First, we show the claim on the stopping time property. By the previous discussion, we only need to consider the case where 0 < ˜ t$_{α}$ ( λ ) < 1 and ˜ π$_{0}$ ( λ ) > 0. Define where (Ω , A , P ) is the probability space with Ω the sample space, A the sigma-algebra, and P the probability measure. Then F$_{t}$ = σ ( X ( m ) s ( ω ) , 1 ≥ s ≥ t ) , t ∈ [0 , 1] and {F$_{t}$ : 0 ≤ t ≤ 1 } is a non-increasing sequence of sub-sigma-algebras of A . Write ˜ F DR$_{λ}$ ( t ) = ˜ F DR$_{λ}$ ( t, ω ). By the definition of ˜ t$_{α}$ ( λ ), where A$_{t}$ = { ω ∈ Ω : ˜ F DR$_{λ}$ ( t, ω ) > α } , 1 ≥ t > s . Thus, we only need to show ˜ A$_{s}$ ∈ F$_{s}$$_{∧}$$_{λ}$ . Since either F$_{s}$$_{∧}$$_{λ}$ = F$_{s}$ ⊇ F$_{λ}$ when s ≤ λ or F$_{s}$$_{∧}$$_{λ}$ = F$_{λ}$ ⊇ F$_{s}$ when s ≥ λ , the stopping time property holds once we prove ˜ A$_{s}$ ∈ F$_{s}$ . Let Q be the set of all rational numbers. Since 0 < α < 1 and ˜ π$_{0}$ ( λ ) > 0, the following decompositions are valid: A$_{t}$ = ⋃ r ∈ $_{Q}$( A$_{t,r}$ ∩ B$_{r}$ ) and where A$_{t,r}$ = { ω ∈ Ω : t m − $^{1}$max { R ( t ) , 1 } ≥ r } and B$_{r}$ = { ω ∈ Ω : α ˜ π$_{0}$ ( λ ) < r } . Thus, it suffices to show We now move to show Define I$_{i}$ = [ p$_{(}$$_{i}$$_{)}$, p$_{(}$$_{i}$$_{+1)}$ ), i = 1 , ..., n − 1. We will add I$_{0}$ = [ p$_{(0)}$, p$_{(1)}$ ) and I$_{n}$ = [ p$_{(}$$_{n}$$_{)}$, p$_{(}$$_{n}$$_{+1)}$ ] when p$_{(}$$_{n}$$_{)}$ < 1. When p$_{(}$$_{n}$$_{)}$ = 1, we take I$_{n}$$_{−}$$_{1}$ to be [ p$_{(}$$_{n}$$_{−}$$_{1)}$, p$_{(}$$_{n}$$_{)}$ ] . Obviously there must be a unique j ∗ with 0 ≤ j ∗ ≤ n such that s ∈ I$_{j}$ ∗ . Given R (1) = m and ˜ π$_{0}$ ( λ ) ∈ [0 , 1], the properties of L ( · ) 9 A ∗ s,r = ⋂ { t : s<t ≤ 1 } A$_{t,r}$ ∈ F$_{s}$ . (A.4) ⋂ { t : s<t ≤ 1 } ( A$_{t,r}$ ∩ B$_{r}$ ) = ⋂ { t : s<t ≤ 1 } A$_{t,r}$ ∩ B$_{r}$ ∈ F$_{s}$ , (A.3) ˜ A$_{s}$ = ⋂ { t : s<t ≤ 1 } ⋃ r ∈ Q ( A$_{t,r}$ ∩ B$_{r}$ ) = ⋃ r ∈ Q ⋂ { t : s<t ≤ 1 } ( A$_{t,r}$ ∩ B$_{r}$ ) , { ω ∈ Ω : ˜ t$_{α}$ ( λ ) ≤ s } = ⋂ { t : s<t ≤ 1 } A$_{t}$ = ˜ A$_{s}$ , (A.2) X ( m ) t ( ω ) = ( 1$_{{}$$_{p}$$_{1}$$_{≤}$$_{t}$$_{}}$ ( ω ) , ..., 1$_{{}$$_{p}$$_{m}$$_{≤}$$_{t}$$_{}}$ ( ω ) , ω ∈ Ω,