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 2010

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Hitting Times and Maximum Values

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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QuestionofDay

Question of the Day

What is the probability that a simple random walk with $p = 1 ∕ 2 = q$ starting at the origin will hit value $a > 0$ before it hits value $- b < 0$ , where $b > 0$ ? What do you expect in analogy for the standard Wiener process and why?

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Key Concepts

With the Reflection Principle, we can derive the p.d.f of the hitting time $T_{a}$ .
With the hitting time, we can derive the c.d.f. of the maximum of the Wiener Process on the interval $0 \leq u \leq t$ .

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Vocabulary

The Reflection Principle for the Wiener process reflected about a first passage has the same distribution as the original motion.
The hitting time $T_{a}$ is the first time the Wiener process assumes the value $a$ . Specifically in notation from analysis $T_{a} = inf {t > 0 : W (t) = a} .$

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Mathematical Ideas

Hitting Times

Consider the standard Wiener process $W (t)$ , which starts at $W (0) = 0$ . Let $a > 0$ . Let us denote the hitting time $T_{a}$ be the first time the Wiener process hits $a$ . Specifically in notation from analysis

T_{a} = inf {t > 0 : W (t) = a} .

Note the very strong analogy with the duration of the game in the gambler’s ruin.

Some Wiener process sample paths will hit $a > 0$ fairly directly. Others will make an excursion (for example, to negative values) and take a long time to finally reach $a$ . Thus $T_{a}$ will have a probability distribution. We will determine that distribution by a heuristic procedure similar to the first step analysis we made for coin-flipping fortunes.

Specifically, we will consider a probability by conditioning, that is, conditioning on whether or not $T_{a} \leq t$ , for some given value of $t$ .

ℙ [W (t) \geq a] = ℙ [W (t) \geq a | T_{a} \leq t] ℙ [T_{a} \leq t] + ℙ [W (t) \geq a | T_{a} > t] ℙ [T_{a} > t]

Now note that the second conditional probability is $0$ because it is an empty event. Therefore:

ℙ [W (t) \geq a] = ℙ [W (t) \geq a | T_{a} \leq t] ℙ [T_{a} \leq t] .

Now, consider Wiener process “started over” again the time $T_{a}$ when it hits $a$ . By the shifting transformation from the previous section, this would have the distribution of Wiener process again, and so

\begin{array}{l} ℙ [W (t) \geq a | T_{a} \leq t] & = ℙ [W (t) \geq a | W (T_{a}) = a, T_{a} \leq t] \\ = ℙ [W (t) - W (T_{a}) \geq 0 | T_{a} \leq t] \\ = 1 ∕ 2 . \end{array}

This argument is a specific example of the Reflection Principle for the Wiener process. It says that the Wiener process reflected about a first passage has the same distribution as the original motion.

Actually, this argument contains a serious logical gap, since $T_{a}$ is a random time, not a fixed time. That is, the value of $T_{a}$ is different for each sample path, it varies with $ω$ . On the other hand, the shifting transformation defined in the prior section depends on having a fixed time, called $h$ in that section. In order ti fix this logical gap, we must make sure that “random times” act like fixed times. Under special conditions, random times can act like fixed times. Specifically, this proof can be fixed and made completely rigorous by showing that the standard Wiener process has the strong Markov property and that $T_{a}$ is a Markov time corresponding to the event of first passage from $0$ to $a$ .

Thus

ℙ [W (t) \geq a] = (1 ∕ 2) ℙ [T_{a} \leq t] .

\begin{array}{l} ℙ [T_{a} \leq t] & = 2 ℙ [W (t) \geq a] \\ = \frac{2}{\sqrt{2 π t}} \int_{a}^{\infty} exp (- u^{2} ∕ (2 t)) d u \\ = \frac{2}{\sqrt{2 π}} \int_{a ∕ \sqrt{t}}^{\infty} exp (- v^{2} ∕ 2) d v \end{array}

(note the change of variables $v = u ∕ \sqrt{t}$ in the second integral) and so we have derived the c.d.f. of the hitting time random variable. One can easily differentiate to obtain the p.d.f

f_{T_{a}} (t) = \frac{a}{\sqrt{2 π}} t^{- 3 ∕ 2} exp (- a^{2} ∕ (2 t)) .

Note that this is much stronger than the analogous result for the duration of the game until ruin in the coin-flipping game. There we were only able to derive an expression for the expected value of the hitting time, not the probability distribution of the hitting time. Now we are able to derive the probability distribution of the hitting time fairly intuitively (although strictly speaking there is a gap). Here is a place where it is simpler to derive a quantity for Wiener process than it is to derive the corresponding quantity for random walk.

Let us now consider the probability that the Wiener process hits $a > 0$ , before hitting $- b < 0$ , where $b > 0$ . To compute this we will make use of the interpretation of Standard Wiener process as being the limit of the symmetric random walk. Recall from the exercises following the section on the gambler’s ruin in the fair ( $p = 1 ∕ 2 = q$ ) coin-flipping game that the probability that the random walk goes up to value $a$ before going down to value $b$ when the step size is $Δ x$ is

ℙ [to a before - b] = \frac{b Δ x}{(a + b) Δ x} = \frac{b}{a + b}

Thus, the probability of $a > 0$ , before hitting $- b < 0$ does not depend on the step size, and also does not depend on the time interval. Therefore in passing to the limit the probabilities should remain the same. Here is a place where it is easier to derive the result from the coin-flipping game and pass to the limit than to derive the result directly from Wiener process principles.

The Distribution of the Maximum

Let $t$ be a given time, let $a > 0$ be a given value, then

\begin{array}{l} ℙ [max_{0 \leq u \leq t} W (u) \geq a] & = ℙ [T_{a} \leq t] \\ = \frac{2}{\sqrt{2 π}} \int_{a ∕ \sqrt{t}}^{\infty} exp (- y^{2} ∕ 2) d y \end{array}

Sources

This section is adapted from: Probability Models, by S. Ross, and A First Course in Stochastic Processes Second Edition by S. Karlin, and H. Taylor, Academic Press, 1975.

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Problems to Work

Problems to Work for Understanding

Differentiate the c.d.f. of $T_{a}$ to obtain the expression for the p.d.f of $T_{a}$ .
Show that $E [T_{a}] = \infty$ for $a > 0$ .
Suppose that the fluctuations of a share of stock of a certain company are well described by a Wiener process. Suppose that the company is bankrupt if ever the share price drops to zero. If the starting share price is $A (0) = 5$ , what is the probability that the company is bankrupt by $t = 25$ ? What is the probability that the share price is above $10$ at $t = 25$ ?.
Suppose you own one share of stock whose price changes according to a Wiener process. Suppose you purchased the stock at a price $b + c$ , $c > 0$ and the present price is $b$ . You have decided to sell the stock either when it reaches the price $b + c$ or when an additional time $t$ goes by, whichever comes first. What is the probability that you do not recover your purchase price?

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Books

Reading Suggestion:

References

[1] S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.

[2] Sheldon M. Ross. Introduction to Probability Models. Elsevier, 8th edition edition, 2003.

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Links

Outside Readings and Links:

Russell Gerrard, City University, London, Stochastic Modeling . Notes for the MSc in Actuarial Science, 2003-2004. Contributed by S. Dunbar October 30, 2005.
Yuval Peres, University of California Berkeley, Department of Statistics . Notes on sample paths of Brownian Motion. Contributed by S. Dunbar, October 30, 2005.

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Last modified: Processed from LATEX source on November 8, 2010