The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 418 Wednesday, February 4, 2004 1:08 PM

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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