Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

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Rating

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Question of the Day

Suppose you know the graph

y = f (x)

of the function

f (x)

. What is the effect on the graph of the transformation

f (x + h) - f (h)

? What is the effect on the graph of the transformation

f (1 ∕ x)

? Consider the function

f (x) = sin (x)

as an example.

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Key Concepts

Vocabulary

Mathematical Ideas

Transformations of the Wiener Process

A set of transformations of the Wiener process produce the Wiener process again. Since these transformations result in the Wiener process, each tells us something about the “shape” and “characteristics” of the Wiener process. These results prove especially helpful when studying the properties of the Wiener process sample paths. The first of these transformations is a time homogeneity which says that the Wiener process can be re-started anywhere. The second says that the Wiener process can be rescaled in time and space. The third is an inversion. Roughly, each of these says the Wiener process is self-similar in various ways. See the comments after the proof for more detail.

Proof. We have to systematically check each of the defining properties of the Wiener process in turn for each of the transformed processes.

$W_{shift} (t) = W (t + h) - W (h)$
1. The increment $W_{shift} (t + s) - W_{shift} (s) = [W (t + s + h) - W (h)] - [W (s + h) - W (h)] = W (t + s + h) - W (s + h)$
  which is by definition normally distributed with mean $0$ and variance $t$ .
2. The increment $W_{shift} (t_{4}) - W_{shift} (t_{3}) = W (t_{4} + h) - W (t_{3} + h)$
  is independent from
  $W (t_{2} + h) - W (t_{1} + h) = W_{shift} (t_{2}) - W_{shift} (t_{1}),$
  by the property of independence of disjoint increments of $W (t)$ .
3. $W_{shift} (0) = W (0 + h) - W (h) = 0 .$
4. As the composition and difference of continuous functions, $W_{shift}$ is continuous.
$W_{scale} (t) = c W (t ∕ c^{2})$
1. The increment $W_{scale} (t) - W_{scale} (s) = c W ((t) ∕ c^{2}) - c W (s ∕ c^{2}) = c (W (t ∕ c^{2}) - W (s ∕ c^{2}))$
  is normally distributed because it is a multiple of a normally distributed random variable. Since the increment $W (t ∕ c^{2}) - W (s ∕ c^{2})$ has mean zero, then
  $W_{scale} (t) - W_{scale} (s) = c (W (t ∕ c^{2}) - W (s ∕ c^{2}))$
  must have mean zero. The variance is
  $\begin{array}{l} E [{(W_{scale} (t) - W (s))}^{2}] & = E [{(c W ((t) ∕ c^{2}) - c W (s ∕ c^{2}))}^{2}] \\ = c^{2} E [{(W (t ∕ c^{2}) - W (s ∕ c^{2}))}^{2}] \\ = c^{2} (t ∕ c^{2} - s ∕ c^{2}) = t - s . \end{array}$
2. Note that if $t_{1} < t_{2} \leq t_{3} < t_{4}$ , then $t_{1} ∕ c^{2} < t_{2} ∕ c^{2} \leq t_{3} ∕ c^{2} < t_{4} ∕ c^{2}$ , and the corresponding increments $W (t_{4} ∕ c^{2}) - W (t_{3} ∕ c^{2})$ and $W (t_{2} ∕ c^{2}) - W (t_{1} ∕ c^{2})$ are independent. Then the multiples of each by $c$ are independent and so $W_{scale} (t_{4}) - W_{scale} (t_{3})$ and $W_{scale} (t_{2}) - W_{scale} (t_{1})$ are independent.
3. $W_{scale} (0) = c W (0 ∕ c^{2}) = c W (0) = 0$ .
4. As the composition of continuous functions, $W_{scale}$ is continuous.

□

Proof. To show that

W_{inv} (t) = t W (1 ∕ t)

is a Wiener process by the four definitional properties requires another fact which is outside the scope of the text. The fact is that any Gaussian process $X (t)$ with mean $0$ and $Cov [X (s), X (t)] = min (s, t)$ must be the Wiener process. See the references and outside links for more information. Using this information, we present a partial proof:

$W_{inv} (t) - W_{inv} (s) = t W (1 ∕ t) - s W (1 ∕ s)$
which will be the difference of normally distributed random variables each with mean $0$ , so the difference will be normal with mean $0$ . It remains to check that the normal random variable has the correct variance.
$\begin{array}{l} E [{(W_{inv} (t) - W_{inv} (s))}^{2}] & = E [{(s W (1 ∕ s) - t W (1 ∕ t))}^{2}] \\ = E [{(s W (1 ∕ s) - s W (1 ∕ t) + s W (1 ∕ t) - t W (1 ∕ t) - (s - t) W (0))}^{2}] \\ = s^{2} E [{(W (1 ∕ s) - W (1 ∕ t))}^{2}] + s (s - t) E [(W (1 ∕ s) - W (1 ∕ t)) (W (1 ∕ t) - W (0))] + {(s - t)}^{2} E [{(W (1 ∕ t) - W (0))}^{2}] \\ = s^{2} E [{(W (1 ∕ s) - W (1 ∕ t))}^{2}] + {(s - t)}^{2} E [{(W (1 ∕ t) - W (0))}^{2}] \\ = s^{2} (1 ∕ s - 1 ∕ t) + {(s - t)}^{2} (1 ∕ t) \\ = t - s \end{array}$
Note the use of independence of $W (1 ∕ s) - W (1 ∕ t)$ from $W (1 ∕ t) - W (0)$ at the third equality.
It seems to be hard to show the independence of increments directly. Instead rely on the fact that a Gaussian process with mean $0$ and covariance function $min (s, t)$ is a Wiener process, and thus prove it indirectly.
Note that
$Cov [W_{inv} (s), W_{inv} (t)] = s t min (1 ∕ s, 1 ∕ t) = min (s, t) .$
By definition, $W_{inv} (0) = 0$ .
The argument that $lim_{t \to 0} W_{inv} (t) = 0$ is equivalent to showing that $lim_{t \to \infty} W (t) ∕ t = 0$ . To show this requires use of Kolmogorov’s inequality for the Wiener process and clever use of the Borel-Cantelli lemma and is beyond the scope of this course. Use the translation property in the third statement of this theorem to prove continuity at every value of $t$ .

□

The following comments are adapted from Stochastic Calculus and Financial Applications by J. Michael Steele. Springer, New York, 2001, page 40. These laws tie the Wiener process to three important groups of transformations on

[0, \infty)

, and a basic lesson from the theory of differential equations is that such symmetries can be extremely useful. On a second level, the laws also capture the somewhat magical fractal nature of the Wiener process. The scaling law tells us that if we had even one-billionth of a second of a Wiener process path, we could expand it to a billions years’ worth of an equally valid Wiener process path! The translation symmetry is not quite so startling, it merely says that Wiener process can be restarted anywhere, that is any part of a Wiener process captures the same behavior as at the origin. The inversion law is perhaps most impressive, it tells us that the first second of the life of a Wiener process path is rich enough to capture the behavior of a Wiener process path from the end of the first second until the end of time.

Sources

This section is adapted from: A First Course in Stochastic Processes by S. Karlin, and H. Taylor, Academic Press, 1975, pages 351–353 and Financial Derivatives in Theory and Practice by P. J. Hunt and J. E. Kennedy, John Wiley and Sons, 2000, pages 23–24.

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Problems to Work for Understanding

Reading Suggestion:

Outside Readings and Links:

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