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# Brownian Motion

Brownian Motion. Chuan -Hsiang Han November 24, 2010. Symmetric Random Walk. Given ; let and , and denotes the outcome of th toss. Define the r.v. 's that for each A S.R.W. is a process such that and. Independent Increments of S.R.W.

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## Brownian Motion

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1. Brownian Motion Chuan-Hsiang Han November 24, 2010

2. Symmetric Random Walk Given ; let and , and denotes the outcome of th toss. Define the r.v.'sthat for each A S.R.W. is a process such that and

3. Independent Increments of S.R.W. Choose , the r.v.sare independent, where the increment is defined by Note: (1) Increments are independent. (2) The increment has mean 0 and variance .(Stationarity)

4. Martingale Property of S.R.W. For any nonnegative integers , contains all the information of the first coin tosses. If R.W. is not symmetric, it is not a martingale.

5. Markov Property of S.R.W. For any nonnegative integers and any integrable function If R.W. is not symmetric, it is still a Markov process.

6. Quadratic Variation of S.R.W. The quadratic variation up to time is defined to be Note the difference between (an average over all paths), and (pathwise property)

7. Scaled S.R.W. Goal: to approximate Brownian Motion 1. new time interval is "very small" of instead of 1 2. magnitude is "small" of instead of 1. For any can be defined as a linear interpolation between the nearest such that .

8. Properties of Scaled S.R.W. (i) independent increments: for any are independent. Each is an integer. (ii) Stationarity: for any ,

9. (iii) Martingale property (iv) Markov Property: for any function , these exists a function so that (v) Quadratic variation: for any ,

10. Limiting (Marginal) Distribution of S.R.W. Theorem 3.2.1. (Central Limit Theorem) For any fixed , in dist. or Proof: shown in class.

11. A Numerical Example

12. Log-Normality as the Limit of the Binomial Model Theorem 3.2.2. (Central Limit Theorem) For any fixed , in the distribution sense, where , ,and

13. What is Brownian Motion? "If is a continuous process with independent increments that are normally distributed, then is a Brownian motion."

14. Standard Brownian Motions check Definition 3.3.1 in the text. Definition of SBM: Let the stochastic process under a probability space be continuous and satisfy: 1. 2. 3. is independent of for .

15. Covariance Matrix Check for any nonnegative and For any vector with In fact,

16. Joint Moment-Generating Function of BM

17. Alternative Characteristics of BrownianMotion (Theorem 3.3.2) For any continuous process with , the following three properties are equivalent. (i) increments are independent and normally distributed. (ii) For any , are jointly normally distributed. (ii) has the joint moment-generating function as before. If any of the three holds, then , is a SBM.

18. Filtration for B.M. Definition 3.3.3 Let be a probability space on which the B.M. is defined. A filtration for the B.M. is a collection of -algebras , satisfying (i) (Information accumulates) For , . (ii) (Adaptivity) each is -measurable. (iii) (Independence of future increments) , the incrementis independent of . [Note, this property leads to Efficient Market Hypothesis.]

19. Martingale property Theorem 3.3.4 B.M. is a martingale. Proof:

20. Levy's Characteristics of Brownian Motion The process is SBM iffthe conditional characterizationfunction is

21. Variations: First-Order (Total) Variation Given a function defined on , the total variation is defined by where the partition and

22. If is differentiable, for some . Then .

23. Quadratic Variation Def. 3.4.1 The quadratic variation of up to time is defined by

24. If is continuous differentiable, for some . Then

25. Quadratic Variation of B.M. Thm. 3.4.3 Let be a Brownian Motion. Then for all a.s.. B.M. accumulates quadratic variation at rate one per unit time. Informal notion: , ,

26. Geometric Brownian Motion The geometric Brownian motion is a process of the following form where is the current value, is a B.M., is the drift andis the volatility. For each partition, define the log returns

27. Volatility Estimation of GBM The realized variance is defined by which converges to as

28. BM is a Markov process Thm. 3.5.1 Let be a B.M. and be a filtration for this B.M.. Then (1)Wt0 is a Markov process. Thm. 3.6.1. (2) is martingale. (We call exponential martingale.) Note:

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