$T^\sharp$ is a stopping time (after Eq. (8))

Let $T^\sharp$ be the time index of (8) built from the noises of a run. For every period $\tau$ the event $\{T^\sharp\ge\tau\}$ is determined by the history up to and including period $\tau$. Equivalently, in forward time, $T+1-T^\sharp$ is a stopping time of the run's filtration.

This is what allows optional stopping at $T^\sharp$ in the proofs of Lemma 2 and of (18).

Formalization Note The paper's $\mathcal F_\tau$ is the filtration's index $T+1-\tau$; $T^\sharp\le T+1$, so the subtraction does not truncate.

Differences from the source:

• generalization: T♯ built from the noises of the resolving run → T♯ built from the noises of any run, for any x and any model; reason: the claim uses only that the noises are adapted; the resolving run is one such run
• encoding: T♯ is a stopping time (backward time: {T♯ ≥ τ} ∈ ℱ_τ) → ω ↦ T + 1 − T♯(ω) is an ℱ-stopping time in forward time; reason: ℱ_τ (paper) is ℱ (T + 1 − τ) here and T♯ ≤ T + 1; the two statements are equivalent
• encoding: T♯ := max{τ ≥ 1 : |ξ̄ʳ→τ−1| > γ} ∨ ⌈2|r″(x_T)|B_ξ/r′(x_T) + 1⌉ (8) → the maximum runs over 2 ≤ τ ≤ T (empty maximum 0); ceilings are Nat.ceil; reason: τ = 1 would involve ξ̄→0, which contains ξ_1/0, and never changes T♯ because the second term is at least 1
• reading: A4: the conditional distribution of ξ_t given p_t is Q(p_t) → the conditional law of ξ_t given the past ℱ_{t+1} is Q(p_t) (condExp of indicators of Borel sets); reason: the Bellman equation (5) and the recursion (7), p.3543, take the expectation over ξ ~ Q(price) given the current state, and the network analogue B4 (p.3549) states independence from past noise; under the weaker reading Q does not determine R^π(T, x_T)
• encoding: prices p ∈ [p̲, p̄] with demand f(p) and noise law Q(p) → demand rates λ ∈ [d̲, d̄] with price g(λ) = f⁻¹(λ) and noise law Q(λ) := Q(f⁻¹(λ)); |f(p) − f(p′)| = |λ − λ′| in A5; reason: the paper's own parametrization in (5); f is a bijection [p̲, p̄] → [d̲, d̄], so the instances are the same