2. Decomposition of a Bessel process.
After reviewing the first version of our paper, Jim Pitman pointed out to us
that ideas similar to the path decomposition developed in this section have already
appeared in literature; see, e.g., Neveu and Pitman (1989) or Imhof (1992).
We need some detailed estimates, which we obtain by means of a path de-
composition of a Bes(3), Zt. We will repeatedly use Williams' path decompositions
(Williams (1979), 111.79). See also Millar (1978) for an alternate proof.
Proposition 2.1. Let Zt be a Bes(3) started at L 0.
(a) If M inf{Zt : t 0}, then M is uniformly distributed on [0, L].
(b) Let T = inf{£ : Zt = M}. Given {M = m), the law of {Zut T} is the
same as the law of a Brownian motion started at L and run until the first
time it hits m.
(c) Given {M = m}, the law of {Zt+T, t 0} is the same as the law of Zt + m,
where Zt is a Bes(3) started at 0.
(d) The processes {Zt,t T} and {Zt+r Zr,t 0} are independent.
Proposition 2.1(a) can be rephrased as saying
PLBES(Zt hits M) = ¥-. (2.1)
A related path decomposition is the following.
Proposition 2.2. Let Xt be a Brownian motion started at L 0 and killed on
hitting 0.
(a) IfV = sup{Xt : t 0}, then FL(L' A) = L/X.
(b) Let S = inf{t : Xt = L'}. Given {V = a}, the law of {Xut S} is the
same as the law of a Bes(3) started at L and run until it hits a.
(c) Given {V = a], the law of {X
t
+s,£ 0} is the same as the law of a Zt,
where Zt is a Bes(3) started at 0 and run until it hits a.
(d) The processes {Xt,t S} and {Xt+s Xs,t 0} are independent.
Proof. The proof is very similar to that of Proposition 2.1. We follow Millar's
approach. To see (a),
P L (L' A) - P L (X
t
hits A before hitting 0) - L/X.
If Mt = s u p
s t
X
t
, then the pair (Xt,Mt) is a strong Markov process, and S
sup{£ : Xt = Mt}. For any strong Markov process Yt, the law of Yt up to the last
time U it is in a set A and given {Yu = y} is the same as Yt conditioned to hit y,
while the law of Yt after U is the same as for Yt starting from Yu and conditioned
never to return to A] moreover, given {Yu = y}, the processes {Ytlt U} and
Yt+u,t 0} are independent (see Meyer, Smythe, and Walsh (1972)). (b), (c), and
(d) follow easily from this fact with Yt = (Xt, Mt) and A = {(x,m) : x = m}.
7
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