How do investors' expectations drive asset prices?

Proof. We have the following system:

the forward stochastic differential equation (FSDE) for the information
process I_t and its volatility process σ{

( 11 - ( ^х‘^ц ) - ( ^X»'’ ) + /7    ⁰    1 *

U - ( ʌ ' - (   )41 --,^ )

[^t ( X<¹>X<²>         0

⁺ √o ∖    0     σ^y(s,X_β^w,X<^2})

⁴-----------------_v---------------

σ

and the backward stochastic differential equation (BSDE) for the forward
price F of the asset

f^τ /                                                    ∖

^γ_t - X⅜ - J (λ<¹ * (s, X  X<²>) X<¹> + λ<²> (s, X<¹>,X<²>) Z<²<) ds

The coupled system (11) and (12) is a forward-backward stochastic differen-
tial equation (FBSDE). We use the Four-Step-Scheme given in Ma, Protter,
Yong [23], more precisely we apply the version given in Ma, Yong [24], to
find the solution.

Step 1 : We define the function z(t, x, y, w) — σ^τ(t, x, y)w for (t, x, y, w) ∈
R × R² × R × R². With this definition we have the 2-dimensional function

z ʌ   Z Zl ʌ (₊ x   ( DX2 W₁ ʌ

z(t,X,y,W) —         (t,X,y,W) —     ₁< _λ .

^v Z  '^y' Z    ∖z₂J^y'  '^y' ^!   ∖σ^v (t,x₁,x₂ ) W₂ )

Step 2 : With the function z we solve the partial differential equation
(PDE)

Xi + J -ʌɑ( (s,X1,X₂) Zi (s,x,u (s,x), vu(s,x))
-λ^²) (s, Xi, X₂) Z₂ (s, X, u (s, x), vu(s, x))
₊1 _tr∖ ( xx ⁰ ʌ^u U ^u^

2   [∖ 0 σ^y(s,x) J у u_xsxι

+(( ₆(θχ) 1^, ( :)) ds^,

15

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