Theorem 1 Assume that the information process It is governed by the sto-
chastic differential equation
I — I0 + ItIβσ1sdWt , 0 ≤ t ≤ T,
J0
where the volatility process σ1t is given by
σt — σ10 + [ b(s,Is,σ1s) ds + ∕* σy (s,Ia,σ1a) dW^ , 0 ≤ t ≤ T,
J0 J0
with deterministic smooth functions b and σv. Then the forward price Ft of
the asset admits the following representation under the probability measure
P
Γτ Γτ
Ft — it
- λ<1 >(s,Iβ,σf)Z^ds - λ^∖s,Isσ1))Z^ds
Jt Jt
rτ rτ
- Z^dW1s - Z^dWsv,
Jt Jt
0 ≤ t ≤ T,
with
z — Z Zff ) — ( iσ σ(t°ιt,σi) ) vti>Λ), 0≤t≤t■
and
yu(t,x) — (uxfft,x"),ux2(t,x))r,
where u : [0,T] × R2 → R is the solution of the partial differential equation
0 — ut(t,x1,x2) - ʌɑ) (t,xι,x2) X1X2¾ 1(t,xι,x2) (10)
-A(2( (t,x1,x2) σy (t,xi,x2) ux2(t,xi,x2)
+ 2 (x?x2 ux±x±(t,x1,x2 ) + (σv (t,x1,x2 ))2
ux2x2(t,x1,x2 ɔ
+b(t,xι,x2) uxffffx1,x2)
u(T,xι,x2 ) — x1
for 0 ≤ t ≤ T, with indices on the function u indicating partial derivatives.
Moreover Ft is given by
Ft — u(t,It,σ1t ) , 0 ≤ t ≤ T.
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