How do investors' expectations drive asset prices?

Theorem 1 Assume that the information process I_t is governed by the sto-
chastic differential equation

I — I₀ + I^tI_βσ¹_sdWt ,   0 ≤ t ≤ T,

J₀

where the volatility process σ¹_t is given by

^σt — σ¹₀ + [ b(s,I_s,σ¹_s) ds + ∕* σ^y (s,I_a,σ¹_a) dW^ ,   0 ≤ t ≤ T,

J₀                 J₀

with deterministic smooth functions b and σ^v. Then the forward price F_t of
the asset admits the following representation under the probability measure
P

Γ^τ                     Γ^τ

-    λ<¹ >(s,I_β,σf)Z^ds -    λ^∖s,I_sσ¹))Z^ds

Jt                          Jt

r^τ            r^τ

- Z^dW¹_s - Z^dW_s^v,

Jt               Jt

0 ≤ t ≤ T,

with

^{z — Z} Zff ) ^— ( ^iσ σ(t°ι_t,σ_i) ) v^ti>Λ), ⁰≤^t≤^t■

and

yu(t,x) — (u_xfft,x"),u_x2(t,x))^r,

where u : [0,T] × R² → R is the solution of the partial differential equation

0 — u_t(t,x₁,x₂) - ʌɑ⁾ (t,xι,x₂) X1X2¾ ₁(t,xι,x₂)       (10)

-A⁽²( (t,x₁,x₂) σ^y (t,x_i,x₂) u_x2(t,x_i,x₂)

+ 2 (x?x2 ^ux±x±(t,x₁,x₂ ) + (σ^v (t,x₁,x₂ ))²

^ux2x2(t,x₁,x₂ ɔ

+b(t,xι,x₂) u_xffffx₁,x₂)

u(T,xι,x₂ ) — x₁

for 0 ≤ t ≤ T, with indices on the function u indicating partial derivatives.
Moreover F_t is given by

F_t — u(t,I_t,σ¹_t ) ,   0 ≤ t ≤ T.

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