3.5 Markov Property 107When the maximum step size llllll = maxj=O, ... ,m-l(tj+l -tj) is small,
then the first term on the right-hand side of (3.4.15) is approximately equal to
its limit, which is a^2 times the amount of quadratic variation accumulated by
Brownian motion on the interval [T 1 1 T 2 ], which is T 2 - T 1. The second term
on the right-hand side of (3.4.15) is (a-!a^2 )
2
times the quadratic variation
of t , which was shown in Remark 3.4.5 to be zero. The third term on the
right-hand side of (3.4.1'5) is 2a (a-!a^2 ) times the cross variation of W(t )
and t, which was also shown in Remark 3.4.5 to be zero. We conclude that
when the maximum step size llllll is small, the right-hand side of (3.4.15) is
approximately equal to a^2 (T 2 - Tl), and hence
(3.4.16)If the asset price S(t) really is a geometric Brownian motion with constant
volatility a, then a can be identified from price observations by computing the
left-hand side of (3.4.16) and taking the square root. In theory, we can make
this approximation as accurate as we like by decreasing the step size. In prac
tice, there is a limit to how small the step size can be. Between trades, there
is no information about prices, and when a trade takes place, it is sometimes
at the bid price and sometimes at the ask price. On small time intervals, the
difference in prices due to the bid-ask spread can be as large as the difference
due to price fluctuations during the time interval.
3.5 Markov Property
In this section, we show that Brownian motion is a Markov process and discuss
its transition density.
Theorem 3.5.1. Let W(t), t � 0, be a Brownian motion and let :F(t), t � 0,
be a filtration for this Brownian motion (see Definition 3.3.3). Then W(t),
t � 0, is a Markov process.
PROOF: According to Definition 2.3.6, we must show that whenever 0 � s �
t and f is a Borel-measurable function, there is another Borel-measurable
function g such that
lE[f(W(t))j:F( s)] = g(W(s)). (3.5.1)
To do this, we write
lE[f(W(t)) j:F(s)] = lE[/((W(t)-W(s)) + W(s)) j:F(s)]. (3.5.2)
The random variable W(t) -W(s) is independent of :F(s), and the random
variable W(s) is :F(s)-measurable. This permits us to apply the Independence