CUTTING BROWNIAN PATHS

9

Remark. We will need the following property, which is analogous to the strong

Markov property, but which can be applied at random times which are not stopping

times. We call it the pseudo-strong Markov property. The justification is based

on the repeated application of Proposition 2.1. Given L\ and Mi, the processes

{Zt,0 t U\} and {Z(Ui + t) — Mi,t 0} are independent and the second

one is a Bes(3) starting from L\ — M\. The processes {Zt,0 t Ti} and

{Z(T\ +1) — M\,t 0} are independent and the second one is a Bes(3) starting

from 0. By induction, if Li and Mi are given, then the processes {Zt,0 t Ui}

and {Z(Ui +t) — Mi,t0} are independent and the second one is a Bes(3) starting

from Li - M{. The processes {Zt,0 t Ti} and {Z(T{ + t) - M{,t 0} are

independent and the second one is a Bes(3) starting from 0.

We have the following facts about the distribution of the Mi and Li. These

are straightforward applications of Propositions 2.1 and 2.2.

Proposition 2.3. (a) Mi is uniformly distributed on [0,Li].

(b) Given Mi and Li, the distribution of Mi+i is uniform on [Mi, Li].

(c) log[(Z^—M*)/(Mi+i — Mi)] is an exponential random variable with parameter

1.

(d)

P

L

i ( L ,

+

i A | L , , M

m

) =

L

; y

+ 1

.

(e) log[(Li4-i — Mi+i)/(Li — Mi+i)] is an exponential random variable with pa-

rameter 1.

Proof, (a) This is immediate by Proposition 2.1(a).

(b) By Proposition 2.1(c) and the pseudo-strong Markov property,

P(Mi+i m | Li, Mi) = P(M

i +

i-Mi m-Mi \ Li, Mi) = F(Zt ever hits m-Mi),

where Zt is a Bes(3) starting from Li — Mi. As in (a), the distribution of the

minimum of Zt is uniform on [0, Li — Mi].

(c) If U is uniform on [0,1], then — log U is an exponential random variable

with parameter 1. This and (b) show that that the conditional distribution of

\og[(Li — Mi)/(Mj+i — Mi)] is exponential with parameter 1. Since the conditional

distribution does not depend on Li and Mi, we obtain (c).

(d) By the pseudo-strong Markov property, Zt = Ztjr\ji — Mi is a Bes(3)

started at Li — Mi. By Proposition 2.1(b), the law of Zt for t Ti+i — Ui, given

that the minimum of Zt is M

i +

i, is that of a Brownian motion killed on hitting

Mi+i. Therefore by Proposition 2.2(a) and translation invariance,

P ^ ( sup ZtX\Li,Mi+l) = L'-^i+l.

(e) If P(V A') = £/y, then log(V/^) is an exponential random variable

with parameter 1. We condition on the values of Li and Mi+i and apply (d) and the