Proof. We have the following system:
the forward stochastic differential equation (FSDE) for the information
process It and its volatility process σ{
( 11 - ( х‘ц ) - ( 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