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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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**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.'sthat for each A S.R.W. is a process such that and**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)**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.**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.**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)**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 .**Properties of Scaled S.R.W.**(i) independent increments: for any are independent. Each is an integer. (ii) Stationarity: for any ,**(iii) Martingale property**(iv) Markov Property: for any function , these exists a function so that (v) Quadratic variation: for any ,**Limiting (Marginal) Distribution of S.R.W.**Theorem 3.2.1. (Central Limit Theorem) For any fixed , in dist. or Proof: shown in class.**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**What is Brownian Motion?**"If is a continuous process with independent increments that are normally distributed, then is a Brownian motion."**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 .**Covariance Matrix**Check for any nonnegative and For any vector with In fact,**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.**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.]**Martingale property**Theorem 3.3.4 B.M. is a martingale. Proof:**Levy's Characteristics of Brownian Motion**The process is SBM iffthe conditional characterizationfunction is**Variations: First-Order (Total) Variation**Given a function defined on , the total variation is defined by where the partition and**If is differentiable,**for some . Then .**Quadratic Variation**Def. 3.4.1 The quadratic variation of up to time is defined by**If is continuous differentiable,**for some . Then**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: , ,**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**Volatility Estimation of GBM**The realized variance is defined by which converges to as**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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