Optimality

164 D. M. Dabrowska
so that in both cases it is enough to consider the determinants for ordered sequences
s = (s1, . . . ,sm), s1 < s2 < . . . < sm of points in (0, τ] m .
For any such sequence s, the matrix dm(s) has a simple pattern:
⎛
c(s1)
⎜ c(s1)
¯dm(s)
⎜
= ⎜ c(s1)
⎜
⎝ .
c(s1)
c(s2)
c(s2)
c(s1)
c(s2)
c(s3)
. . .
. . .
. . .
c(s1)
c(s2)
c(s3)
.
⎞
⎟ .
⎟
⎠
c(s1) c(s2) c(s3) . . . c(sm)
We have ¯ dm(s) = A T mCm(s)Am where Cm(s) is a diagonal matrix of increments
Cm(s) = diag [c(s1)−c(s0), c(s2)−c(s1), . . . c(sm)−c(sm−1)],
(c(s0) = 0, s0 = 0) and Am is an upper triangular matrix
⎛
1
⎜ 0
⎜
Am = ⎜ .
⎜ .
⎝ 0
1
1
0
. . . 1
. . . 1
. . . 1
1
1
.
1
0 0 . . . 0 1
To see this it is enough to note that Brownian motion forms a process with independent
increments, and the kernel k(s, t) = c(s∧t) is the covariance function of a
time transformed Brownian motion.
Apparently, det Am = 1. Therefore
and
det ¯ dm(s) =
⎞
⎟ .
⎟
⎠
m�
[c(sj)−c(sj−1)]
det ¯ Dm(t, u;s) = det ¯ dm(s)[c(t∧u)−Um(t;s)[ ¯ dm(s)] −1 Vm(s;u)]
j=1
= det ¯ dm(s)[c(t∧u)−Um(t;s)A −1
m C −1
m (s)(A T m) −1 Vm(s;u)].
The inverse A−1 m is given by Jordan matrix
⎛
1 −1 0 . . . 0 0
⎜ 0 1 −1 . . . 0 0
A −1
m =
⎜
⎝
and a straightforward multiplication yields
det ¯ Dm(t, u;s) = c(t∧u)
−
×
.
.
.
.
0 0 0 . . . 1 −1
0 0 0 . . . 0 1
m�
[c(sj)−c(sj−1)]
j=1
⎞
⎟
⎠
m�
[c(t∧si)−c(t∧si−1)][c(u∧si)−c(u∧si−1)]
i=1
m�
j=1,j�=i
[c(sj)−c(sj−1)].