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Table 1 A list of interesting maxima

From: On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called “meander option”

BM P ( · A ) RW P(·<A)
sup u θ B u θ sup u 1 B u 1−eλA sup u θ Z u 1−αA
sup u g θ B u g θ sup u 1 b u 1−e−2λA sup u g θ Z u 1−α2A
b:brownian bridge(b.b.)    
sup g θ u θ B u ε θ g θ sup u 1 m u 1 1 + e λA sup g θ u θ Z u 1 1 + α A
m:brownian meander    
sup u d θ B u 1 1 e 2 λA 2 λA sup u d θ Z u 1 1 A 1 α 1 α 1 α 2 A
sup θ u d θ B u 1 1 e λA 2 λA sup θ u d θ Z u 1 2 α 1 α 1 α A A
sup g θ u d θ B u ε d θ g θ sup u 1 e u 1 1 e 2 λA 1 2 λA sup g θ u d θ Z u 1 1 α 2 A 1 A 1 α 1 α
e:normalized excursion    
  1. where
  2. ∙ for BM, θExp(λ2/2), i.e., its density is f θ (x)= 1 ( 0 , ) (x) λ 2 2 exp λ 2 x 2 , and P(ε=1)=P(ε=0)=1/2.
  3. ∙ for RW, θGeom(1−q), i.e., P(θ=k)=(1q) q k ,(k=0,1,2,), α= 1 1 q 2 q .
  4. ∙ for RW, g t =sup{ut: Z u =0}, d t =inf{u>t: Z u =0}.