On the exponential Orlicz norms of stopped Brownian motion

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 K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying

E (τ^{k}) \leq C {(E τ)}^{k} k^{k}

for all k ≥ 1 with some fixed constant C > 0. The method of proof relies upon Taylor expansion, Burkholder-Gundy’s inequality, best constants in Doob’s maximal inequality, Davis’ best constants in the

L^{p}

-inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).

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Peškir, Goran. "On the exponential Orlicz norms of stopped Brownian motion." Studia Mathematica 117.3 (1996): 253-273. <http://eudml.org/doc/216255>.

@article{Peškir1996,
abstract = {Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_p(x) = exp(|x|^p)-1$ with 0 < p ≤ 2) of $max_\{0≤t≤τ\}|B_t|$ or $|B_τ|$ to be finite, where $B = (B_t)_\{t≥0\}$ 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 $∥max_\{0≤t≤τ\}|B_t|∥_\{ψ_1\} < ∞$ as soon as $E(τ^\{k\}) = O(C^\{k\}k^\{k\})$ for some constant C > 0 as k → ∞ (or equivalently $∥τ∥_\{ψ_1\} < ∞$). In particular, if τ ∼ Exp(λ) or $|N(0,σ^2)|$ then the last condition is satisfied, and we obtain $∥max_\{0≤t≤τ\}|B_t|∥_\{ψ_1\} ≤ K √\{E(τ)\}$ with some universal constant K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying $E(τ^\{k\}) ≤ C(Eτ)^\{k\}k^\{k\}$ for all k ≥ 1 with some fixed constant C > 0. The method of proof relies upon Taylor expansion, Burkholder-Gundy’s inequality, best constants in Doob’s maximal inequality, Davis’ best constants in the $L^p$-inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).},
author = {Peškir, Goran},
journal = {Studia Mathematica},
keywords = {Brownian motion (Wiener process); stopping time; exponential Young function; exponential Orlicz norm; Doob's maximal inequality for martingales; Burkholder-Gundy's inequality; Davis' best constants; Hermite polynomial; continuous (local) martingale; Ito's integral; the quadratic variation process; time change (of Brownian motion); Kahane-Khinchin's inequalities; Brownian motion; martingale inequalities; continuous local martingale},
language = {eng},
number = {3},
pages = {253-273},
title = {On the exponential Orlicz norms of stopped Brownian motion},
url = {http://eudml.org/doc/216255},
volume = {117},
year = {1996},
}

TY - JOUR
AU - Peškir, Goran
TI - On the exponential Orlicz norms of stopped Brownian motion
JO - Studia Mathematica
PY - 1996
VL - 117
IS - 3
SP - 253
EP - 273
AB - Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_p(x) = exp(|x|^p)-1$ with 0 < p ≤ 2) of $max_{0≤t≤τ}|B_t|$ or $|B_τ|$ to be finite, where $B = (B_t)_{t≥0}$ 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 $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} < ∞$ as soon as $E(τ^{k}) = O(C^{k}k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥τ∥_{ψ_1} < ∞$). In particular, if τ ∼ Exp(λ) or $|N(0,σ^2)|$ then the last condition is satisfied, and we obtain $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} ≤ K √{E(τ)}$ with some universal constant K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying $E(τ^{k}) ≤ C(Eτ)^{k}k^{k}$ for all k ≥ 1 with some fixed constant C > 0. The method of proof relies upon Taylor expansion, Burkholder-Gundy’s inequality, best constants in Doob’s maximal inequality, Davis’ best constants in the $L^p$-inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).
LA - eng
KW - Brownian motion (Wiener process); stopping time; exponential Young function; exponential Orlicz norm; Doob's maximal inequality for martingales; Burkholder-Gundy's inequality; Davis' best constants; Hermite polynomial; continuous (local) martingale; Ito's integral; the quadratic variation process; time change (of Brownian motion); Kahane-Khinchin's inequalities; Brownian motion; martingale inequalities; continuous local martingale
UR - http://eudml.org/doc/216255
ER -

References

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[12] G. Peškir, Maximal inequalities of Kahane-Khintchine's type in Orlicz spaces, Math. Proc. Cambridge Philos. Soc. 115 (1994), 175-190. Zbl0809.60005
[13] G. Peškir, Best constants in Kahane-Khintchine inequalities for complex Steinhaus variables, Proc. Amer. Math. Soc., to appear. Zbl0841.41024
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[16] G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991), 579-586. Zbl0719.60050

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