Brownian Motion
2015-10-28 18:52:03 0 举报AI智能生成
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Brownian Motion
Example
Xi~P({1})=P({-1})=1/2
St is the standardized partial sum of Xi
Then St convergence in distribution to Brownian motion
CLT
Definition
B0
0
increments
Stationary
Independence
Bt
sample path
continuous
Property
Multivariate Gaussian distribution
expectation=0
Bt-Bs~B(t-s)
Sample paths
Continuous
Nowhere differentiable
classical integration methods failed
Unbounded variation on any finite interval
What does this mean?
irregularity
B is 1/2-self-similar
H-self-similar
Comparison vs. Poisson process
Levy Process
drift+B+jump component
Extension
Brownian motion with drift
Standard Brownian motion
Geometric Brownian motion
Gaussian White and Colored noise
Review: Convergence
almost surely converge
converge in mean square
converge in probability
converge in distribution
Functional Central Limited Theorem
Simple version (iid)
Stochastic Process
Finite dimensional distribution converge
Uniform tightness
Stochastic compactness
MD array
Approximating Brownian Motion
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