The Best Doob-Type Bounds for the Maximum of Brownian Paths

Abstract

Let B = (B _t)_t≥o be standard Brownian motion started at zero. Then the following inequality is shown to be satisfied:

$$ E\left( {\mathop {0 \le t \le \tau \,|{B_t}{|^p}}\limits^{\max } } \right) \le \mathop \gamma \nolimits_{p,q}^* {\left( {E\int_0^\tau {|{B_t}{|^{q - 1}}dt} } \right)^{p/\left( {q + 1} \right)}} $$

for all stopping times τ for B, all 0 < p < 1 + q, and all q > 0, with the best possible value for the constant being equal:

$$ \Upsilon _{p,q}^* = (1 + k)\left( {\frac{{s_* }} {{k^k }}} \right)^{1/(1 + k)} $$

where k = p/(q-p + l), and s* is the zero point of the (unique) maximal solution s ↦,g* (s) of the differential equation:

$$ {g^\alpha }\left( s \right)\left( {{s^\beta } - {g^\beta }\left( s \right)} \right){{dg} \over {ds}}\left( s \right) = K $$

satisfying 0 < g*(s) < s for all s > s*, where α = q/p — 1, β = 1/p and K = p/2. This solution is also characterized by g*(s)/s → 1 for s → ∞. The equality above is attained at the stopping time:

$${\tau _*} = \inf \left\{ {t > 0|{X_t} = {g_*}\left( {{S_t}} \right)} \right\}$$

where X _t = |B _t|^p and ${S_t} = {\max _{0 \le r \le t}}|{B_r}{|^p}$. In the case p = 1 the closed form for s ↦ g*(s) is found. This yields $\mathop \gamma \nolimits_{1,q}^* = {\left( {q\left( {q + 1} \right)/2} \right)^{1/\left( {q + 1} \right)}}{\left( {\Gamma \left( {1 + \left( {q + 1} \right)/q} \right)} \right)^{q/\left( {q + 1} \right)}}$ for all q > 0. In the case p ≠ 1 no closed form for s ↦ g*(s) seems to exist. The inequality above holds also in the case p = q +1 (Doob’s maximal inequality). In this case the equation above ( with K = p/2c ) admits g* (s) = λs as the maximal solution, and the equality is attained only in the limit through the stopping times τ* = τ* (c) when c tends to the best value $\mathop \gamma \nolimits_{q + 1,q}^* = {\left( {q + 1} \right)^{q + 2}}/2{q^q}$ from above. The method of proof relies upon the principle of smooth fit of Kolmogorov and the maximality principle. The results obtained extend to the case when B starts at any given point, as well as to all non-negative submartingales.