1 Continuous Time Processes - IPM

From (??) and (1.5.6), the desired relation (1.5.5) and the semi-group property
follow. An important feature of continuous time Markov chains is that
Pt satisfied the the Kolmogorov forward and backward equations. In view of
the semi-group property the same is true for Brownian motion and we will
explain in example 1.5.2 below what the infinitesimal generator of Brownian
motion is. With some of the fundamental definitions of Brownian motion in
place we now calculate some quantities of interest.
Example 1.5.1 Since the random variables Zs and Zt are dependent, it is
reasonable to calculate their covaraince. Assume s < t, then we have
Cov(Zs, Zt) = Cov(Zs, Zt − Zs + Zs)
(By independence of increments) = Cov(Zs, Zs)
= sσ2 .
This may appear counter-intuitive at first sight since one expects Zs and Zt
to become more independent as t − s increases while the covariance depends
only on min(s, t) = s. However, if we divide Cov(Zs, Zt) by Var[Zs]Var[Zt]
we see that the correlation tends to 0 as t increases for fixed s. ♠
Example 1.5.2 One of the essential features of continuous time Markov
chains was the existence of the infinitesimal generator. In this example we
derive a formula for the infinitesimal generator of Brownian motion. For a
function ψ on the state space R, the action of the semi-group Pt is given by
(1.5.4). We set uψ(t, x) = u(t, x) = Ptψ. The Gaussian pt(x; σ) has variance
tσ 2 and therefore it tends to the δ-function supported at the origin as t → 0,
i.e.,
It is straightforward to verify that
lim(Ptψ)(x)
= lim u(t, x) = ψ(x).
t→0 t→0
∂u
∂t
= σ2
2
∂2u . (1.5.7)
∂x2 Therefore from the validity of Kolomogorov’s backward equation ( dPt = APt)
dt
we conclude that the infinitesimal generator of Brownian motion is given by
A = σ2
2
d2 . (1.5.8)
dx2 Thus the matrix A is now replaced by a differential operator. ♠
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