The Maximal Drawdown of the Brownian Meander Yueyun Hu, Zhan Shi, Marc Yor

The maximal drawdown of the Brownian meander Yueyun Hu, Zhan Shi, Marc Yor

To cite this version:

Yueyun Hu, Zhan Shi, Marc Yor. The maximal drawdown of the Brownian meander. Elec- tronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (39), 10.1214/ECP.v20-3945. hal-01301802

HAL Id: hal-01301802 https://hal.archives-ouvertes.fr/hal-01301802 Submitted on 13 Apr 2016

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by Yueyun Hu, Zhan Shi and Marc Yor† UniversitéParis XIII, UniversitéParis VI & Université Paris VI

Summary. 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 Brow- nian meander.

Keywords. Brownian meander, Bessel process, maximal drawdown.

2010 Mathematics Subject Classiﬁcation. 60J65.

1 Introduction

Let (X(t), t ∈ [0, 1]) be a random process. Its maximal drawdown on [0, 1] is deﬁned by X#(1) := sup [ X(s) − X(s)] , s∈[0, 1] where X(s) := supu∈[0,s] X(u). There has been some recent research interest on the study of drawdowns from probabilistic point of view ([7], [8]) as well as applications in insurance and ﬁnance ([1], [2], [3], [10], [12]). 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]); E 1 in particular, the value of ( m#(1) ) is the normalizing constant in the density function of

Partly supported by ANR project MEMEMO2 (2010-BLAN-0125).

1 E 1 this limiting distribution. The sole aim of the present note is to compute ( 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 |W (g + s(1 − g))| m(s) := , s ∈ [0, 1] . (1 − g)1/2 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].

Theorem 1.1. Let (m(s), s ∈ [0, 1]) be a standard Brownian meander. We have 1 π 1/2 (1.1) E = , sups∈[0, 1][ m(s) − m(s)] 2 where m(s) := supu∈[0,s] m(u).

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 (1949–2014), 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.

2 Proof

Let R := (R(t), t ≥ 0) be a three-dimensional Bessel process with R(0) = 0, i.e., the Eu- clidean modulus of a standard three-dimensional Brownian motion. The proof of Theorem 1.1 relies on an absolute continuity relation between (m(s), s ∈ [0, 1]) and (R(s), s ∈ [0, 1]), recalled as follows.

Fact 2.1. (Imhof [5]) Let (m(s), s ∈ [0, 1]) be a standard Brownian meander. Let (R(s), s ∈ [0, 1]) be a three-dimensional Bessel process with R(0) = 0. For any measurable and non-negative functional F , we have π 1/2 1 E F (m(s), s ∈ [0, 1]) = E F (R(s), s ∈ [0, 1]) . h i 2 hR(1) i

2 We now proceed to the proof of Theorem 1.1. Let 1 L := E . sups∈[0, 1][ m(s) − m(s)]

Write R(t) := supu∈[0, t] R(u) for t ≥ 0. By Fact 2.1, π 1/2 1 1 L = E 2 hR(1) sups∈[0, 1][ R(s) − R(s)]i π 1/2 ∞ 1 = E 1 1 da , {sups∈[0, 1][ R(s)−R(s)]< a } 2 Z0 hR(1) i the last equality following from the Fubini–Tonelli theorem. By the scaling property, E 1 E a [ 1 1 ]= [ 2 1 ] for all a> 0. So by means R(1) {sups∈[0, 1][ R(s)−R(s)]< a } R(a ) {supu∈[0,a2][ R(u)−R(u)]<1} of a change of variables b = a2, we obtain: π 1/2 ∞ 1 L = E 1 db . {supu∈[0, b][ R(u)−R(u)]<1} 8 Z0 hR(b) i Deﬁne, for any random process X,

X τ1 := inf{t ≥ 0 : X(t) − X(t) ≥ 1} ,

with X(t) := sups∈[0, t] X(s). For any b> 0, the event {supu∈[0, b][ R(u) − R(u)] < 1} means R {τ1 > b}, so ∞ τ R π 1/2 1 π 1/2 1 1 L = E 1 R db = E db , {τ1 >b} 8 Z0 hR(b) i 8 Z0 R(b) 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 1 XI.1.11, applied to the parameters p = q = 2 and ν = 2 ), 1 t 1 R(t)= U db , t ≥ 0 , 4 Z0 R(b) where U := (U(s), s ≥ 0) is a four-dimensional squared Bessel process with U(0) = 0; in other words, U is the square of the Euclidean modulus of a standard four-dimensional Brownian motion. 1 t 1 Let us introduce the increasing functional σ(t) := 4 0 R(b) db, t ≥ 0. We have R = U ◦σ, and R

R τ1 = inf{t ≥ 0 : R(t) − R(t) ≥ 1} = inf{t ≥ 0 : U(σ(t)) − U(σ(t)) ≥ 1} = inf{σ−1(s): s ≥ 0 and U(s) − U(s) ≥ 1}

3 −1 U U R which is σ (τ1 ). So τ1 = σ(τ1 ), i.e.,

R τ1 1 U db =4τ1 , Z0 R(b) which implies that 1/2 E U L = (2π) (τ1 ) .

U The Laplace transform of τ1 is determined by Lehoczky [6], from which, however, it E U does not seem obvious to deduce the value of (τ1 ). Instead of using Lehoczky’s result E U directly, we rather apply his method to compute (τ1 ). By Itˆo’s formula, (U(t)−4t, t ≥ 0) t is a continuous martingale, with quadratic variation 4 0 U(s) ds; so applying the Dambis– Dubins–Schwarz theorem (Revuz and Yor [9], TheoremR V.1.6) to (U(t) − 4t, t ≥ 0) yields the existence of a standard Brownian motion B =(B(t), t ≥ 0) such that

t U(t)=2B( U(s) ds)+4t , t ≥ 0 . Z0

U Taking t := τ1 , we get U τ1 U U U(τ1 )=2B( U(s) ds)+4τ1 . Z0 We claim that

U τ1 (2.1) E B( U(s) ds) =0 . h Z0 i E U 1 E U Then (τ1 )= 4 [U(τ1 )]; hence π (2.2) L = (2π)1/2 E(τ U )=( )1/2 E[U(τ U )] . 1 8 1 E U Let us admit (2.1) for the moment, and prove the theorem by computing [U(τ1 )] using U Lehoczky [6]’s method; in fact, we determine the law of U(τ1 ).

U Lemma 2.2. The law of U(τ1 ) is given by

P U −a {U(τ1 ) > a} =(a + 1)e , ∀a> 0.

In particular, ∞ E U −a [U(τ1 )] = (a + 1)e da =2. Z0 π 1/2 E U π 1/2 Since L =( 8 ) [U(τ1 )] (see (2.2)), this yields L =( 2 ) as stated in Theorem 1.1.

4 The rest of the note is devoted to the proof of Lemma 2.2 and (2.1).

P U Proof of Lemma 2.2. Fix b > 1. We compute the probability {U(τ1 ) > b} which, due U U P U to the equality U(τ1 ) = U(τ1 ) + 1, coincides with {U(τ1 ) > b − 1}. By applying the U strong Markov property at time σ0 := inf{t ≥ 0 : U(t)=1}, we see that the value of P U {U(τ1 ) > b} does not change if the squared Bessel process U starts at U(0) = 1. Indeed, U U U U observing that σ ≤ τ , U(σ ) = 1 and that U(τ ) = sup U U U(s), we have 0 1 0 1 s∈[σ0 , τ1 ]

P U P P U {U(τ1 ) > b} = sup U(s) > b = 1{U(τ1 ) > b} , U U n s∈[σ0 , τ1 ] o

the subscript 1 in P1 indicating the initial value of U. More generally, for x ≥ 0, we write

Px(•) := P(•| U(0) = x); so P = P0.

Let b0 = 1 < b1 < ··· < bn := b be a subdivision of [1, b] such that max1≤i≤n(bi − U bi−1) → 0, n → ∞. Consider the event {U(τ1 ) > b}: since U(0) = 1, this means U hits U Z position b before time τ1 ; for all i ∈ [1, n − 1] ∩ , starting from position bi, U must hit bi+1 before hitting bi − 1 (caution: not to be confused with bi−1). More precisely, let U U σi := inf{t ≥ 0 : U(t)= bi} and let Ui(s) := U(s + σi ), s ≥ 0; then

n−1 U {U(τ1 ) > b} ⊂ {Ui hits bi+1 before hitting bi − 1} . i\=1

By the strong Markov property, the events {Ui hits bi+1 before hitting bi − 1}, 1 ≤ i ≤

n − 1, are independent (caution : the processes (Ui(s), s ≥ 0), 1 ≤ i ≤ n − 1, are not independent). Hence

n−1 P U P (2.3) 1{U(τ1 ) > b} ≤ bi {U hits bi+1 before hitting bi − 1} . Yi=1

Conversely, let ε> 0, and if max1≤i≤n(bi − bi−1) <ε, then we also have

n−1 P U P 1{U(τ1+ε) > b} ≥ bi {U hits bi+1 before hitting bi − 1} , Yi=1

U U with τ1+ε := inf{t ≥ 0 : U(t)−U(t) ≥ 1+ε}. By scaling, U(τ1+ε) has the same distribution U as (1+ ε)U(τ1 ). So, as long as max1≤i≤n(bi − bi−1) <ε, we have

n−1 b P {U(τ U ) > b} ≤ P {U hits b before hitting b − 1} ≤ P {U(τ U ) > } . 1 1 bi i+1 i 1 1 1+ ε Yi=1

5 1 Since x is a scale function for U, we have

1 − 1 1 − 1 P {U hits b before hitting b − 1} = bi−1 bi =1 − bi bi+1 . bi i+1 i 1 − 1 1 − 1 bi−1 bi+1 bi−1 bi+1

If limn→∞ max0≤i≤n−1(bi+1 − bi) = 0, then for n →∞,

n−1 1 − 1 n−1 bi bi+1 bi − 1 1 1 = (bi+1 − bi)+ o(1) − bi Xi=1 bi−1 bi+1 Xi=1 b r − 1 → dr Z1 r = b − 1 − log b .

Therefore,

n−1 P −(b−1−log b) −(b−1) lim bi {U hits bi+1 before hitting bi − 1} = e = b e . n→∞ Yi=1 Consequently, P U −(b−1) {U(τ1 ) > b} = b e , ∀b> 1.

U U We have already noted that U(τ1 )= U(τ1 ) − 1. This completes the proof of Lemma 2.2.

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 (Gr)r≥0-Brownian motion (Revuz and Yor [9], Theorem V.1.6), where, for r ≥ 0,

t −1 Gr := FC(r), C(r) := A (r), A(t) := U(s) ds , Z0

−1 and A denotes the inverse of A. [We mention that FC(r) is well deﬁned because C(r) is

an (Ft)t≥0-stopping time.] As such,

U τ1 U U(s) ds = A(τ1 ) . Z0 U U F G U F For all r ≥ 0, {A(τ1 ) >r} = {τ1 >C(r)} ∈ C(r) = r (observing that τ1 is an ( t)t≥0- τ U U G U 1 stopping time), which means that A(τ1 ) is a ( r)r≥0-stopping time. If A(τ1 )= 0 U(s) ds has a ﬁnite expectation, then we are entitled to apply the (ﬁrst) Wald identityR to see that E U [B(A(τ1 ))] = 0 as claimed in (2.1).

6 E U It remains to prove that [A(τ1 )] < ∞. Recall that U is the square of the Euclidean modulus of an R4-valued Brownian motion. By considering only the ﬁrst coordinate of this Brownian motion, say β, we have

P sup U(s) < a1−ε ≤ P sup |β(s)| < a(1−ε)/2 = P sup |β(s)| < a−ε/2 ; n s∈[0, a] o n s∈[0, a] o n s∈[0, 1] o so by the small ball probability for Brownian motion, we obtain:

1−ε ε P sup U(s) < a ≤ exp(−c1 a ) , n s∈[0, a] o

for all a ≥ 1 et all ε ∈ (0, 1), with some constant c1 = c1(ε) > 0. On the event 1−ε U 1−ε Z {sups∈[0, a] U(s) ≥ a }, if τ1 > a, then for all i ∈ [1, a − 1] ∩ , the squared Bessel process U, starting from i, must ﬁrst hit position i + 1 before hitting i − 1 (which, for each i, can be realized with probability ≤ 1 − c2, where c2 ∈ (0, 1) is a constant that does not depend on i, nor on a). Accordingly,1

P 1−ε U ⌊a1−ε−1⌋ 1−ε sup U(s) ≥ a , τ1 > a ≤ (1 − c2) ≤ exp(−c3 a ) , n s∈[0, a] o with some constant c3 > 0, uniformly in a ≥ 2. We have thus proved that for all a ≥ 2 and all ε ∈ (0, 1), P U 1−ε ε {τ1 > a} ≤ exp(−c3 a ) + exp(−c1 a ).

1 Taking ε := 2 , we see that there exists a constant c4 > 0 such that

P U 1/2 {τ1 > a} ≤ exp(−c4 a ), ∀a ≥ 2.

On the other hand, U being a squared Bessel process, we have, for all a> 0 and all b ≥ a2,

b b 2 P A a b P A P U s −c5 b/a , { ( ) ≥ } = { (1) ≥ 2 } ≤ sup ( ) ≥ 2 ≤ e a n s∈[0, 1] a o

2 for some constant c5 > 0. Hence, for b ≥ a and a ≥ 2,

2 P U P U P 1/2 −c5 b/a {A(τ1 ) ≥ b} ≤ {τ1 > a} + {A(a) ≥ b} ≤ exp(−c4 a ) + e .

Taking a := b2/5 gives that

P U 1/5 {A(τ1 ) ≥ b} ≤ exp(−c6 b ) ,

E U for some constant c6 > 0 and all b ≥ 4. In particular, [A(τ1 )] < ∞ as desired. 1 This is the special case bi := i of the argument we have used to obtain (2.3).

7 References

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Yueyun Hu Zhan Shi Marc Yor† Département de Mathématiques LPMA, Case 188 LPMA, Case 188 & IUF UniversitéParis XIII UniversitéParis VI UniversitéParis VI 99avenueJ-BClément 4placeJussieu 4placeJussieu F-93430 Villetaneuse F-75252 Paris Cedex 05 F-75252 Paris Cedex 05 France France France [email protected] [email protected] [email protected]