9 Brownian Motion

# 9 Brownian Motion - (2) Brownian Motion Definition. A...

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Unformatted text preview: (2) Brownian Motion Definition. A standard Brownian motion (BM) is a stochastic process W = { W t , t ≥ } satisfying (i) W = 0 (ii) for any 0 ≤ t < t 1 < ··· < t k , the rv’s W t k- W t k − 1 (increments) are independent, (iii) W t- W s ∼ N (0 , ( t- s )), where t > s . Note: Any linear transformation of the standard BM of the form ˜ W t = μt + σW t where μ ∈ R , σ > 0 are constants is called a Brownian motion with drift μ and diffusion coefficient (or volatility) σ . tim e in d e x (m u = 0 ) Brownian motion 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0-2-1 1 2 Trajectories of a B row nian m otion Trajectories of a B row nian m otion tim e in d e x (m u = 2 ) Brownian motion 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0-0.5 0.0 0.5 1.0 1.5 2.0 2 Properties of the BM : 1. The paths of a BM are almost surely continuous. 2. W is a Gaussian process. That is, for all 0 ≤ t 1 < t 2 < . . . < t k , the vector ( W t 1 , . . . , W t k ) has a multinormal distribution characterized by E( W t i ) = 0 and Cov(...
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## This note was uploaded on 10/24/2010 for the course ACTSC 445 taught by Professor Christianelemieux during the Fall '09 term at Waterloo.

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9 Brownian Motion - (2) Brownian Motion Definition. A...

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