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

Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2009

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Approximation of Brownian Motion by Coin-Flipping Sums

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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QuestionofDay

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 $f (a x)$ where $a > 1$ ? What is the effect on the graph of the transformation $(1 ∕ a) f (x)$ where $a > 1$ ? What about the transformation $f (a x) ∕ b$ where $a > 1$ and $b > 1$ .

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Key Concepts

Brownian motion can be approximated by a properly scaled “random fortune” process (i.e. random walk).
Brownian motion is the limit of “random fortune” discrete time processes (i.e. random walks), properly scaled. The study of Brownian motion is therefore an extension of the study of random fortunes.

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Vocabulary

We define approximate Brownian Motion $Ŵ_{N} (t)$ to be the rescaled random walk with steps of size $1 ∕ \sqrt{N}$ taken every $1 ∕ N$ time units where $N$ is a large integer.

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Mathematical Ideas

Approximation of Brownian Motion by Fortunes

As we have now assumed many times, $i \geq 1$ let

Y_{i} = \{\begin{matrix} + 1 with probability 1/2 \\ - 1 with probability 1/2 \end{matrix}

be a sequence of independent, identically distributed Bernoulli random variables. Note that $Var [Y_{i}] = 1$ , which we will need to use in a moment. Let $Y_{0} = 0$ for convenience and set

T_{n} = \sum_{i = 0}^{n} Y_{i}

be the sequence of sums, which represent the successive net fortunes of our notorious gambler. As usual, we will sketch the graph of $T_{n}$ versus time using linear interpolation between the points $(n - 1, T_{n - 1})$ and $(n, T_{n})$ to obtain a continuous, piecewise linear function. Since this interpolation defines a function for all time, we could write $Ŵ (t)$ , and then for instance, $Ŵ (n) = T_{n}$ . Now $Ŵ (t)$ is a function defined on $[0, \infty)$ . This function is piecewise linear with segments of length $\sqrt{2}$ . The notation $Ŵ (t)$ reminds us of the piecewise linear nature of the function.

Now we will compress time, and rescale the space in a special way. Let $N$ be a large integer, and consider the rescaled function

Ŵ_{N} (t) = (\frac{1}{\sqrt{N}}) Ŵ (N t) .

This has the effect of taking a step of size $\pm 1 ∕ \sqrt{N}$ in $1 ∕ N$ time unit. That is,

Ŵ_{N} (1 ∕ N) = (\frac{1}{\sqrt{N}}) Ŵ (N \cdot 1 ∕ N) = \frac{T_{1}}{\sqrt{N}} = \frac{Y_{1}}{\sqrt{N}} .

Now consider

Ŵ_{N} (1) = \frac{Ŵ (N)}{\sqrt{N}} = \frac{T_{N}}{\sqrt{N}} .

According to the Central Limit Theorem, this quantity is approximately normally distributed, with mean zero, and variance 1. More generally,

Ŵ_{N} (t) = \frac{Ŵ (N t)}{\sqrt{N}} = \sqrt{t} \frac{Ŵ (N t)}{\sqrt{N t}}

If $N t$ is an integer, this will be normally distributed with mean $0$ and variance $t$ . Furthermore, $Ŵ_{N} (0) = 0$ and $Ŵ_{N} (t)$ is a continuous function, and so is continuous at $0$ . Altogether, this should be a strong suggestion that $Ŵ_{N} (t)$ is an approximation to Standard Brownian Motion. We will define the very jagged piecewise linear function $Ŵ_{N} (t)$ as approximate Brownian Motion.

Theorem 1. The joint distributions of $Ŵ_{N} (t)$ converges to the joint normal distribution

f (x_{1}, t_{1}; x_{2}, t_{2}; \dots; x_{n}, t_{n}) = p (x_{1}, t) p (x_{2} - x_{1}, t_{2} - t_{1}) \dots p (x_{n} - x_{n - 1}, t_{n} - t_{n - 1})

of the Standard Brownian Motion.

With some additional foundational work, a mathematical theorem establishes that the rescaled fortune processes actually converge to the mathematical object called the Standard Brownian Motion as defined in the previous section. The proof of this mathematical theorem is beyond the scope of a text of this level, but the theorem above should strongly suggest how this can happen, and give some intuitive feel for the approximation of Brownian motion through the rescaled coin-flip process.

Sources

This section is adapted from Probability by Leo Breiman, Addison-Wesley, Reading MA, 1968, Section 12.2, page 251. This section also benefits from ideas in W. Feller, in Introduction to Probability Theory and Volume I, Chapter III and An Introduction to Stochastic Modeling 3rd Edition, H. M. Taylor, S. Karlin, Academic Press, 1998.

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Problems to Work

Problems to Work for Understanding

Flip a coin $25$ times, recording whether it comes up Heads or Tails each time. Scoring $Y_{i} = + 1$ for each Heads and $Y_{i} = - 1$ for each flip, also keep track of the accumulated sum $T_{n} = \sum_{i = 1}^{n} T_{i}$ for $i = 1 \dots 25$ representing the net fortune at any time. Plot the resulting $T_{n}$ versus $n$ on the interval $[0, 25]$ . Finally, using $N = 5$ plot the rescaled approximation $Ŵ_{5} (t) = (1 ∕ \sqrt{5}) T (5 t)$ on the interval $[0, 5]$ on the same graph.

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Books

Reading Suggestion:

References

[1] Leo Breiman. Probability. SIAM, 1992.

[2] William Feller. An Introduction to Probability Theory and Its Applications, Volume I, volume I. John Wiley and Sons, third edition, 1973. QA 273 F3712.

[3] H. M. Taylor and Samuel Karlin. An Introduction to Stochastic Modeling. Academic Press, third edition, 1998.

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Links

Outside Readings and Links:

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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1

Email to Steve Dunbar, sdunbar1 at unl dot edu

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