One-sided sub-exponential uniform boundary

Assume the Axiom of Choice and fix a scale $c>0$. For $0\le \lambda <1\mathrm{/}c$, define ${\psi }_{E,c}(\lambda ):=\frac{-\log (1-c\lambda )-c\lambda }{c^2}\mathrm{.}$ On a discrete filtered probability space $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbb{N}}_0},P)$, a pair of real processes $(S_t,V_t{)}_{t\in {\mathbb{N}}_0}$ is a one-sided sub-exponential pair at scale $c$ when $S_0=V_0=0$, $V_t\ge 0$, and both processes are ${\mathcal{F}}_t$-measurable at each time, and for every $\lambda \in [0,1\mathrm{/}c)$ there is a real process $(L_t^{\lambda }{)}_{t\in {\mathbb{N}}_0}$ that is nonnegative, integrable, and ${\mathcal{F}}_t$-measurable at each time, satisfies $\mathbb{E}{L}_0^{\lambda }\le 1$ and $\mathbb{E}[L_t^{\lambda }\mid {\mathcal{F}}_{t-1}]\le L_{t-1}^{\lambda }$ almost surely for each $t\ge 1$, and obeys $\exp \text{\hspace{0.17em}⁣}(\lambda S_t-{\psi }_{E,c}(\lambda )V_t)\le L_t^{\lambda }\text{almost surely for every }t\ge 0.$

For $\alpha \in (0,1)$, a Borel-measurable function $u:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a one-sided sub-exponential uniform boundary with crossing probability $\alpha$ and scale $c$ if, for every discrete filtered probability space and every one-sided sub-exponential pair at scale $c$, $P\text{\hspace{0.17em}⁣}(\mathrm{\exists }t\ge 1:S_t\ge u(V_t))\le \alpha \mathrm{.}$

Filtration and filtered probability space: the filtration consists of sigma-algebras ${\mathcal{F}}_t\subseteq {\mathcal{F}}_{t+1}$ on one probability space, with time starting at zero.

Adapted and integrable stochastic process: a real process is adapted when each value is measurable with respect to the corresponding ${\mathcal{F}}_t$; integrability is imposed at every fixed time.

Nonnegative test supermartingales: the auxiliary process $L^{\lambda }$ is nonnegative, adapted, integrable, satisfies the conditional supermartingale inequalities, and has initial expectation at most one.

The Axiom of Choice: AC is explicitly assumed for the conditional-expectation convention used by the supplied supermartingale definitions.

A Howard et al., Definition 1 (sub-$\psi$ condition), Definition 2 (uniform boundary), and ${\psi }_{E,c}$, §2

A Tropp, Lecture 26, §§26.4.3–26.4.4