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bmotion1 - Brownian Motion Definition A Brownian motion is...

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Unformatted text preview: Brownian Motion Definition. A Brownian motion is a stochastic process B = { B t , t ≥ } satisfying (i) B = 0 (ii) for any 0 ≤ t < t 1 < ··· < t k , the rv’s B t k- B t k- 1 (increments) are independent, (iii) B t- B s ∼ N ( μ ( t- s ) , σ 2 ( t- s )), where t > s , μ ∈ R , σ > 0. The parameter μ is called the drift coefficient and the pa- rameter σ 2 is called the diffusion coefficient. A Brownian motion with μ = 0 and σ = 1 is called the standard Brown- ian motion. 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 T ra je c to rie s o f a B ro w n ia n m o tio n T ra je c to rie s o f a B ro w n ia n m o tio n 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 1 Properties : 1. Brownian motion exists. 2. From the definition, it is easy to find the transition prob- abilities of standard Brownian motion. For 0 = t < t 1 < ··· < t n- 1 < t n : P ( B t n ≤ b n | B t i = b i , i = 1 ,..,n- 1) = = P ( B t n- B t n- 1 ≤ b n- b n- 1 | B t n- 1 = b n- 1 ) = P ( B t n- B t n- 1 ≤ b n- b n- 1 ) = Z b n- b n- 1-∞ 1 r 2 π ( t n- t n- 1 ) e- x 2 2( t n- t n- 1 ) dx....
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bmotion1 - Brownian Motion Definition A Brownian motion is...

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