14-Arbitrage Page 418 Wednesday, February 4, 2004 1:08 PM
418
1
The Mathematics of Financial Modeling and Investment Management
1 pω
1 = ---------- ∑ P({}ω Akt)πs()ωRts, = ---------- ∑ -----------------π ()ωRts
π π ω ∈A
kt
PAkt)
s
(
,
Akt ω ∈Akt Akt
1 ≤k ≤Mt
Substituting in the previous equation, we obtain, for each interval (t,T),
πA
kt
R 0 ,t
ξAkt = (Et[]ξT)A = ---------------------
kt π
A 10
which we can rewrite in the usual notation as
πtR 0 ,t
ξt = Et[]ξT = ----------------
π 10
We can now state the following result. Consider any ℑj-measurable
variable xj. This condition can be expressed equivalently stating that xj
assumes constant values on each set of the partition Ij. Then the follow-
ing relationship holds:
E P^1
t xj
Q[]= E
t ----[ξjxj]
ξt
To see this, consider the following demonstration, which hinges on the
fact that xj assumes a constant value on each Ahj and, therefore, can be
taken out of sums. In addition, as demonstrated above, from
1 = -----^1 E
t[πsRts, ]
πt
the following relationship holds:
PA( kt)πA ω ,
kt
= ∑ pωπ ()s Rts
ω ∈Akt
1 ≤k ≤Mt