An invariance principle for Azéma martingales

Nathanaël Enriquez

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Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_{p} (x) = {e x p (| x |}^{p}) - 1$ with 0 < p ≤ 2) of $m a x_{0 \leq t \leq τ} | B_{t} |$ or $| B_{τ} |$ to be finite, where $B = {(B_{t})}_{t \geq 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} < \infty$ as soon as $E (τ^{k}) = O (C^{k} k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥ τ ∥_{ψ_{1}} < \infty$ ). In particular, if τ ∼ Exp(λ) or $| N (0, σ^{2}) |$ then the last condition is satisfied, and we obtain $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} \leq K \sqrt E (τ)$ with some universal constant...