How do investors' expectations drive asset prices?

where the filtration is the one generated by the information process I. Be-
cause of the equality of F_τ and It the following relationship holds.

F_t = E^p[F_τ ∣σ {I_uI 0 ≤ и ≤ t}] = E^p[I_τ ∣σ {I_u∣ 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 λ_s = (λ[,¹∖λ[,²))^z which defines the martingale (∣∣ ∙ ∣∣ is the euclidean
R²-norm)

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 F_τ and
the process

W w_t' \._( 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

[ 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, I_t and σ{: λ (t, I_t, σ¹_t ).

In the following theorem we give a formula for the forward price F_t in
terms of the information process and the market price of risk λ (t, I_t, σ¹_t ).
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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