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

Pacific Journal of Mathematics for Industry

Table 2 A list of joint distributions

BM	$P (\cdot ≦ A, B_{θ} \in dx)$
$P (sup_{u ≦ θ} B_{u} ≦ A, B_{θ} \in dx)$	$(\frac{λ}{2} e^{- λ \| x \|} - \frac{λ}{2} e^{λx} e^{- 2 λ max (A, x)}) dx$
$P {(sup_{u ≦ g_{θ}} B_{u} ≦ A, B_{θ} \in dx)}^{*}$	$\frac{λ}{2} e^{- λ \| x \|} (1 - e^{- 2 λA}) dx$
$P (sup_{g_{θ} ≦ u ≦ θ} B_{u} ≦ A, B_{θ} \in dx)$	$\frac{λ}{2} \frac{1}{1 - e^{- 2 λA}} 1_{x ≦ A} (e^{- λ \| x \|} - e^{λx - 2 λA}) dx$
$P (sup_{u ≦ d_{θ}} B_{u} ≦ A, B_{θ} \in dx)$	$(1 - \frac{x^{+}}{A}) 1_{x ≦ A} \frac{λ}{2} (e^{- λ \| x \|} - e^{λx - 2 λA}) dx$
$P (sup_{θ ≦ u ≦ d_{θ}} B_{u} ≦ A, B_{θ} \in dx)$	$(1 - \frac{x^{+}}{A}) 1_{x ≦ A} \frac{λ}{2} e^{- λ \| x \|} dx$
$P (sup_{g_{θ} ≦ u ≦ d_{θ}} B_{u} ≦ A, B_{θ} \in dx)$	$\frac{1 - \frac{x^{+}}{A}}{1 - e^{- 2 λA}} 1_{x ≦ A} \frac{λ}{2} (e^{- λ \| x \|} - e^{λx - 2 λA}) dx$

(∗)Note: We see on this line that $sup_{u ≦ g_{θ}} B_{u}$ and B_θ are independent.