I would like to understand whether (and if not, under which conditions) the following two definitions of local boundedness of a stochastic process $ H = (H_t)_{t \geq 0} $ on a filtered probability space $(Ω, \mathcal F, (\mathcal F_t)_{t \geq 0}, ℙ)$ coincide.
- (the sup-definition) an adapted stochastic process $H$ is said to be locally bounded, if $$ ∀ t \geq 0, \quad \sup_{0 \leq s \leq t} |H_s| < ∞, \quad \text{a.s.} $$
- (the localization definition) an adapted stochastic process $H$ is said to be locally bounded if there is a sequence of stopping times $τ_n$ almost surely increasing to infinity, such that $ 1_{\{τ_n > 0\}} H^{τ_n} $ are uniformly bounded processes.
The first definition comes from §5.1.3 of J.F. Le Gall's Brownian Motion, Martingales, and Stochastic Calculus (where the definition also requires $H$ progressive), the second one is found e.g. on planetmath.
Some observations: The localization definition seems to imply the sup definition. A continuous adapted process $H$ is locally bounded in the sense of both of the above definitions, with $τ_n = \inf\{ t \geq 0 : |H_t| \geq n \}$ as the localizing sequence. I think that a càdlàg process will be locally bounded in the first sense, but perhaps (?) not always in the second.