4.1. Wiener process - Brownian motion

• Introduction: history and relation with other problems (diffusion?)

• Definition and some theory

• Simulation of Wiener process, demonstration of properties shown in theory section

4.1.1. Definition

Definition 4.1 (Wiener process - Brownian motion)

A Wiener process is a random process \(W(t)\) with

1. initial condition, almost surely

\[W(0) = 0\]

2. increments with zero-mean normal distribution

\[W(t) - W(s) \sim \mathscr{N}(0,|t-s|)\]

3. \(W\) has independent increments: \(W(t) - W(t+u)\) is independent from \(W_s\), \(s < t\)

4. in \(W(t)\) is almost surely continuous in \(t\)

Almost sure convergence in statistics

“Almost surely” here means almost sure converngece and it is explained in the section dealing with convergence in statistics, and used below to prove some properties of a Wiener process.

4.1.2. Properties

Property 4.1 (Covariance of increments)

Covariance of an increment follows the definition of Wiener process and the definition of normal distribution,

(4.1)\[\mathbb{E}\left[ \left( W(t) - W(s) \right)^2 \right] = \mathbb{E}\left[ \mathscr{N}(0, |t-s|) \right] = |t - s| \ .\]

Covariance of independent increments - on non-overlapping ranges - is zero, as independence implies no correlation, i.e. zero covariance. Thus, if \(a \le b \le c \le d\), \(W(b)-W(a)\) and \(W(d)-W(c)\) are independent by property \((3)\) in Definition 4.1 of Wiener process, and thus their covariance - and correlation - is zero,

(4.2)\[\mathbb{E}\left[ ( W(b) - W(a) ) ( W(d) - W(c) ) \right] = 0\]

Covariance of two generic increments reads

(4.3)\[\mathbb{E}\left[ \left( W(t_1) - W(s_1) \right) \left( W(t_2) - W(s_2) \right) \right] = \big| [s_1, t_1] \cap [s_2, t_2]\big|\]

as it’s proved below.

Proof of the covariance of two generic increments

If \(s_1 \le s_2 \le t_2 \le t_1\),

\[\begin{split}\begin{aligned} & \mathbb{E}\left[ \left( W(t_1) - W(s_1) \right) \left( W(t_2) - W(s_2) \right) \right] = \\ & \quad = \mathbb{E}\left[ \left( W(t_1) - W(t_2) + W(t_2) - W(s_2) + W(s_2) - W(s_1) \right) \left( W(t_2) - W(s_2) \right) \right] = \\ & \quad = \underbrace{\mathbb{E}\left[ \left( W(t_1) - W(t_2) \right) (W(t_2) - W(s_2) ) \right]}_{=0} + \underbrace{\mathbb{E}\left[ \left( W(t_2) - W(s_2) \right) (W(t_2) - W(s_2) ) \right]}_{=|t_2 - s_2|} + \\ & \quad + \underbrace{\mathbb{E}\left[ \left( W(s_2) - W(s_1) \right) (W(t_2) - W(s_2) ) \right]}_{= 0} = \\ & \quad = 0 + |t_2 - s_2| + 0 = \big| [s_1, t_1] \cap [s_2, t_2] \big| \ . \end{aligned}\end{split}\]

Similarly, if \(s_1 \le s_2 \le t_1 \le t_2 \),

\[\begin{split}\begin{aligned} & \mathbb{E}\left[ \left( W(t_1) - W(s_1) \right) \left( W(t_2) - W(s_2) \right) \right] = \\ & \quad = \mathbb{E}\left[ \left( W(t_1) - W(s_2) + W(s_2) - W(s_1) \right) \left( W(t_2) - W(t_1) + W(t_1) - W(s_2) \right) \right] = \\ & \quad = \underbrace{\mathbb{E}\left[ \left( W(t_1) - W(s_2) \right) (W(t_2) - W(t_1) ) \right]}_{=0} + \underbrace{\mathbb{E}\left[ \left( W(t_1) - W(s_2) \right) (W(t_1) - W(s_2) ) \right]}_{=|t_1 - s_2|} + \\ & \quad + \underbrace{\mathbb{E}\left[ \left( W(s_2) - W(s_1) \right) (W(t_2) - W(t_1) ) \right]}_{= 0} + \underbrace{\mathbb{E}\left[ \left( W(s_2) - W(s_1) \right) (W(t_1) - W(s_2) ) \right]}_{= 0} = \\ & \quad = 0 + |t_1 - s_2| + 0 = \big| [s_1, t_1] \cap [s_2, t_2] \big| \ . \end{aligned}\end{split}\]

All the other situations can be proved in the same way.

Property 4.2 (Statistics of maximum)

For \(a \ge 0\),

\[P\left( M(t) \ge a \right) = 2 P \left( W(t) \ge a \right) = 2 - 2 \, \phi\left( \frac{a}{\sqrt{t}} \right) \ ,\]

with

\[M(t) = \max_{0 \le \tau \le t} W(\tau)\]

and

\[\phi(x) = \int_{y = -\infty}^{x} p_{\mathscr{N}(0,1)}(x) \, dx\]

is the cumulative probability function of a normal distribution \(\mathscr{N}(0,1)\).

Proof.

The second inequality immediately follows from the very definition of Wiener process with initial conditions \(W(0)\),

\[\begin{split}\begin{aligned} P \left( W(t) - W(0) \ge a \right) & = P \left( \mathscr{N}(0,t) \ge a \right) = && (1) \\ & = P \left( \mathscr{N}\left(0, 1 \right) > \frac{a}{\sqrt{t}} \right) = \\ & = \int_{x=\frac{a}{\sqrt{t}}}^{+\infty} p(y) \, dy = && (2) \\ & = 1 - \int_{x=-\infty}^{\frac{a}{\sqrt{t}}} p(y) \, dy = 1 - \phi \left( \frac{a}{\sqrt{t}} \right) \end{aligned}\end{split}\]

having used \((1)\) scaling rule for transformation of probability functions… todo, and \((2)\) the normalization condition of the probability density \(1 = \int_{x = -\infty}^{+\infty} p(x) \, dx\), and the definition of cumulative probability function.

First inequality. In order to prove the first inequality, it could be useful to introduce the definition of stepping time, \(\tau_a\), as the random variable defined as

\[\tau_a = \min_s \left\{ s: W(s) = a \right\} \ .\]

Using reflection principle, it follows

\[\begin{split}\begin{aligned} P( M(t) \ge a ) & = && (1) \\ & = P( M(t) \ge a, W(t) \ge a) + P( M(t) \ge a, W(t) < a) = && (2) \\ & = P( W(t) \ge a ) + P( M(t) \ge a, W(t) - W(\tau_a) < 0) = && (3) \\ & = P( W(t) \ge a ) + P( M(t) \ge a, W'(t - \tau_a) < 0) = && (4) \\ & = P( W(t) \ge a ) + P( M(t) \ge a ) P ( W'(t - \tau_a) < 0) = && (5) \\ & = P( W(t) \ge a ) + \dfrac{1}{2} P( M(t) \ge a ) \ . \end{aligned}\end{split}\]

haing \((1)\) used “marginalization” to write \(P(A) = P(A,B) + P(A, \overline{B})\), \((2)\) recognized that if \(B: \, W(t) \ge a\) then \(A: \, M(t) \ge a\) or \(B \subseteq A\), and thus \(P(A,B) = P(B)\), and that \(a = W(\tau_a)\), \((3)\) defined the Wiener process \(W'(t - \tau_a) := W(t) - W(\tau_a)\), independent from \(W(s)\), \(0 \le s \le \tau_a\), \((4)\) exploited the independence of the two conditions (todo be more explicit, proof needed?), \((5)\) and the symmetry of Wiener process to get \(P(W'(t-\tau_a) < 0) = \frac{1}{2}\).

Thus, it follows the requied relation

\[P(M(t) \ge a) = 2 P(W(t) \ge a) \ .\]

Property 4.3 (\(W(t)\) is almost surely not differentiable)

For all time \(t\), a Wiener process is almost surely not differentiable, i.e. …todo

Proof.

todo check details

Wiener process is differentiable in \(t\) if the limit

\[\lim_{h \rightarrow 0} \dfrac{W(t+h) - W(t)}{h} = \ell\]

exists finite. Definition of limit reads,

\[\forall \varepsilon > 0 \quad \exists U_{0,\delta} \quad \text{s.t.} \quad \left| \, \frac{W(t+h) - W(t)}{h} - \ell \, \right| < \varepsilon \qquad \forall h \in U_{0,\delta} \backslash \{ 0 \}\]

todo how to go from this definition to the following one?

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Let \(E_{\varepsilon, A, t_0}\) be the event s.t. for a given \(t_0\), \(W(t)\) is differentiable in \(t_0\), i.e. \(\exists\) \(A\), \(\varepsilon_0\) const. s.t. \(\forall \varepsilon\) s.t. \(0 < \varepsilon < \varepsilon_0\), \(W(t) - W(t_0) \le A \varepsilon\) holds for \(\forall \varepsilon\), \(0 < t - t_0 \le \varepsilon\).

Let \(E_{A, t_0} = \cap_{\varepsilon} E_{\varepsilon, A, t_0}\). Then

\[\begin{split}\begin{aligned} P\left( E_{\varepsilon, A, t_0} \right) & = P \left( |W(t) - W(t_0)| \le A \varepsilon \text{ for } \forall t-t_0 \text{ s.t. } 0 < t - t_0 \le \varepsilon \right) = && (1) \\ & = P \left( M(t - t_0)| \le A \varepsilon \right) = && (2) \\ & = 1 - P \left( M(t - t_0)| \ge A \varepsilon \right) = && (3) \\ & = 1 - \left[ 2 - 2 \phi \left( \dfrac{A \varepsilon}{\sqrt{\Delta t}} \right) \right] = \\ & = - 1 + 2 \phi \left( \dfrac{A \varepsilon}{\sqrt{\Delta t}} \right) \ , \end{aligned}\end{split}\]

having used \((1)\)…, \((2)\)…, \((3)\)…

Now, being \(\varepsilon \le \Delta t\), it follows that \(\frac{\varepsilon}{\sqrt{\Delta t}} \le \sqrt{\Delta t}\). As \(\varepsilon \rightarrow 0\), then \(\frac{\varepsilon}{\sqrt{\Delta t}} \rightarrow 0\), and \(\phi\left( \frac{A \varepsilon}{\sqrt{\Delta t}} \right) \rightarrow \frac{1}{2}\), and \(P(E_{\varepsilon,A,t_0}) \rightarrow 0\)