where the filtration is the one generated by the information process I. Be-
cause of the equality of Fτ and It the following relationship holds.
Ft = Ep[Fτ ∣σ {IuI 0 ≤ и ≤ t}] = Ep[Iτ ∣σ {Iu∣ 0 ≤ и ≤ t}] , 0 ≤ t ≤ T.
We assume that the transformation from P to P is given by a Girsanov-
functional. More precisely we assume that there is an adapted R2-valued
process λs = (λ[,1∖λ[,2))z which defines the martingale (∣∣ ∙ ∣∣ is the euclidean
R2-norm)
Φo,t
exp (- i ‘ λ<1W∕ - ∕' ‘ λ<2>dWj’ - Ц‘ ∣∣λ.B2 ds∖ , (9)
0 ≤ t ≤ T,
and the transformed probability measure
4--'
P(A) = E[Φ0,rU] , A ∈ Ft.
With this definition P and P are mutually absolutely continuous on Fτ and
the process
W wt' \._( Wl + J0' λ<*>ds ∖
(, )■=! w'+ʃo' f>λA ,
is a 2-dimensional Brownian motion under P. Hence, we have the represen-
tation for F under the probability measure P
Ft = it -
[ λ<1>ZVds - [ λ^Z^ds - [ ZV>dW18 - [ Z<2>dWj
Jt Jt Jt Jt
for 0 ≤ t ≤ T, where Z = (Z^1∖ Z2ψ is the process given by application of
the martingale representation theorem on F. We assume that λ is a smooth
deterministic function that may depend on t, It and σ{: λ (t, It, σ1t ).
In the following theorem we give a formula for the forward price Ft in
terms of the information process and the market price of risk λ (t, It, σ1t ).
For the derivation we have to solve a forward-backward stochastic differential
equation. This is done by application of mathematical theorems given in Ma,
Protter, Yong [23] and Ma, Yong [24].
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