The maximal drawdown of the Brownian meander

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

We are interested in the maximal drawdown m # (1) of the standard Brownian meander (m(t), t ∈ [0, 1]). Our motivation is the presence of the law of m # (1) in the limiting distribution of randomly biased random walks on supercritical Galton-Watson trees ( [4]); in particular, the value of E( 1 m # (1) ) is the normalizing constant in the density function of this limiting distribution. The sole aim of the present note is to compute E( 1 m # (1) ), which turns out to have a nice numerical value.
Let us first recall the definition of the Brownian meander. Let W := (W (t), t ∈ [0, 1]) be a standard Brownian motion, and let g := sup{t ≤ 1 : W (t) = 0} be the last passage time at 0 before time 1. Since g < 1 a.s., we can define The law of (m(s), s ∈ [0, 1]) is called the law of the standard Brownian meander. For an account of general properties of the Brownian meander, see Yen and Yor [11].
) be a standard Brownian meander. We have The theorem is proved in Section 2.
We are grateful to an anonymous referee for a careful reading of the manuscript and for many suggestions for improvements. N.B. from the first-named coauthors: This note originates from a question we asked our teacher, Professor Marc Yor , who passed away in January 2014, during the preparation of this note. He provided us, in November 2012, with the essential of the material in Section 2.

Proof
Let R := (R(t), t ≥ 0) be a three-dimensional Bessel process with R(0) = 0, i.e., the Eu- We now proceed to the proof of Theorem 1.1. Let .
Write R(t) := sup u∈[0, t] R(u) for t ≥ 0. By Fact 2.1, a } da , the last equality following from the Fubini-Tonelli theorem. By the scaling property, ] for all a > 0. So by means of a change of variables b = a 2 , we obtain: Define, for any random process X, the second identity following from the Fubini-Tonelli theorem. According to a relation between Bessel processes of dimensions three and four (Revuz and Yor [9], Proposition XI.1.11, applied to the parameters p = q = 2 and ν = 1 2 ), in other words, U is the square of the Euclidean modulus of a standard four-dimensional Brownian motion.
Let us introduce the increasing functional σ(t) := 1 which implies that The Laplace transform of τ U 1 is determined by Lehoczky [6], from which, however, it does not seem obvious to deduce the value of E(τ U 1 ). Instead of using Lehoczky's result directly, we rather apply his method to compute E(τ U 1 ). By Itô's formula, (U(t) −4t, t ≥ 0) is a continuous martingale, with quadratic variation 4 t 0 U(s) ds; so applying the Dambis-Dubins-Schwarz theorem (Revuz and Yor [9], Theorem V.1.6) to (U(t) − 4t, t ≥ 0) yields the existence of a standard Brownian motion B = (B(t), t ≥ 0) such that Taking t := τ U 1 , we get We claim that Let us admit (2.1) for the moment, and prove the theorem by computing E[U(τ U 1 )] using Lehoczky [6]'s method; in fact, we determine the law of U(τ U 1 ).
Proof of (2.1). The Brownian motion B being the Dambis-Dubins-Schwarz Brownian motion associated with the continuous martingale (U(t)−4t, t ≥ 0), it is a (G r ) r≥0 -Brownian motion (Revuz and Yor [9], Theorem V.1.6), where, for r ≥ 0, and A −1 denotes the inverse of A. [We mention that F C(r) is well defined because C(r) is an (F t ) t≥0 -stopping time.] As such, so by the small ball probability for Brownian motion, we obtain: Taking a := b 2/5 gives that for some constant c 6 > 0 and all b ≥ 4. In particular, E[A(τ U 1 )] < ∞ as desired.