How do investors' expectations drive asset prices?



Proof. We have the following system:

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

( 11 - ( хц ) - ( X»'’ ) + /7    0    1 *

(H)


U - ( ʌ ' - (   )41 --,^ )

[t ( X<1>X<2>         0

+ √o    0     σy(s,Xβw,X<2})

4-----------------v---------------

σ

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

fτ /                                                    

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

-[(S)Ws "<t«T.


(12)


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) — στ(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    z2Jy'  'y' !   σv (t,x1,x2 ) W2 )

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

u(t, x)


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 ʌu U u^

u,xix2 ) (s,x)l
¾2^2 /        I

0 « t « T,


2   [ 0 σy(s,x) J у uxsxι

+(( 6(θχ) 1, ( :)) ds,

15



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