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Chap3 - Chapter 3 Brownian Motion Definition 3.3.1 Let(Ω F...

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Unformatted text preview: Chapter 3. Brownian Motion Definition 3.3.1. Let (Ω , F , P ) be a proba- bility space. For each w ∈ Ω, suppose there is a continuous function W ( t ) of t ≥ 0 that satisfies W (0) = 0 and that depends on w . Then W ( t ) , t ≥ , is a Brownian motion if for all 0 = t < t 1 < ··· < t m the increments W ( t 1 ) = W ( t 1 )- W ( t ) , W ( t 2 )- W ( t 1 ) , ··· , W ( t m )- W ( t m- 1 ) (3 . 3 . 1) are independent and each of these inrements is normally distributed with E ( W ( t i +1 )- W ( t i )) = 0 (3 . 3 . 2) V ar ( W ( t i +1 )- W ( t i )) = t i +1- t i (3 . 3 . 3) 1 Theorem 3.3.2. Let (Ω , F , P ) be a probabil- ity space. For each w ∈ Ω, suppose there is a continuous function W ( t ) of t ≥ 0 that sat- isfies W (0) = 0 and that depends on w . The following three properties are equivalent; (i) For all 0 = t < t 1 < ··· < t m , the incre- ments W ( t 1 ) = W ( t 1 )- W ( t ) , W ( t 2 )- W ( t 1 ) , ··· , W ( t m )- W ( t m- 1 ) are independent and each of these increments is normally distributed with mean abd variance given by (3.3.2) and (3.3.3) (ii) For all 0 = t < t 1 < ··· < t...
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Chap3 - Chapter 3 Brownian Motion Definition 3.3.1 Let(Ω F...

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