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Pacific Journal of Mathematics for Industry

Table 2 A list of joint distributions

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

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

(∗)Note: We see on this line that $sup u ≦ g θ B u$ and B_θ are independent.

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