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
More intriguing information
1. Unilateral Actions the Case of International Environmental Problems2. Should informal sector be subsidised?
3. Literary criticism as such can perhaps be called the art of rereading.
4. The Interest Rate-Exchange Rate Link in the Mexican Float
5. Stillbirth in a Tertiary Care Referral Hospital in North Bengal - A Review of Causes, Risk Factors and Prevention Strategies
6. The name is absent
7. Short report "About a rare cause of primary hyperparathyroidism"
8. How do investors' expectations drive asset prices?
9. Regulation of the Electricity Industry in Bolivia: Its Impact on Access to the Poor, Prices and Quality
10. News Not Noise: Socially Aware Information Filtering