Conditionally sub-Gaussian increments

Assume the Axiom of Choice (AC) The Axiom of Choice for the conditional-expectation existence convention. Let $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbb{N}}_0},P)$ be a discrete filtered probability space Filtration and filtered probability space, and let $(X_t{)}_{t\ge 1}$ be a real-valued adapted process Adapted and integrable stochastic process. Fix $\mu \in \mathbb{R}$ and a proxy ${\sigma }^2>0$. The increments are conditionally sub-Gaussian around $\mu$ with proxy ${\sigma }^2$ when, for every integer $t\ge 1$ and every fixed $\lambda \in \mathbb{R}$, $e^{\lambda (X_t-\mu )}$ is integrable and ${\mathbb{E}}_P[e^{\lambda (X_t-\mu )}\mid {\mathcal{F}}_{t-1}]\le e^{{\lambda }^2{\sigma }^2\mathrm{/}2}P\text{-almost surely},$ where conditional expectation means the class of integrable versions of Conditional expectation as an ae class represented as in Conditional expectation given a sigma algebra. The exceptional null set may depend on $t$ and $\lambda$; no common null set over the uncountable set $\mathbb{R}$ is assumed.

The integrability of $X_t$ follows from the two cases $\lambda =1$ and $\lambda =-1$: conditional expectation preserves the integral, so both $e^{X_t-\mu }$ and $e^{-(X_t-\mu )}$ have finite expectation, and $\mathrm{\mid }{X}_t-\mu \mathrm{\mid }\le e^{X_t-\mu }+e^{-(X_t-\mu )}$. The proxy is an upper bound in the conditional moment-generating-function inequality; it is not asserted to be a variance. Independence and identical distribution of the increments are not part of this definition.