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
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