1 Continuous Time Processes - IPM

that the Brownian motion has at least one 0 in the interval (t1, t2) can be
written as
P =
1 ∞
a2
−
√ P [Ta < t1 − t◦]e 2t◦σ
2πtσ −∞
2 da. (1.5.12)
Let us explain the validity of this assertion. At time t◦, Zt◦ can be at any
a2
1 −
point a ∈ R. The Gaussian exponential factor √ e 2t◦σ
2πtσ 2 is the density
function for Zt◦. The factor P [Ta < t1 − t◦] is equal to the probability
that starting at a, the Brownian motion will assume value 0 in the ensuing
time interval (t1 − t◦). The validity of the assertion follows from these facts
put together. In view of the symmetry between a and −a, and the density
function for Ta which we obtained in example 1.5.3, (1.5.12) becomes
∞
a2
2 P = √
2πt◦σ ◦ e− 2t◦σ √2πσ a ( t1−t◦
a2
− 1
◦ e 2u
u √ udu)da 1 = πσ2√ t1−t◦
3
−
t◦ ◦ u 2 ( a2 1 1
∞
◦ ae− 2σ2 ( + u t◦ ) da)du
= √ t◦ t1−t◦ du
π ◦ (u+t◦) √ u
The substitution √ u = x yields
P = 2
π tan−1
t1 − t◦
t◦
= 2
π cos−1
t1
t◦
, (1.5.13)
for the probability of at least one crossing of 0 betwen times t◦ and t1. It
follows that the probability of no crossing in the time interval (t◦, t1) is
2
π sin−1
t1
which is the arc-sine law for Brownian motion. ♠
So far we have only considered Brownian motion in dimension one. By
looking at m copies of independent Brownian motions Zt = (Zt;1, · · · , Zt;m)
we obtain Brownian motion in R m . While m-dimensional Brownian motion
is defined in terms of coordinates, there is no preference for any direction in
space. To see this more clearly, let A = (Ajk) be an m×m orthogonal matrix.
This means that AA ′ = I where superscript ′ denotes the transposition of
the matrix, and geometrically it means that the linear transformation A
preserves lengths and therefore necessarily angles too. Let
Yt;j =
t◦
m
AjkZt;k.
k=1
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