X_t$ almost surely. This is true from the Markov inequality above, which gives convergence in probability (hence subsequential convergence almost surely, which matches the limit above).

Corollary: If $X$ is a pre-Brownian motion, then it has a modification which is almost surely Hölder continuous with exponent $\alpha=\frac{1}{2}-\delta$ for every $\delta \in (0, \frac{1}{2})$.

Finally! We arrive at the definition of Brownian motion.

A process $(B_t)_{t\geq 0}$ is a Brownian motion if it is a pre-Brownian motion and all sample paths are continuous.

More is to be discussed in the next post, including the Wiener measure, properties of sample paths (Blumenthal’s 0-1 law), and the strong Markov property.