Random Walks

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Chapter 8

• Random Walks

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8.1 First Passage Time

• Toss a coin infinitely many times. Assume the
  tosses are independent, and on each toss, the
  probability of H is ½, as is the probability of
  T.
• Define

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• The process is a symmetric random
  walk (see Fig. 8.1)

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• Its analogue in continuous time is Brownian
  motion.
• Define
• The random variable is called the first passage
  time to 1.
• Note that (1) It is the first time the number of
  heads exceeds by one the number of tails.
• (2) If never gets to 1 (e.g.,
  ,) then

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8.2 is almost surely finite

• and are martingales,
  where
• Since and a stopped martingale is a
  martingale, we have

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• (See Fig. 8.2 for an illustration of the various
  functions involved).

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8.3 The moment generating function for
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8.4 Expectation of

• Recall that

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8.5 The Strong Markov Property
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8.6 General First Passage Times

• Define

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8.7 Example Perpetual American Put
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• How do we recognize the value of an American
  derivative security ?
• There are three parts to the proof of the
  conjecture. We must show

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8.8 Difference Equation
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8.9 Distribution of First Passage Times
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8.10 The Reflection Principle
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