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URLhttps://dlsun.github.io/probability/brownian-motion.html
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Meta TitleLesson 49 Brownian Motion | Introduction to Probability
Meta DescriptionIntroduction to probability textbook.
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The videos above discussed Brownian motion of particles moving in two or three dimensions; for simplicity, we will only consider Brownian motion in one dimension. This can be used to model, among other things, a particle moving along a line. Shown below are 5 sample paths of Brownian motion. Definition 49.1 (Brownian Motion) Brownian motion { B ( t ) ; t ≥ 0 } is a continuous-time process with the following properties: The “particle” starts at the origin at t = 0 : B ( 0 ) = 0 . The displacement over any interval ( t 0 , t 1 ) , denoted by B ( t 1 ) − B ( t 0 ) , follows a Normal ( μ = 0 , σ = α ( t 1 − t 0 ) ) distribution. Independent increments: The displacements over non-overlapping intervals are independent. The parameter α controls the scale of Brownian motion. These three properties allow us to calculate most probabilities of interest. Example 49.1 (Motion of a Pollen Grain) The horizontal position of a grain of pollen suspended in water can be modeled by Brownian motion with scale α = 4 mm 2 / s . What is the probability the pollen grain moves by more than 10 mm (in the horizontal direction) between 1 and 4 seconds? If the particle is more than 5 mm away from the origin after 1 second, what is the probability the pollen grain moves by more than 10 mm between 1 and 4 seconds? Solution. . The displacement of the particle between 1 and 4 seconds is represented by B ( 4 ) − B ( 1 ) , which we know follows a Normal ( μ = 0 , σ = 4 ( 4 − 1 ) ⏟ 12 ) distribution. Since the question does not indicate whether the displacement is positive or negative, we’re really interested in determining P ( | B ( 4 ) − B ( 1 ) | > 10 ) . Because the distribution of B ( 4 ) − B ( 1 ) is symmetric about 0 , this is the same as 2 P ( B ( 4 ) − B ( 1 ) > 10 ) or 2 P ( B ( 4 ) − B ( 1 ) < − 10 ) . P ( | B ( 4 ) − B ( 1 ) | > 10 ) = 2 P ( B ( 4 ) − B ( 1 ) > 10 ) = 2 P ( B ( 4 ) − B ( 1 ) − 0 12 ⏟ Z > 10 − 0 12 ) = 2 ( 1 − Φ ( 10 12 ) ) ≈ .0039 . b. The interval ( 0 , 1 ) does not overlap with the interval ( 1 , 4 ) . Therefore, by the independent increments property, B ( 1 ) = B ( 1 ) − B ( 0 ) is independent of B ( 4 ) − B ( 1 ) , and P ( | B ( 4 ) − B ( 1 ) | > 10   |   | B ( 1 ) | > 5 ) = P ( | B ( 4 ) − B ( 1 ) | > 10 ) = .0039 . Brownian Motion as the Limit of a Random Walk Brownian motion is the extension of a (discrete-time) random walk { X [ n ] ; n ≥ 0 } to a continuous-time process { B ( t ) ; t ≥ 0 } . The recipe is as follows: Suppose the steps of the random walk happens at intervals of Δ t seconds. That is, X ( t ) = X [ t Δ t ] We let Δ t → 0 . Since each step happens so quickly, it does not make sense to take steps of the same size. We should also rescale the size of the steps to be commensurate with how quickly they happen. The right rescaling is: B ( t ) = Δ t X ( t ) . If we start with a discrete-time random walk and rescale time and step sizes in this way, we get Brownian motion. The animation below illustrates this. Essential Practice Brownian motion is used in finance to model short-term asset price fluctuation. Suppose the price (in dollars) of a barrel of crude oil varies according to a Brownian motion process; specifically, suppose the change in a barrel’s price t days from now is modeled by Brownian motion B ( t ) with α = .15 . Find the probability that the price of a barrel of crude oil has changed by more than $1, in either direction, after 5 days. Repeat (a) for a time interval of 10 days. Given that the price has increased by $1 after one week (7 days), what is the probability that the price has increased by at least $2 after two weeks (14 days)?
Markdown
Type to search - [Introduction to Probability](https://dlsun.github.io/probability/) - [Preface](https://dlsun.github.io/probability/index.html) - **I Basics of Probability** - [**1** Probability and Counting](https://dlsun.github.io/probability/counting.html) - [Motivating Example](https://dlsun.github.io/probability/counting.html#motivating-example) - [Theory](https://dlsun.github.io/probability/counting.html#theory) - [Examples](https://dlsun.github.io/probability/counting.html#examples) - [Additional Exercises](https://dlsun.github.io/probability/counting.html#additional-exercises) - [**2** The Factorial](https://dlsun.github.io/probability/factorial.html) - [Motivating Example](https://dlsun.github.io/probability/factorial.html#motivating-example-1) - [Theory](https://dlsun.github.io/probability/factorial.html#theory-1) - [Examples](https://dlsun.github.io/probability/factorial.html#examples-1) - [Additional Exercises](https://dlsun.github.io/probability/factorial.html#additional-exercises-1) - [**3** Box Models and Combinations](https://dlsun.github.io/probability/box-models.html) - [Motivating Example](https://dlsun.github.io/probability/box-models.html#motivating-example-2) - [Theory](https://dlsun.github.io/probability/box-models.html#theory-2) - [Essential Practice](https://dlsun.github.io/probability/box-models.html#essential-practice) - [Additional Practice](https://dlsun.github.io/probability/box-models.html#additional-practice) - [**4** Sampling With Replacement](https://dlsun.github.io/probability/replacement.html) - [Motivating Example](https://dlsun.github.io/probability/replacement.html#motivating-example-3) - [Discussion](https://dlsun.github.io/probability/replacement.html#discussion) - [Examples](https://dlsun.github.io/probability/replacement.html#examples-2) - [Bonus Material](https://dlsun.github.io/probability/replacement.html#bonus-material) - [**5** Double Counting](https://dlsun.github.io/probability/double-counting.html) - [Motivating Example](https://dlsun.github.io/probability/double-counting.html#motivating-example-4) - [Theory](https://dlsun.github.io/probability/double-counting.html#theory-3) - [Examples](https://dlsun.github.io/probability/double-counting.html#examples-3) - [**6** Conditional Probability](https://dlsun.github.io/probability/conditional.html) - [Motivating Example](https://dlsun.github.io/probability/conditional.html#motivating-example-5) - [Theory](https://dlsun.github.io/probability/conditional.html#theory-4) - [Examples](https://dlsun.github.io/probability/conditional.html#examples-4) - [Additional Exercises](https://dlsun.github.io/probability/conditional.html#additional-exercises-2) - [**7** Independence](https://dlsun.github.io/probability/independence.html) - [Motivating Example](https://dlsun.github.io/probability/independence.html#motivating-example-6) - [Theory](https://dlsun.github.io/probability/independence.html#theory-5) - [Examples](https://dlsun.github.io/probability/independence.html#examples-5) - [Additional Exercises](https://dlsun.github.io/probability/independence.html#additional-exercises-3) - [**8** Law of Total Probability](https://dlsun.github.io/probability/ltp.html) - [Motivating Example](https://dlsun.github.io/probability/ltp.html#motivating-example-7) - [Theory](https://dlsun.github.io/probability/ltp.html#theory-6) - [Examples](https://dlsun.github.io/probability/ltp.html#examples-6) - [**9** Bayes’ Theorem](https://dlsun.github.io/probability/bayes.html) - [**9\.1** Motivating Example](https://dlsun.github.io/probability/bayes.html#motivating-example-8) - [**9\.2** Theory](https://dlsun.github.io/probability/bayes.html#theory-7) - [**9\.3** Examples](https://dlsun.github.io/probability/bayes.html#examples-7) - **II Discrete Probability** - [**10** Random Variables](https://dlsun.github.io/probability/rv.html) - [Motivating Example](https://dlsun.github.io/probability/rv.html#motivating-example-9) - [Theory](https://dlsun.github.io/probability/rv.html#theory-8) - [Essential Practice](https://dlsun.github.io/probability/rv.html#essential-practice-1) - [Additional Exercises](https://dlsun.github.io/probability/rv.html#additional-exercises-4) - [**11** Cumulative Distribution Functions](https://dlsun.github.io/probability/cdf.html) - [Theory](https://dlsun.github.io/probability/cdf.html#theory-9) - [Examples](https://dlsun.github.io/probability/cdf.html#examples-8) - [**12** Hypergeometric Distribution](https://dlsun.github.io/probability/hypergeometric.html) - [Motivating Example](https://dlsun.github.io/probability/hypergeometric.html#motivating-example-10) - [Theory](https://dlsun.github.io/probability/hypergeometric.html#theory-10) - [Visualizing the Distribution](https://dlsun.github.io/probability/hypergeometric.html#visualizing-the-distribution) - [Calculating Hypergeometric Probabilities on the Computer](https://dlsun.github.io/probability/hypergeometric.html#calculating-hypergeometric-probabilities-on-the-computer) - [Another Formula for the Hypergeometric Distribution (optional)](https://dlsun.github.io/probability/hypergeometric.html#another-formula-for-the-hypergeometric-distribution-optional) - [Essential Practice](https://dlsun.github.io/probability/hypergeometric.html#essential-practice-2) - [Additional Exercises](https://dlsun.github.io/probability/hypergeometric.html#additional-exercises-5) - [**13** Binomial Distribution](https://dlsun.github.io/probability/binomial.html) - [Motivating Example](https://dlsun.github.io/probability/binomial.html#motivating-example-11) - [Theory](https://dlsun.github.io/probability/binomial.html#theory-11) - [Visualizing the Distribution](https://dlsun.github.io/probability/binomial.html#visualizing-the-distribution-1) - [Calculating Binomial Probabilities on the Computer](https://dlsun.github.io/probability/binomial.html#calculating-binomial-probabilities-on-the-computer) - [Essential Practice](https://dlsun.github.io/probability/binomial.html#essential-practice-3) - [Additional Exercises](https://dlsun.github.io/probability/binomial.html#additional-exercises-6) - [**14** Geometric Distribution](https://dlsun.github.io/probability/geometric.html) - [Motivating Example](https://dlsun.github.io/probability/geometric.html#motivating-example-12) - [Theory](https://dlsun.github.io/probability/geometric.html#theory-12) - [Visualizing the Distribution](https://dlsun.github.io/probability/geometric.html#visualizing-the-distribution-2) - [Essential Practice](https://dlsun.github.io/probability/geometric.html#essential-practice-4) - [**15** Negative Binomial Distribution](https://dlsun.github.io/probability/negative-binomial.html) - [Motivating Example](https://dlsun.github.io/probability/negative-binomial.html#motivating-example-13) - [Theory](https://dlsun.github.io/probability/negative-binomial.html#theory-13) - [Visualizing the Distribution](https://dlsun.github.io/probability/negative-binomial.html#visualizing-the-distribution-3) - [Calculating Negative Binomial Probabilities on the Computer](https://dlsun.github.io/probability/negative-binomial.html#calculating-negative-binomial-probabilities-on-the-computer) - [Essential Practice](https://dlsun.github.io/probability/negative-binomial.html#essential-practice-5) - [Additional Exercises](https://dlsun.github.io/probability/negative-binomial.html#additional-exercises-7) - [**16** Poisson Distribution](https://dlsun.github.io/probability/poisson.html) - [Motivating Example](https://dlsun.github.io/probability/poisson.html#motivating-example-14) - [Theory](https://dlsun.github.io/probability/poisson.html#theory-14) - [Visualizing the Distribution](https://dlsun.github.io/probability/poisson.html#visualizing-the-distribution-4) - [Calculating Poisson Probabilities on the Computer](https://dlsun.github.io/probability/poisson.html#calculating-poisson-probabilities-on-the-computer) - [Essential Practice](https://dlsun.github.io/probability/poisson.html#essential-practice-6) - [**17** Poisson Process](https://dlsun.github.io/probability/poisson-process.html) - [Motivating Example](https://dlsun.github.io/probability/poisson-process.html#motivating-example-15) - [Theory](https://dlsun.github.io/probability/poisson-process.html#theory-15) - [Why Poisson?](https://dlsun.github.io/probability/poisson-process.html#why-poisson) - [Essential Practice](https://dlsun.github.io/probability/poisson-process.html#essential-practice-7) - [**18** Joint Distributions](https://dlsun.github.io/probability/joint-discrete.html) - [Motivating Example](https://dlsun.github.io/probability/joint-discrete.html#motivating-example-16) - [Theory](https://dlsun.github.io/probability/joint-discrete.html#theory-16) - [Essential Practice](https://dlsun.github.io/probability/joint-discrete.html#essential-practice-8) - [Additional Exercises](https://dlsun.github.io/probability/joint-discrete.html#additional-exercises-8) - [**19** Marginal Distributions](https://dlsun.github.io/probability/marginal-discrete.html) - [Motivating Example](https://dlsun.github.io/probability/marginal-discrete.html#motivating-example-17) - [Theory](https://dlsun.github.io/probability/marginal-discrete.html#theory-17) - [Essential Practice](https://dlsun.github.io/probability/marginal-discrete.html#essential-practice-9) - [Additional Exercises](https://dlsun.github.io/probability/marginal-discrete.html#additional-exercises-9) - [**20** Conditional Distributions](https://dlsun.github.io/probability/conditional-discrete.html) - [Motivating Example](https://dlsun.github.io/probability/conditional-discrete.html#motivating-example-18) - [Theory](https://dlsun.github.io/probability/conditional-discrete.html#theory-18) - [Essential Practice](https://dlsun.github.io/probability/conditional-discrete.html#essential-practice-10) - [Additional Exercises](https://dlsun.github.io/probability/conditional-discrete.html#additional-exercises-10) - [**21** Sums of Random Variables](https://dlsun.github.io/probability/sums-discrete.html) - [Theory](https://dlsun.github.io/probability/sums-discrete.html#theory-19) - [Essential Practice](https://dlsun.github.io/probability/sums-discrete.html#essential-practice-11) - [Additional Exercises](https://dlsun.github.io/probability/sums-discrete.html#additional-exercises-11) - [**22** Expected Value](https://dlsun.github.io/probability/expected-value.html) - [Motivating Example](https://dlsun.github.io/probability/expected-value.html#motivating-example-19) - [Theory](https://dlsun.github.io/probability/expected-value.html#theory-20) - [Essential Practice](https://dlsun.github.io/probability/expected-value.html#essential-practice-12) - [Additional Exercises](https://dlsun.github.io/probability/expected-value.html#additional-exercises-12) - [**23** Expected Value and Infinity](https://dlsun.github.io/probability/ev-infinity.html) - [**23\.1** Pascal’s Wager](https://dlsun.github.io/probability/ev-infinity.html#pascals-wager) - [**23\.2** St. Petersburg Paradox](https://dlsun.github.io/probability/ev-infinity.html#st-petersburg) - [**24** LOTUS](https://dlsun.github.io/probability/lotus.html) - [Motivating Example](https://dlsun.github.io/probability/lotus.html#motivating-example-20) - [Theory](https://dlsun.github.io/probability/lotus.html#theory-21) - [Essential Practice](https://dlsun.github.io/probability/lotus.html#essential-practice-13) - [Additional Exercises](https://dlsun.github.io/probability/lotus.html#additional-exercises-13) - [**25** 2D LOTUS](https://dlsun.github.io/probability/lotus2d.html) - [Theory](https://dlsun.github.io/probability/lotus2d.html#theory-22) - [Essential Practice](https://dlsun.github.io/probability/lotus2d.html#essential-practice-14) - [**26** Linearity of Expectation](https://dlsun.github.io/probability/linearity.html) - [Theory](https://dlsun.github.io/probability/linearity.html#theory-23) - [Essential Practice](https://dlsun.github.io/probability/linearity.html#essential-practice-15) - [Additional Practice](https://dlsun.github.io/probability/linearity.html#additional-practice-1) - [**27** Expected Value of a Product](https://dlsun.github.io/probability/ev-product.html) - [Theory](https://dlsun.github.io/probability/ev-product.html#theory-24) - [Essential Practice](https://dlsun.github.io/probability/ev-product.html#essential-practice-16) - [**28** Variance](https://dlsun.github.io/probability/variance.html) - [Motivating Example](https://dlsun.github.io/probability/variance.html#motivating-example-21) - [Theory](https://dlsun.github.io/probability/variance.html#theory-25) - [Essential Practice](https://dlsun.github.io/probability/variance.html#essential-practice-17) - [Additional Practice](https://dlsun.github.io/probability/variance.html#additional-practice-2) - [**29** Covariance](https://dlsun.github.io/probability/covariance.html) - [Theory](https://dlsun.github.io/probability/covariance.html#theory-26) - [Essential Practice](https://dlsun.github.io/probability/covariance.html#essential-practice-18) - [Additional Practice](https://dlsun.github.io/probability/covariance.html#additional-practice-3) - [**30** Properties of Covariance](https://dlsun.github.io/probability/cov-properties.html) - [Optional Video](https://dlsun.github.io/probability/cov-properties.html#optional-video) - [Theory](https://dlsun.github.io/probability/cov-properties.html#theory-27) - [Essential Practice](https://dlsun.github.io/probability/cov-properties.html#essential-practice-19) - [Additional Practice](https://dlsun.github.io/probability/cov-properties.html#additional-practice-4) - [**31** Random Walk](https://dlsun.github.io/probability/random-walk.html) - [Theory](https://dlsun.github.io/probability/random-walk.html#theory-28) - [Essential Practice](https://dlsun.github.io/probability/random-walk.html#essential-practice-20) - [**32** Law of Large Numbers](https://dlsun.github.io/probability/lln.html) - [Motivating Example](https://dlsun.github.io/probability/lln.html#motivating-example-22) - [Theory](https://dlsun.github.io/probability/lln.html#theory-29) - [Essential Practice](https://dlsun.github.io/probability/lln.html#essential-practice-21) - **III Continuous Probability** - [**33** Continuous Random Variables](https://dlsun.github.io/probability/continuous.html) - [Motivating Example](https://dlsun.github.io/probability/continuous.html#motivating-example-23) - [Theory](https://dlsun.github.io/probability/continuous.html#theory-30) - [Optional Video](https://dlsun.github.io/probability/continuous.html#optional-video-1) - [Essential Practice](https://dlsun.github.io/probability/continuous.html#essential-practice-22) - [**34** Uniform Distribution](https://dlsun.github.io/probability/uniform.html) - [Motivation](https://dlsun.github.io/probability/uniform.html#motivation) - [Theory](https://dlsun.github.io/probability/uniform.html#theory-31) - [Essential Practice](https://dlsun.github.io/probability/uniform.html#essential-practice-23) - [**35** Exponential Distribution](https://dlsun.github.io/probability/exponential.html) - [Motivating Example](https://dlsun.github.io/probability/exponential.html#motivating-example-24) - [Theory](https://dlsun.github.io/probability/exponential.html#theory-32) - [Essential Practice](https://dlsun.github.io/probability/exponential.html#essential-practice-24) - [**36** Transformations](https://dlsun.github.io/probability/transformations.html) - [Motivating Example](https://dlsun.github.io/probability/transformations.html#motivating-example-25) - [Theory](https://dlsun.github.io/probability/transformations.html#theory-33) - [Essential Practice](https://dlsun.github.io/probability/transformations.html#essential-practice-25) - [Additional Practice](https://dlsun.github.io/probability/transformations.html#additional-practice-5) - [**37** Expected Value of Continuous Random Variables](https://dlsun.github.io/probability/ev-continuous.html "37 Expected Value of Continuous Random Variables") - [Theory](https://dlsun.github.io/probability/ev-continuous.html#theory-34) - [Essential Practice](https://dlsun.github.io/probability/ev-continuous.html#essential-practice-26) - [**38** LOTUS for Continuous Random Variables](https://dlsun.github.io/probability/lotus-continuous.html "38 LOTUS for Continuous Random Variables") - [Theory](https://dlsun.github.io/probability/lotus-continuous.html#theory-35) - [Essential Practice](https://dlsun.github.io/probability/lotus-continuous.html#essential-practice-27) - [**39** Variance of Continuous Random Variables](https://dlsun.github.io/probability/var-continuous.html "39 Variance of Continuous Random Variables") - [Theory](https://dlsun.github.io/probability/var-continuous.html#theory-36) - [Essential Practice](https://dlsun.github.io/probability/var-continuous.html#essential-practice-28) - [Additional Practice](https://dlsun.github.io/probability/var-continuous.html#additional-practice-6) - [**40** Normal Distribution](https://dlsun.github.io/probability/normal.html) - [Motivation](https://dlsun.github.io/probability/normal.html#motivation-1) - [Standard Normal Distribution](https://dlsun.github.io/probability/normal.html#standard-normal-distribution) - [(General) Normal Distribution](https://dlsun.github.io/probability/normal.html#general-normal-distribution) - [Essential Practice](https://dlsun.github.io/probability/normal.html#essential-practice-29) - [**41** Joint Continuous Distributions](https://dlsun.github.io/probability/joint-continuous.html) - [Theory](https://dlsun.github.io/probability/joint-continuous.html#theory-37) - [Worked Examples](https://dlsun.github.io/probability/joint-continuous.html#worked-examples) - [Essential Practice](https://dlsun.github.io/probability/joint-continuous.html#essential-practice-30) - [Additional Practice](https://dlsun.github.io/probability/joint-continuous.html#additional-practice-7) - [**42** Marginal Continuous Distributions](https://dlsun.github.io/probability/marginal-continuous.html) - [Motivating Example](https://dlsun.github.io/probability/marginal-continuous.html#motivating-example-26) - [Theory](https://dlsun.github.io/probability/marginal-continuous.html#theory-38) - [Essential Practice](https://dlsun.github.io/probability/marginal-continuous.html#essential-practice-31) - [**43** Expectations of Joint Continuous Distributions](https://dlsun.github.io/probability/ev-joint-continuous.html "43 Expectations of Joint Continuous Distributions") - [Theory](https://dlsun.github.io/probability/ev-joint-continuous.html#theory-39) - [Essential Practice](https://dlsun.github.io/probability/ev-joint-continuous.html#essential-practice-32) - [**44** Covariance of Continuous Random Variables](https://dlsun.github.io/probability/cov-continuous.html "44 Covariance of Continuous Random Variables") - [Theory](https://dlsun.github.io/probability/cov-continuous.html#theory-40) - [Essential Practice](https://dlsun.github.io/probability/cov-continuous.html#essential-practice-33) - [**45** Sums of Continuous Random Variables](https://dlsun.github.io/probability/sums-continuous.html "45 Sums of Continuous Random Variables") - [Theory](https://dlsun.github.io/probability/sums-continuous.html#theory-41) - [Essential Practice](https://dlsun.github.io/probability/sums-continuous.html#essential-practice-34) - [**46** Central Limit Theorem](https://dlsun.github.io/probability/clt.html) - [Motivation](https://dlsun.github.io/probability/clt.html#motivation-2) - [Theory](https://dlsun.github.io/probability/clt.html#theory-42) - [Worked Examples](https://dlsun.github.io/probability/clt.html#worked-examples-1) - [Essential Practice](https://dlsun.github.io/probability/clt.html#essential-practice-35) - [Additional Practice](https://dlsun.github.io/probability/clt.html#additional-practice-8) - **IV Random Processes** - [**47** Random Processes](https://dlsun.github.io/probability/random-process.html) - [Motivation](https://dlsun.github.io/probability/random-process.html#motivation-3) - [Theory](https://dlsun.github.io/probability/random-process.html#theory-43) - [Essential Practice](https://dlsun.github.io/probability/random-process.html#essential-practice-36) - [**48** Examples of Random Processes](https://dlsun.github.io/probability/random-process-examples.html) - [**49** Brownian Motion](https://dlsun.github.io/probability/brownian-motion.html) - [Brownian Motion as the Limit of a Random Walk](https://dlsun.github.io/probability/brownian-motion.html#brownian-motion-as-the-limit-of-a-random-walk "Brownian Motion as the Limit of a Random Walk") - [Essential Practice](https://dlsun.github.io/probability/brownian-motion.html#essential-practice-37) - [**50** Mean Function](https://dlsun.github.io/probability/mean-function.html) - [Theory](https://dlsun.github.io/probability/mean-function.html#theory-44) - [Essential Practice](https://dlsun.github.io/probability/mean-function.html#essential-practice-38) - [**51** Variance Function](https://dlsun.github.io/probability/var-function.html) - [Theory](https://dlsun.github.io/probability/var-function.html#theory-45) - [Essential Practice](https://dlsun.github.io/probability/var-function.html#essential-practice-39) - [**52** Autocovariance Function](https://dlsun.github.io/probability/cov-function.html) - [Theory](https://dlsun.github.io/probability/cov-function.html#theory-46) - [Essential Practice](https://dlsun.github.io/probability/cov-function.html#essential-practice-40) - [**53** Stationary Processes](https://dlsun.github.io/probability/stationary.html) - [Motivation](https://dlsun.github.io/probability/stationary.html#motivation-4) - [Theory](https://dlsun.github.io/probability/stationary.html#theory-47) - [Essential Practice](https://dlsun.github.io/probability/stationary.html#essential-practice-41) - **V Random Signal Processing** - [**54** Autocorrelation Function](https://dlsun.github.io/probability/autocorrelation.html) - [Theory](https://dlsun.github.io/probability/autocorrelation.html#theory-48) - [Essential Practice](https://dlsun.github.io/probability/autocorrelation.html#essential-practice-42) - [**55** Power of a Stationary Process](https://dlsun.github.io/probability/power.html) - [Motivation](https://dlsun.github.io/probability/power.html#motivation-5) - [Theory](https://dlsun.github.io/probability/power.html#theory-49) - [Essential Practice](https://dlsun.github.io/probability/power.html#essential-practice-43) - [**56** Power Spectral Density](https://dlsun.github.io/probability/psd.html) - [Motivation](https://dlsun.github.io/probability/psd.html#motivation-6) - [Theory](https://dlsun.github.io/probability/psd.html#theory-50) - [Essential Practice](https://dlsun.github.io/probability/psd.html#essential-practice-44) - [**57** LTI Filters in the Time Domain](https://dlsun.github.io/probability/lti-time.html) - [Motivation](https://dlsun.github.io/probability/lti-time.html#motivation-7) - [Review](https://dlsun.github.io/probability/lti-time.html#review) - [Theory](https://dlsun.github.io/probability/lti-time.html#theory-51) - [Essential Practice](https://dlsun.github.io/probability/lti-time.html#essential-practice-45) - [**58** LTI Filters in the Frequency Domain](https://dlsun.github.io/probability/lti-frequency.html) - [Motivation](https://dlsun.github.io/probability/lti-frequency.html#motivation-8) - [Theory](https://dlsun.github.io/probability/lti-frequency.html#theory-52) - [Essential Practice](https://dlsun.github.io/probability/lti-frequency.html#essential-practice-46) - **Appendix** - [**A** Distribution Tables](https://dlsun.github.io/probability/distribution-table.html) - [**A.1** Discrete Distributions](https://dlsun.github.io/probability/distribution-table.html#discrete-distributions) - [**A.2** Continuous Distributions](https://dlsun.github.io/probability/distribution-table.html#continuous-distributions) - [**B** Complex Numbers](https://dlsun.github.io/probability/complex.html) - [Motivation](https://dlsun.github.io/probability/complex.html#motivation-9) - [Theory](https://dlsun.github.io/probability/complex.html#theory-53) - [Essential Practice](https://dlsun.github.io/probability/complex.html#essential-practice-47) - [**C** Fourier Transforms](https://dlsun.github.io/probability/fourier.html) - [Continuous-Time Fourier Transforms](https://dlsun.github.io/probability/fourier.html#continuous-time-fourier-transforms) - [Discrete-Time Fourier Transforms](https://dlsun.github.io/probability/fourier.html#discrete-time-fourier-transforms) - [Essential Practice](https://dlsun.github.io/probability/fourier.html#essential-practice-48) - [**D** Fourier Tables](https://dlsun.github.io/probability/fourier-table.html) - [**D.1** Continuous-Time Fourier Transforms](https://dlsun.github.io/probability/fourier-table.html#ctft) - [**D.2** Discrete-Time Fourier Transforms](https://dlsun.github.io/probability/fourier-table.html#dtft) - [**D.3** Fourier Properties](https://dlsun.github.io/probability/fourier-table.html#fourier-properties) - [Published with bookdown](https://github.com/rstudio/bookdown) Facebook Twitter LinkedIn Weibo Instapaper A A Serif Sans White Sepia Night PDF EPUB # [Introduction to Probability](https://dlsun.github.io/probability/) # Lesson 49 Brownian Motion The videos above discussed Brownian motion of particles moving in two or three dimensions; for simplicity, we will only consider Brownian motion in one dimension. This can be used to model, among other things, a particle moving along a line. Shown below are 5 sample paths of Brownian motion. ![](https://dlsun.github.io/probability/bookdown-demo_files/figure-html/unnamed-chunk-178-1.png) **Definition 49.1 (Brownian Motion)** **Brownian motion** {B(t);t≥0} { B ( t ) ; t ≥ 0 } is a continuous-time process with the following properties: 1. The “particle” starts at the origin at t\=0 t \= 0 : B(0)\=0 B ( 0 ) \= 0 . 2. The displacement over any interval (t0,t1) ( t 0 , t 1 ) , denoted by B(t1)−B(t0) B ( t 1 ) − B ( t 0 ) , follows a Normal(μ\=0,σ\=√α(t1−t0)) Normal ( μ \= 0 , σ \= α ( t 1 − t 0 ) ) distribution. 3. **Independent increments:** The displacements over non-overlapping intervals are independent. The parameter α α controls the scale of Brownian motion. These three properties allow us to calculate most probabilities of interest. **Example 49.1 (Motion of a Pollen Grain)** The horizontal position of a grain of pollen suspended in water can be modeled by Brownian motion with scale α\=4mm2/s α \= 4 mm 2 / s. 1. What is the probability the pollen grain moves by more than 10 mm (in the horizontal direction) between 1 and 4 seconds? 2. If the particle is more than 5 mm away from the origin after 1 second, what is the probability the pollen grain moves by more than 10 mm between 1 and 4 seconds? *Solution.* . 1. The displacement of the particle between 1 and 4 seconds is represented by B(4)−B(1) B ( 4 ) − B ( 1 ) , which we know follows a Normal(μ\=0,σ\=√4(4−1)√12) Normal ( μ \= 0 , σ \= 4 ( 4 − 1 ) ⏟ 12 ) distribution. Since the question does not indicate whether the displacement is positive or negative, we’re really interested in determining P(\|B(4)−B(1)\|\>10) P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ) . Because the distribution of B(4)−B(1) B ( 4 ) − B ( 1 ) is symmetric about 0 0 , this is the same as 2P(B(4)−B(1)\>10) 2 P ( B ( 4 ) − B ( 1 ) \> 10 ) or 2P(B(4)−B(1)\<−10) 2 P ( B ( 4 ) − B ( 1 ) \< − 10 ) . P(\|B(4)−B(1)\|\>10)\=2P(B(4)−B(1)\>10)\=2P(B(4)−B(1)−0√12Z\>10−0√12)\=2(1−Φ(10√12))≈.0039. P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ) \= 2 P ( B ( 4 ) − B ( 1 ) \> 10 ) \= 2 P ( B ( 4 ) − B ( 1 ) − 0 12 ⏟ Z \> 10 − 0 12 ) \= 2 ( 1 − Φ ( 10 12 ) ) ≈ .0039 . b. The interval (0,1) ( 0 , 1 ) does not overlap with the interval (1,4) ( 1 , 4 ) . Therefore, by the independent increments property, B(1)\=B(1)−B(0) B ( 1 ) \= B ( 1 ) − B ( 0 ) is independent of B(4)−B(1) B ( 4 ) − B ( 1 ) , and P(\|B(4)−B(1)\|\>10 ∣∣ \|B(1)\|\>5)\=P(\|B(4)−B(1)\|\>10)\=.0039. P ( \| B ( 4 ) − B ( 1 ) \| \> 10 \| \| B ( 1 ) \| \> 5 ) \= P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ) \= .0039 . ### Brownian Motion as the Limit of a Random Walk Brownian motion is the extension of a (discrete-time) random walk {X\[n\];n≥0} { X \[ n \] ; n ≥ 0 } to a continuous-time process {B(t);t≥0} { B ( t ) ; t ≥ 0 }. The recipe is as follows: 1. Suppose the steps of the random walk happens at intervals of Δt Δ t seconds. That is, X(t)\=X\[tΔt\] X ( t ) \= X \[ t Δ t \] 2. We let Δt→0 Δ t → 0 . Since each step happens so quickly, it does not make sense to take steps of the same size. We should also rescale the size of the steps to be commensurate with how quickly they happen. The right rescaling is: B(t)\=√ΔtX(t). B ( t ) \= Δ t X ( t ) . If we start with a discrete-time random walk and rescale time and step sizes in this way, we get Brownian motion. The animation below illustrates this. ## Essential Practice 1. Brownian motion is used in finance to model short-term asset price fluctuation. Suppose the price (in dollars) of a barrel of crude oil varies according to a Brownian motion process; specifically, suppose the change in a barrel’s price t t days from now is modeled by Brownian motion B(t) B ( t ) with α\=.15 α \= .15. 1. Find the probability that the price of a barrel of crude oil has changed by more than \$1, in either direction, after 5 days. 2. Repeat (a) for a time interval of 10 days. 3. Given that the price has increased by \$1 after one week (7 days), what is the probability that the price has increased by at least \$2 after two weeks (14 days)?
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The videos above discussed Brownian motion of particles moving in two or three dimensions; for simplicity, we will only consider Brownian motion in one dimension. This can be used to model, among other things, a particle moving along a line. Shown below are 5 sample paths of Brownian motion. ![](https://dlsun.github.io/probability/bookdown-demo_files/figure-html/unnamed-chunk-178-1.png) **Definition 49.1 (Brownian Motion)** **Brownian motion** { B ( t ) ; t ≥ 0 } is a continuous-time process with the following properties: 1. The “particle” starts at the origin at t \= 0 : B ( 0 ) \= 0 . 2. The displacement over any interval ( t 0 , t 1 ) , denoted by B ( t 1 ) − B ( t 0 ) , follows a Normal ( μ \= 0 , σ \= α ( t 1 − t 0 ) ) distribution. 3. **Independent increments:** The displacements over non-overlapping intervals are independent. The parameter α controls the scale of Brownian motion. These three properties allow us to calculate most probabilities of interest. **Example 49.1 (Motion of a Pollen Grain)** The horizontal position of a grain of pollen suspended in water can be modeled by Brownian motion with scale α \= 4 mm 2 / s. 1. What is the probability the pollen grain moves by more than 10 mm (in the horizontal direction) between 1 and 4 seconds? 2. If the particle is more than 5 mm away from the origin after 1 second, what is the probability the pollen grain moves by more than 10 mm between 1 and 4 seconds? *Solution.* . 1. The displacement of the particle between 1 and 4 seconds is represented by B ( 4 ) − B ( 1 ) , which we know follows a Normal ( μ \= 0 , σ \= 4 ( 4 − 1 ) ⏟ 12 ) distribution. Since the question does not indicate whether the displacement is positive or negative, we’re really interested in determining P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ). Because the distribution of B ( 4 ) − B ( 1 ) is symmetric about 0, this is the same as 2 P ( B ( 4 ) − B ( 1 ) \> 10 ) or 2 P ( B ( 4 ) − B ( 1 ) \< − 10 ). P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ) \= 2 P ( B ( 4 ) − B ( 1 ) \> 10 ) \= 2 P ( B ( 4 ) − B ( 1 ) − 0 12 ⏟ Z \> 10 − 0 12 ) \= 2 ( 1 − Φ ( 10 12 ) ) ≈ .0039 . b. The interval ( 0 , 1 ) does not overlap with the interval ( 1 , 4 ). Therefore, by the independent increments property, B ( 1 ) \= B ( 1 ) − B ( 0 ) is independent of B ( 4 ) − B ( 1 ), and P ( \| B ( 4 ) − B ( 1 ) \| \> 10 \| \| B ( 1 ) \| \> 5 ) \= P ( \| B ( 4 ) − B ( 1 ) \| \> 10 ) \= .0039 . ### Brownian Motion as the Limit of a Random Walk Brownian motion is the extension of a (discrete-time) random walk { X \[ n \] ; n ≥ 0 } to a continuous-time process { B ( t ) ; t ≥ 0 }. The recipe is as follows: 1. Suppose the steps of the random walk happens at intervals of Δ t seconds. That is, X ( t ) \= X \[ t Δ t \] 2. We let Δ t → 0 . Since each step happens so quickly, it does not make sense to take steps of the same size. We should also rescale the size of the steps to be commensurate with how quickly they happen. The right rescaling is: B ( t ) \= Δ t X ( t ) . If we start with a discrete-time random walk and rescale time and step sizes in this way, we get Brownian motion. The animation below illustrates this. ## Essential Practice 1. Brownian motion is used in finance to model short-term asset price fluctuation. Suppose the price (in dollars) of a barrel of crude oil varies according to a Brownian motion process; specifically, suppose the change in a barrel’s price t days from now is modeled by Brownian motion B ( t ) with α \= .15. 1. Find the probability that the price of a barrel of crude oil has changed by more than \$1, in either direction, after 5 days. 2. Repeat (a) for a time interval of 10 days. 3. Given that the price has increased by \$1 after one week (7 days), what is the probability that the price has increased by at least \$2 after two weeks (14 days)?
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