S TOCHASTIC M ODELS L ECTURE 4 B ROWNIAN M OTIONS Nan Chen MSc Program in Financial Engineering The Chinese University of Hong Kong (Shenzhen) Nov 11, 2015

Outline 1.Martingale Theory 2.Definition of Brownian Motions 3.Hitting Times and Maximum Variable

4.1 B ASIC M ARTINGALE T HEORY

A Motivating Example A gambler plays a game repeatedly. At each play he wins $1 with prob. 50%, and loses $1 with prob. 50%. The plays are independent of each other. Let be the outcome of each play and what is the value of the following conditional expectation?

Martingales A stochastic process is said to be a martingale process if A martingale can be viewed as a generalized version of a fair game. We have

Example I: Product of Random Numbers Let be a sequence of independent random variables with Then, is a martingale.

Example II: Branching Process Consider a branching process (See Section 1.7) and let denote the size of the nth generation. If is the mean number of offspring per individual, then is a martingale when

Stopping Times The positive integer-valued (possibly infinite) random variable is said to be a stopping time for a process if the occurrence of event is determined by the value of

Example III: Hitting Time of a Process Consider a process Let i.e., is the first time that the value of the process is larger than or equal to Then, such is a stopping time.

Example IV: Maximum Time Consider the above process again. But, let i.e., is the first time that the process reaches the largest value of Is a stopping time?

Martingale Stopping Theorem Theorem: If either – is bounded, or; – is bounded, then

Example V: Gambler Ruin’s Problem Revisited Consider an individual who starts at position and at each step either moves 1 position to the right with probability 0.5 or 1 position to the left with probability 0.5 Assume that the successive movements are independent. Suppose that he will stop moving when he reaches either or What is the probability that he stops at

Example VI: Expected Number of Steps Consider the preceding example. We calculated the expected number of steps this individual takes to reach or in Exercise 59 of Chapter 1. Here we can use the martingale stopping theorem to derive the same result.

Example VII: Unfair Games Now suppose that the individual moves to his right with a higher probability. That is, let with Let Verify that is a martingale. Using it and the martingale stopping theorem, we can compute the probability that he reaches earlier than

4.2 D EFINITION OF B ROWNIAN M OTION

Symmetric Random Walk We know that the symmetric random walk can be represented by with being a sequence of independent random variables whose distribution is given by

Symmetric Random Walk as Markov and Martingale Process The symmetric random walk is not only a Markov process but also a martingale process.

Independent and Stationary Increments Take a sequence of integers: Then, are independent (independent increments). Furthermore, the distribution of does not depend on (stationary increments).

Scaled Random Walk Now, let us rescale time and space for the random walk model, speeding up this process by taking smaller and smaller steps in smaller and smaller time intervals. That is, let

Mean and Variance of Scaled Random Walk Mean Variance:

Limits of Scaled Random Walk We now let and go to in a way such that Un-scaled RWScaled RW

Limits of Scaled Random Walk (Continued) Based on the central limit theorem, we have

Brownian Motion A stochastic process is said to be a Brownian motion process if – – it has stationary and independent increments – for every is normally distributed with mean 0 and variance

Brownian Motion as Markov and Martingale Process The independent increment assumption implies that Brownian motion is a Markov process: It also implies that Brownian motion is a martingale:

Covariance of Brownian Motion Covariance: if

Example VIII: Brownian Motion For compute

4.3 H ITTING T IMES AND M AXIMUM V ARIABLES

Distribution of Hitting Time Let be the first time the Brownian motion process hits a level We intend to derive the probability of based on the following observations: –

Distribution of Hitting Time We can see that which implies that the probability density function of is

Maximum Value of a Brownian Motion For

Example IX: Hitting Probability of Brownian Motion Note that Brownian motion is a martingale process. We still can use the martingale stopping theorem to compute where is the first time the Brownian motion hits or