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 , B θ ∈ dx )
P sup u ≦ θ B u ≦ A , B θ ∈ dx
λ 2 e − λ | x | − λ 2 e λx e − 2 λ max ( A , x ) dx
P sup u ≦ g θ B u ≦ A , B θ ∈ dx ∗
λ 2 e − λ | x | 1 − e − 2 λA dx
P sup g θ ≦ u ≦ θ B u ≦ A , B θ ∈ dx
λ 2 1 1 − e − 2 λA 1 x ≦ A e − λ | x | − e λx − 2 λA dx
P sup u ≦ d θ B u ≦ A , B θ ∈ dx
1 − x + A 1 x ≦ A λ 2 e − λ | x | − e λx − 2 λA dx
P sup θ ≦ u ≦ d θ B u ≦ A , B θ ∈ dx
1 − x + A 1 x ≦ A λ 2 e − λ | x | dx
P sup g θ ≦ u ≦ d θ B u ≦ A , B θ ∈ dx
1 − x + A 1 − e − 2 λA 1 x ≦ A λ 2 e − λ | x | − e λx − 2 λA dx