108156.pdf

3. Risky Assets 69

We shall use the Central Limit Theorem^2 to obtain the limit asN→∞of
the random walkwN(t). To this end we put

x(n)=

k(n)−mτ

σ

√

τ

for eachn=1, 2 ,..., which is a sequence of independent identically distributed
random variables, each with expectation 0 and variance 1. The Central Limit
Theorem implies that

x(1) +x(2) +···+x(n)

√

n

→X

in distribution asn→∞,whereXis a random variable with standard normal
distribution (mean 0 and variance 1).
Let us fix anyt>0. Because the random walkwNis only defined at discrete
times being whole multiples of the stepτ=N^1 ,weconsiderwN(tN), wheretN
is the whole multiple ofN^1 nearest tot. Then, clearly,NtNis a whole number
for eachN, and we can write

wN(tN)=

√

tN

x(1) +x(2) +···+x(NtN)

√

NtN

.

AsN→∞,wehavetN→tandNtN→∞,sothat

wN(tN)→W(t)

in distribution, whereW(t)=

√

tX. The last equality means thatW(t)is
normally distributed with mean 0 and variancet.
This argument, based on the Central Limit Theorem, works for any single
fixed timet>0. It is possible to extend the result to obtain a limit for all
timest≥ 0 simultaneously, but this is beyond the scope of this book. The limit
W(t) is called theWiener process(orBrownian motion). It inherits many of
the properties of the random walk, for example:

1.W(0) = 0, which corresponds towN(0) = 0.

2.E(W(t)) = 0, corresponding toE(wN(t)) = 0 (see the solution of Exer-

cise 3.25).

3. Var(W(t)) =t, with the discrete counterpart Var(wN(t)) =t(see the solu-
   tion of Exercise 3.25).


4. The incrementsW(t 3 )−W(t 2 )andW(t 2 )−W(t 1 ) are independent for 0≤
   t 1 ≤t 2 ≤t 3 ; so are the incrementswN(t 3 )−wN(t 2 )andwN(t 2 )−wN(t 1 ).

(^2) See, for example, Capi ́nski and Zastawniak (2001).