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Unformatted text preview: n
1X
Pp
Xm x ! P ( x)
(6.25)
n m=1
where has a standard normal distribution. That is,
Zx
2
1
p e y /2 dy
P ( x) =
2⇡
1 The conclusion in (6.25) is often written as
n
1X
p
Xm )
n m=1 where ) is read “converges in distribution to.” Recalling that if we multiply
a standard normal by a constant c then the result has a normal distribution
with mean 0 and variance 2 , we see that
p n
p
1X
t· p
Xm ) t
n m=1 and the limit is a normal with mean 0 and variance t.
This motivates the following deﬁnition: 199 6.6. BLACKSCHOLES FORMULA Deﬁnition. B (t) is a standard Brownian motion if B (0) = 0 and it satisﬁes
the following conditions:
(a) Independent increments. Whenever 0 = t0 < t1 < . . . < tk
B (t1 ) B (t0 ), . . . , B (tk ) B (tk 1) are independent. (b) Stationary increments. The distribution of Bt Bs is normal (0, t s). (c) t ! Bt is continuous.
To explain (a) note that if ni = ti /h then the sums ni X Xm i = 1, . . . k 1 <mni are independent. For (b) we note that that the distribution of the sum only
depends on the number of ter...
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 Spring '10
 DURRETT
 The Land

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