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On maximum increase and decrease of Brownian motion
Paavo Salminen
a, Pierre Vallois
b,∗aÅbo Akademi University, Mathematical Department, FIN-20500 Åbo, Finland bUniversité Henri Poincaré, Département de Mathématique, 54506 Vandoeuvre les Nancy, France
Received 7 December 2005; accepted 15 September 2006 Available online 13 January 2007
Abstract
The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is com- puted. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three- dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be−0.47936. . ..
©2007 Elsevier Masson SAS. All rights reserved.
Résumé
Dans cet article nous déterminons la loi conjointe de la plus grande montée et de la plus grande descente d’un mouvement brownien arrêté en un temps exponentiel indépendant. La preuve repose sur la décomposition de la trajectoire brownienne aux instants où le processus atteint son maximum, resp. son minimum, avant le temps exponentiel. La loi de la plus grande descente d’un processus de Bessel, de dimension trois, issu de 0 et arrêté lorsqu’il atteint un niveau fixé, joue également un rôle important.
Le coefficient de corrélation linéaire de la grande montée et de la plus grande descente d’un mouvement brownien arrêté en temps fixe est déterminé :−0.47936. . ..
©2007 Elsevier Masson SAS. All rights reserved.
MSC:60J60; 60J65; 60G17; 62P05
Keywords:h-transform; Time reversal; Path decompositions; Brownian motion with drift; Excursion process; Maximum process; Itô measure;
Maximum drawdown; Covariance; Catalan’s constant
1. Introduction and notation
1. In this paper we are interested in the joint distribution of the maximum increase and decrease for a standard Brownian motion, for short BM. Let us start with some notation. Let Ω:=C(R+,R)be the space of continuous functionsω:R+→RandXt(ω)=ω(t ),t0, the coordinate mappings. With everyω we associate its lifetime
* Corresponding author.
E-mail addresses:phsalmin@abo.fi (P. Salminen), vallois@iecn.u-nancy.fr, vallois@antares.iecn.u-nancy.fr (P. Vallois).
0246-0203/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpb.2006.09.007
ζ (ω)∈(0,∞]and considerXtto be defined fort < ζ (ω). The standard notationFtis used for theσ-algebra generated by the coordinate mappings up to timet, and we setF:=F∞.
Further,Pμx andEμx denote the probability measure and the expectation operator on(Ω,F)under which the co- ordinate processX= {Xt: t0}is a Brownian motion with driftμstarted fromx, for short BM(μ). For simplicity, PxandExstand for the corresponding objects for BM.
The maximum increase before timetis defined as D+t := sup
0uvt
(Xv−Xu) (1.1)
and, analogously, the maximum decrease D−t := sup
0uvt
(Xu−Xv). (1.2)
Notice that, e.g., D−t = sup
0vt
sup
0uv
Xu−Xv
. (1.3)
Using the Lévy isomorphism, i.e., underP0
Xv+:= sup
0uv
Xu−Xv: v0 (d)
=
|Xv|: v0
, (1.4)
where(d)=means “to be identical in law with”, it follows from identity (1.3) that
P0(D−t > a)=P0(Ha∧H−a< t ) (1.5)
with
Hb:=inf{t:Xt=b}, b∈R,
the first hitting time ofb(in the canonical setting) with the usual convention thatHb= +∞if the set in the braces is empty. From the equality (1.5) applying, e.g., [4] 1.3.0.2 p. 212, and 3.1.1.4 p. 333 we obtain
P0(D−t > a)= +∞
k=−∞
(−1)k
t
0
ds(2k+1)a
√2π s3/2e−(2k+1)2a2/2s
=1− 1
√2π t
+∞
k=−∞
a
−a
dx
e−(x+4ka)2/2t−e−(x+4ka+2a)2/2t .
Notice also that due to the symmetry of standard Brownian motion we have {Dt−: t0}(d)= {Dt+: t0},
and this holds underPxfor anyx∈R. We refer to Douady et al. [6] for results concerning the distribution of maximum increase and related functionals up to a fixed time in the case of Brownian motion.
For Brownian motion with drift the result corresponding to (1.4) states that underPμthe process{Xv+: v0}is a reflected Brownian motion onR+with drift−μ, for short RBM(−μ) (see, e.g., Harrison [7] p. 49, and McKean [9]
p. 71), more precisely, it is a diffusion onR+with basic characteristics as given in [4] A1.16 p. 129. The probability measure on(Ω,F)associated withX+(underPμ) is denoted byP−μ,+. Clearly, we have for a givena >0
Pμ0(D−t > a)=P−0μ,+(Ha< t ). (1.6)
Similarly, defining X−v :=Xv− inf
0uvXu
it holds underPμthat the processX−is a RBM(μ). LettingPμ,+denote the measure associated withX−we have
Pμ0(D+t > a)=Pμ,0+(Ha< t ). (1.7)
For an explicit expression of thePμ-distribution ofD−t , see Dominé [5] where the method based on spectral repre- sentations is used. In Magdon-Ismail et al. [8] formulas for the mean ofD−t are derived.
2.Unfortunately we are not able to determine explicitly the distribution of(Dt+, D−t ), but replacingt byT, that is, an exponentially distributed random variable independent ofX with mean 1/λ, allows us to find the P-distribution of(D+T, DT−), see Propositions 4.5 and 4.6. We remark that the marginalPμ-distributions ofD+T andDT−are easily computed from (1.7) and (1.6), respectively. Indeed, using standard diffusion theory and some explicit formulas (see e.g. [4] p. 18 and 129) yield
Pμ0(DT−> a)=E−0μ,+
exp(−λHa)
=1/ψλ(a; −μ) (1.8)
and
Pμ0(DT+> a)=1/ψλ(a;μ) (1.9)
with
ψλ(a;ν):=e−νa
ch a
2λ+ν2
+ ν
√2λ+ν2sh a
2λ+ν2 .
In our approach for finding the joint distribution we consider first the case where the infimum is attained before the supremum. In this case it is clear that the maximum increase is nothing but the difference of the supremum and the infimum, and, in a sense, we have reduced the problem to the problem for finding the distribution of the maximum decrease. The opposite case where the supremum is attained before the infimum is clearly treated using symmetry.
It is natural when the infimum is attained before the supremum to decompose the exponentially stopped Brownian path into three parts:
– the first part is up to the hitting time of the infimum,
– the second part is from the hitting time of the infimum to the hitting time of the supremum, – the third part is from the hitting time of the supremum to the exponential time.
We prove in Theorem 3.5 that these three parts are conditionally independent given the infimum and the supremum, and find their distributions in terms of the three-dimensional Bessel processes with drift. Our approach is mainly based on theh-transform techniques, excursion theory and path decompositions of Brownian motion with drift.
The above described path decomposition up to T permits us to determine the joint distribution of (D+T, DT−) since nowD+T =ST−IT and, under this decomposition,D−T is the maximum of the maximum decreases of the three conditionally independent fragments. To find the distribution of the maximum decrease for the first and the third part is fairly straightforward diffusion theory. To compute the maximum decrease for the second part is equivalent for finding the distribution of the maximum decrease for a three-dimensional Bessel process with drift (see Proposition 2.5).
Although the distribution and the density function of(DT+, DT−)are complicated it is possible to determine by the scaling property of BM the covariance betweenDt+andDt−and this is given by
E(D+t D−t )=
1−2 log 2+2β(2) t, where
β(2):=∞
k=0
(−1)k(2k+1)−2=0.91596. . .
is Catalan’s constant. Hence, the correlation coefficientρbetweenDT+andDT−is easily obtained to be ρ:=E(Dt+D−t )−(E(Dt+))2
Var(D+t ) = −0.47936. . . .
3.One motivation to study the maximum decrease and increase comes from mathematical finance where the maximum decrease, also called maximum drawdown (MDD), is used to quantify the riskiness of a stock or any other asset.
Related measures used hereby are e.g. the recovery time from MDD and the duration of MDD. Our interest to the problem discussed in the paper arose from a question by Gabor Szekely who asked for an expression for the covariance betweenDt+andDt−.
4.The paper is organized so that in the next section we find the distribution of the maximum decrease of a stopped Brownian motion with positive drift. In fact, we compute this distribution under the restriction that the process does not hit some negative level, and proceed from here to the distribution of the maximum decrease for a three-dimensional Bessel process with drift. In the third section path decompositions are discussed. To prove our main path decompo- sition Theorem 3.5, we first prove a decomposition of the Brownian trajectory{Bt: tT}conditionally onIT (see Theorem 3.2). The fourth section is devoted to computation and analysis of the law of(DT−, DT+).
2. Maximum decrease for stopped Brownian motion with drift
According to (1.2), the maximum decrease up to the first hitting time of a given levelβis defined as D−H
β :=sup{Xu−Xv: 0uvHβ}. In this section we consider thePμ-distribution ofDH−
β under some additional conditions and conditioning. Recall that Sμ(x):= 1
2μ
1−e−2μx
(2.1) is the scale function of BM(μ)and fora < x < b
Pμx(Ha< Hb)=Sμ(b)−Sμ(x)
Sμ(b)−Sμ(a). (2.2)
Proposition 2.1.For nonnegativeα, β, andu
Pμ0(D−H
β< u, Hβ< H−α)=
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩ exp
− β S−μ(u)
, uα,
Sμ(α) Sμ(u)exp
−β+α−u S−μ(u)
, αuα+β, Sμ(α)
Sμ(α+β), α+βu.
In particular, Pμ0(D−H
β< u)=exp
− β Sμ(u)
.
For standard Brownian motion, i.e.,μ=0, the above formulas hold withS0(u)=u.
Proof. We assume thatX0=0, and define fora >0 Ha+:=inf{t: Xt> a}.
For a givena >0, in the caseHa+> Ha, let Ξa+(u):=a−Xu+Ha, 0u < Ha+−Ha,
and, ifHa+=Ha, takeΞa+:=∂, where∂is some fictious (cemetery) state. The process Ξ+= {Ξa+: a0}
is called the excursion process, associated withX, for excursions under the running maximum. Let, further, Ma:=sup
Ξa+(u): 0u < Ha+−Ha . An obvious but important fact is that
D−H
β =sup
a<β
Ma. (2.3)
Introduce also foru >0
ξu:=inf{a0: Ma> u},
and
ξu◦:=inf{a0:a−Ma<−u}. Then it holds for positiveα, β, andu
{D−H
βu} =
∀a∈(0, β): Mau
= {ξuβ}, and
{Hβ< H−α} =
∀a∈(0, β): a−Ma>−α
= {ξα◦> β};
hence, for 0< u < α+β Pμ0(DH−
β< u, Hβ< H−α)=Pμ0(ξu> β, ξα◦> β). (2.4) SinceX+underPμ is identical in law with RBM(−μ) it follows that the excursion processΞ+is identical in law with the usual excursion process of RBM(−μ) for excursions from 0 to 0. Consequently, see Pitman and Yor [11],
Π=
(a, Ma): a0
is a homogeneous Poisson point process with the characteristic measure ν(da,dm)=da n(dm),
where form >0 n
(m,+∞)
=1/S−μ(m).
Introduce the sets
A:= [0, β)× [u,+∞) and B:=
(a, m): 0a < β, a−m <−α , and letN denote the counting measure associated withΠ. Now we have
Pμ0(ξu> β, ξα◦> β)=Pμ0
N (A∪B)=0
=exp
−ν(A∪B) .
It is straightforward to computeν(A∪B)for different values onα, β, andu, and we leave this to the reader. Conse- quently, by (2.4), the claimed formula is obtained. 2
Obviously,DH+
ββand forz >0 {D+H
β−β < z} = {Hβ< H−z}.
Consequently, we have from Proposition 2.1 the following corollary giving an expression for the joint distribution of DH+
β andD−H
β. Notice that DH−
βDH+
βD−H
β+β explaining the three cases below.
Corollary 2.2.Forvβandμ0
Pμ0(DH−
β< u, D+H
β < v)=
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩ exp
− β S−μ(u)
, uv−β,
Sμ(v−β) Sμ(u) exp
− v−u S−μ(u)
, v−βuv, Sμ(v−β)
Sμ(v) , vu.
For standard Brownian motion, i.e.,μ=0, the above formulas hold withS0(u)=u.
We proceed by developing the result in Proposition 2.1 for a 3-dimensional Bessel process with driftμ >0, for short BES(3, μ). We recall that BES(3, μ) is a linear diffusion with the generator
GR,μ=1 2
d2
dx2+μcoth(μx)d
dx, x >0. (2.5)
The notationQμx is used for the probability measure on the canonical spaceΩassociated with BES(3, μ) when started fromx0. In the caseμ=0 the corresponding measure is simply denoted byQxand the generator is given by
GR=1 2
d2 dx2+1
x d
dx, x >0. (2.6)
The following lemma is a fairly well-known example onh-transforms. To make the presentation more self con- tained we give a short proof. It is also interesting to compare the result with Lemma 3.1 in the next section.
Lemma 2.3.Let0< x < y be given. The Brownian motion with driftμstarted fromx >0, killed at the first hitting time ofy, and conditioned to hity before0 is identical in law with a3-dimensional Bessel process with drift|μ| started fromx and killed at the first hitting time ofy.
Proof. In our canonical space of continuous functions withX0=xwe have for a givent >0 {t < Hy< H0} = {t < Hy∧H0, Hy◦θt< H0◦θt},
whereθ·is the usual shift operator, i.e.,Xs◦θt=Xs+t. Hence, for anyAt∈Ft
Pμx(At, t < Hy|Hy< H0)=Pμx(At, t < Hy∧H0, Hy◦θt< H0◦θt)/Pμx(Hy< H0)
=Eμx
h1(Xt);At, t < Hy∧H0 / h1(x) by the Markov property, where
h1(x):=Pμx(Hy< H0)=Sμ(x)−Sμ(0)
Sμ(y)−Sμ(0)=1−e−2μx 1−e−2μy.
Consequently, the desired conditioning can be realized by taking the Doobh-transform (withh=h1) of the Brownian motion with driftμkilled atHy∧H0. The generator of theh-transform is
Gh1:=1 2
d2 dx2 +μ d
dx +h(x) h(x)
d dx,
which is easily seen to coincide with (2.5) with|μ|instead ofμ. 2
Remark 2.4.Analogously as above, it can be proved that BM(μ)started fromx >0 and conditioned not to hit 0 is identical in law with BES(3,|μ|)started fromx.
Proposition 2.5.Forβ > u >0 Qμ0(DH−
β< u)=S−μ(β) S−μ(u)exp
− β−u
S−μ(u)−2μ(β−u)
.
For the3-dimensional Bessel process without drift, i.e.,μ=0, the above formula holds withS0(u)=u.
Proof. Note that underQμ0 it holds a.s. on{DH−
β> u}that D−H
β =DH−
β◦θHu.
Therefore, applying the strong Markov property at timeHuyields Qμ0(DH−
β> u)=Qμu(D−H
β > u), and from Lemma 2.3
Qμu(D−H
β> u)=Pμu(D−H
β> u|Hβ< H0)=Pμu(DH−
β> u, Hβ< H0)/Pμu(Hβ< H0)
=Pμ0(D−H
β−u> u, Hβ−u< H−u)/Pμ0(Hβ−u< H−u).
The proof is now easily completed from Proposition 2.1. 2 3. Path decompositions
The main path decomposition results presented in this section are stated in Theorems 3.2 and 3.5. In the first one we consider the decomposition at the global infimum and the second one gives, roughly speaking, the decomposition of the post part of the previous decomposition at its global supremum.
We begin with by stating the following lemma which is proved similarly as Lemma 2.3.
Lemma 3.1.Let0< x < ybe given. The Brownian motion started fromx >0, killed at the first hitting time ofy, and conditioned by the eventHy< H0∧T, whereT is an exponentially distributed random variable with parameter λ independent of the Brownian motion, is identical in law withBES(3,√
2λ)started fromxand killed at the first hitting time ofy.
Proof. We adapt the proof of Lemma 2.3 to our new situation. Fort >0 we have {t < Hy< H0∧T} =
t < Hy∧H0∧T , Hy◦θt< (H0∧T )◦θt . Hence, using the memoryless property ofT we get forAt∈Ft
Px(At, t < Hy|Hy< H0∧T )=Ex
h2(Xt);At, t < Hy∧H0∧T / h2(x) with
h2(x)=Px(Hy< H0∧T )=Px(Hy< H0, Hy< T )=sh(x√
2λ )/sh(y√ 2λ ) (see [4] 1.3.0.5(b) p. 212).
Consequently, the desired conditioning can be realized by taking the Doobh-transform (withh=h2) of a Brownian motion killed at timeHy∧H0∧T. The generator of theh-transform can be computed in the usual way, and is seen to coincide with the generator of BES(3,√
2λ) (see (2.5)). 2
We let, throughout the paper,T denote an exponentially with parameterλdistributed random variable independent ofXunderP0, and define
IT :=inf{Xt: 0tT}, ST :=sup{Xt: 0tT} and
HI:=inf{t: Xt=IT}, HS:=inf{t: Xt=ST}.
Next we discuss the path decomposition at the global infimum for Brownian motion killed atT. In Pitman and Winkel [12] Theorem 3 p. 2205 such a decomposition is proved via discrete approximations. Our description below offers, in addition, an explicit diffusion characterization of the post-HI process. If nothing else is stated the coordinate process is considered underP0.
Theorem 3.2. 1.The processes{Xt: 0t < HI}and{XT−t−XT: 0t < T −HI}are independent and identically distributed.
2.GivenIT =a
(1) the pre-HI-process{Xt: 0t < HI}and the post-HI-process{XHI+t: 0t < T −HI}are independent.
(2) the pre-HI-process is identical in law with aBM(−√
2λ)killed when it hitsa,
(3) the post-HI-process is identical in law with the diffusionZstarted fromaand having the generator GZu(x)=1
2u(x)+h3(x−a)
h3(x−a)u(x)− λ
h3(x−a)u(x), (3.1)
wherex > aand
h3(y):=Py(T < H0)=1−e−y
√2λ, y >0. (3.2)
Moreover,Zis the Doobh-transform withh=h3(· −a)of BMkilled at timeT ∧Ha.
Proof. (a) We have two proofs: the first one is “direct” in a sense that we compute the conditional finite dimensional distributions, and the second one relies on excursion theory of Brownian motion. From our point of view, both proofs contain interesting elements and it seems worthwhile to present these two different approaches.
(b) We begin with thedirect proof of claim 2. Define fors < t Is,t:=inf{Xu: sut}, It:=I0,t
and
HIs,t :=inf
u∈(s, t ): Xu=Is,t
, HIt:=HI0,t.
Letu, v, andt be given such that 0< u < v < t. For positive integersnandmintroduce 0< u1<· · ·< un< uand 0< v1<· · ·< vmwithvm+v < t. Define also
An:= {Xu1∈dx1, . . . , Xun∈dxn}, (3.3)
and
Bm:= {Xv1∈dy1, . . . , Xvm∈dym}. (3.4)
Consider now foru < s < v
P0(An, It∈da, HIt ∈ds, Bm◦θv, Xt∈dz)
=P0(An, Iu> a, Iu,v∈da, HIu,v ∈ds, Iv,t> a, Bm◦θv, Xt∈dz)
=P0
An, Iu> a;P0(Iu,v∈da, HIu,v ∈ds, Iv,t> a, Bm◦θv, Xt∈dz|Fu) . Further,
P0(Iu,v∈da, HIu,v ∈ds, Iv,t> a, Bm◦θv, Xt∈dz|Fu)
=P0
Iu,v∈da, HIu,v∈ds;P0(Iv,t> a, Bm◦θv, Xt∈dz|Fv)|Fu
=P0
Iu,v∈da, HIu,v∈ds;P0(Iv,t> a, Bm◦θv, Xt∈dz|Xv)|Xu
by the Markov property. Lettingp+denote the transition density (with respect to 2 dx) of BM killed when it hitsa and writingXv=z2we have
P0(Iv,t> a, Bm◦θv, Xt∈dz|Xv)=p+(v1;z2, y1)2 dy1· · ·p+(t−v−vm;ym, z)2 dz
=:Fv1,...,vm,t−v(z2, y1, . . . , ym, z)2 dy1· · ·2 dz.
Introduce
ηx(a, α):=Px(Ha∈dα)/dα, (3.5)
and recall the formula due to Lévy:
Px(Iβ∈da, HIβ∈dα, Xβ∈dz)=ηx(a, α)ηz(a, β−α)dα2 dzda, α < β. (3.6) Applying (3.6) and puttingXu=z1we obtain
P0
Iu,v∈da, HIu,v∈ds;P0(Iv,t> a, Bm◦θv, Xt∈dz|Fv)|Fu
=
∞
a
2 dz2ηz1(a, s−u)ηz2(a, v−s)dadsFv1,...,vm,t−v(z2, y1, . . . , ym, z)2 dy1· · ·2 dym2 dz,
and, finally,
P0(An, It∈da, HIt ∈ds, Bm◦θv, Xt∈dz)
=
∞ a
2 dz1
∞ a
2 dz2p+(u1;0, x1)2 dx1· · ·p+(u−un;xn, z1)ηz1(a, s−u)ηz2(a, v−s)dads
×Fv1,...,vm,t−v(z2, y1, . . . , ym, z)2 dy1· · ·2 dym2 dz
=
∞ a
2 dz1
∞ a
2 dz2Fu1,...,un,u(0, x1, . . . , xn, z1)2 dx1· · ·2 dxnηz1(a, s−u)ηz2(a, v−s)dads
×Fv1,...,vm,t−v(z2, y1, . . . , ym, z)2 dy1· · ·2 dym2 dz. (3.7) Replacing in (3.6) the deterministic timeβwith the exponential timeT yields fora <0 anda < z
P0(IT ∈da, HI∈ds, XT ∈dz)=η0(a, s)λe−λsEz
e−λHa
ds2 dzda, (3.8)
and, further,
P0(IT ∈da, HI∈ds)=√
2λe−λsη0(a, s)dads. (3.9)
We operate similarly in (3.7), i.e., introduce the exponential timeT in place oft. After this we integrate overz, and divide with the expression on the r.h.s. in (3.9) and obtain foru < s < v
P0(An, Bm◦θv|IT =a, HI=s)
=Fu1,...,un(x1, . . . , xn;a, s)2 dx1· · ·2 dxnGv1,...,vm(y1, . . . , ym;a;s, v)2 dy1· · ·2 dym, (3.10) with
Fu1,...,un(x1, . . . , xn;a;s)= ˆp+(u1;0, x1)· · · ˆp+(un−un−1;xn−1, xn)ηˆxn(a, s−un) ˆ
η0(a, s) (3.11)
and
Gv1,...,vm(y1, . . . , ym;a;s, v)
=ηˆy1(a, v−s+v1)
√2λ pˆ+(v2−v1;y1, y2)· · · ˆp+(vm−vm−1;ym−1, ym)h3(ym−a), (3.12) whereh3is as in (3.2) and
ˆ
p+(α;x, y):=e−λαp+(α;x, y), ηˆx(a, α):=e−λαηx(a, α).
Because ˆ
ηx(a, α)=Px(Ha∈dα, Ha< T )/dα
it is seen from (3.11) thatFdescribes the finite dimensional distributions ofXconditioned to hitaat timesbeforeT. For the claim concerning the post-process we remark first that (3.12) gives finite dimensional distributions of the announcedh-transform started fromasince
ˆ
ηy1(a, v−s+v1)
√2λ =lim
x↓a
ˆ
p+(v−s+v1;x, y1) h3(x−a) .
Next notice that proceeding as above we can also compute the conditional probabilities forAnandBm◦θvseparately and deduce
P0(An, Bm◦θv|IT =a, HI=s)=P0(An|IT =a, HI=s)P0(Bm◦θv|IT =a, HI=s). (3.13) As is seen from (3.12) the quantity
P0(Bm◦θv|IT =a, HI=s)
is a function of the differencev−sonly, and we find the desired description of the post-process by lettingv↓s(and applying the Lebesgue dominated convergence theorem). To remove the conditioning with respect toHI in (3.13) observe from (3.9) that
P0(HI∈ds|IT =a)=ηˆ0(a, s) ea√2λ ds and, hence,
P0(An|IT =a)= ˆp+(u1;0, x1)2 dx1· · · ˆp+(un−un−1;xn−1, xn)2 dxn
e−(xn−a)
√2λ
e−a√2λ ,
which means that the pre-process is as stated, and, moreover, the claimed conditional independence holds.
It is possible to prove claim 1 also via direct computations with finite dimensional distributions; however, we do not present this proof since, as seen below, the result is in the core of the approach with excursions.
(c)Excursion theoretical proof.The excursion process associated with the excursions above the running minimum is defined similarly as the corresponding process with running maximum in Section 2. Indeed, let fora <0
Ha−:=inf{t: Xt< a}, and, ifHa−> Ha,
Ξa−(u):=Xu+Ha−a, 0u < Ha−−Ha. Then the process
Ξ−= {Ξa−: a0}
is a homogeneous Poisson point process, and is called the excursion process for excursions above the running mini- mum. We remark thatΞ−is identical in law with the excursion process for excursions from 0 of a reflecting Brownian motion. The Itô excursion measure associated withΞ−is denoted byn−(for different descriptions ofn−, see Revuz and Yor [13]).
LetF1andF2be measurable mappings fromC(R+,R)toR+. Now we can write Δ:=E
F1(Xu: uHI)F2(XHI+u−IT: uT −HI)
=E
a<0
F1(Xu: uHa)F2
Ξa−(u): uT −Ha
1{HaT <Ha−}
,
where the sum is over all points ofΞ−(but simplifies, for everyωa.s., only to one term). LetEdenote the excursion space andεa generic excursion. By the compensation formula for Poisson point processes (see Bertoin [2])
Δ=
0
−∞
daE0
F1(Xu: uHa)1{Ha<T} E
F2(εu: uT−Ha)1{T−Ha<ζ}(ε)n−(dε)
=
0
−∞
daE0
F1(Xu: uHa)e−λHa
E
n−(dε)E0
F2(εu: uT )1{T <ζ}(ε)
, (3.14)
where the notationζ (ε)is for the life time ofεand in the second step the fact thatT is an exponentially distributed random variable independent ofXis used. Notice that (3.14) yields, when choosingF1≡1,
E0
F2(XHI+u−IT: uT −HI)
= 1
√2λ
E
n−(dε)E0
F2(εu: uT )1{T <ζ}(ε) .
By absolute continuity, E0
F1(Xu: uHa)e−λHa
=E−
√2λ 0
F1(Xu: uHa) ea
√2λ
and, since−IT is exponentially distributed with parameter√
2λ, we have
Δ=
0
−∞
daE−
√2λ 0
F1(Xu: uHa)
P(IT ∈da) 1
√2λ
E
n−(dε)E0
F2(εu: uT )1{T <ζ}(ε)
. (3.15)
Consequently, the processes {Xu: uHI} and {XHI+u −IT: uT −HI} are independent and, hence, also {Xu: uHI}and{XT−t −XT: uT −HI}are independent. Moreover, {Xu: uHI}givenIT =a is identi- cal in law with BM(−√
2λ) killed at the first hitting time of a. To prove that{XT−t −XT: uT −HI}given XT −IT =bis identical in law with BM(−√
2λ) killed at the first hitting time of−bobserve first that
E
n−(dε)E0
F2(εu: uT )1{T <ζ}(ε)
=λ
∞
0
dte−λtn−
F2(εu: ut )1{t <ζ}(ε) .
Next we claim that
∞
0
dte−λtn−
F2(εu: ut )1{t <ζ}(ε)
=2
∞
0
dbEb
e−λH0F2(XH0−u: uH0)
. (3.16)
Indeed, (3.16) forλ=0 is formula 5 in Biane and Yor [3] Théorème 6.1 p. 79 and the validity forλ >0 is easily verified by inspecting the proof in [3] p. 79. Hence, by spatial symmetry,
E0
F2(XHI+u−IT: uT −HI)
=√ 2λ
∞
0
dbEb
e−λH0F2(XH0−u: uH0)
=√ 2λ
∞
0
dbE0
e−λH−bF2(b+XH−b−u: uH−b)
. (3.17)
Reversing here time and using absolute continuity yield
E0
F2(XT −XT−u: uT−HI)
=√ 2λ
∞
0
dbE0
e−λHbF2(Xs: sHb)
=√ 2λ
∞
0
dbE
√2λ 0
F2(Xs: sHb) e−b
√2λ
which proves the first claim of the theorem.
It remains to verify claim 2 point (3). For a fixedt >0 we obtain from (3.17) Δˆ:=E0
F2(XHI+u−IT: ut )1{tT−HI}
=√ 2λ
∞
0
dbEb
e−λH0F2(XH0−u: ut )1{tH0} .
According to Williams’ time reversal theorem the process {XH0−u: 0u < H0}underPb is identical in law with BES(3) started from 0 and killed at the last exit time atb. Consequently, lettingγbdenote the last exit time we have
Δˆ=√ 2λ
∞
0
dbQ0
e−λγbF2(Xu: ut )1{tγb}
=√ 2λe−λt
∞
0
dbQ0
F2(Xu: ut )QXt
e−λγb1{γb>0}
