A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

( w( t)) t ∈[0
,T ] is a continuous process with values in R , and w(0)
= 0.
For every s < t, w( t) w( s)
is a Gaussian real valued random variable with mean 0 and
variance t s.
Moreover, ∀s < t, ∀f
Ϝ -measurable
n
L 2(0 ,T ; L 2(
,H))
E[( w( t) w( s)) f] = 0 and E[( w( t) w( s)) ] = t s.
About the Itô integration :
2 2
t
∀f ∈ L (0 , T ; L ( , H)) , ( f( s) dw( s))
is a continuous Ϝ -measurable process and
If f ⇀ f then
and
s
t0 t
t t
2
2
E[ f( s) dw( s)] = E[ f( s) ds ] .
T
T
f ( s) dw( s) ⇀ f( s) dw( s)
n
L 2
0 ( , ,P,H)
0
. .
f ( s) dw( s) ⇀ f( s) dw( s ) .
0
L 2(
, ,P, C([0
,T ]))
0
The Fubini Theorem :
2 2
Let f ∈ L (0 , T ; L ( , H)) and h ∈ H with | f| h then
t t
f( s, ., x) dw( s) dx = f( s, ., x) dx dw( s) P a.s.
The Itô formula :
2 2
Let f, g, M ∈ L (0 , T ; L ( , H)) such that ∀t ∈ [0 , T ] ,
t t
M( t) = M + g( s, ., . ) ds + f( s, ., . ) dw( s)
then, for any enough regular ϕ : R → R :
n
0
0
0 0
t
ϕ( t, M( t)) = ϕ (0 , M ) + ∂ ϕ( s, M( s)) g( s, ., . ) ds+
t t
∂ ϕ( s, M( s)) g( s, ., . ) ds + ∂ ϕ( s, M( s)) f( s, ., . ) dw( s)+
0
2
t
1 2
∂2, 2ϕ(
s, M( s)) f( s, ., . ) ds.
2 0
So E ϕ( t, M( t)) dx = E ϕ (0 , M ) dx +
0
0
t
E ∂ ϕ( s, M( s)) g( s, ., . ) ds dx +
0
t
E ∂ ϕ( s, M( s)) g( s, ., . ) ds dx +
0
t
1
2
E ∂2, 2ϕ(
s, M( s)) f( s, ., . ) ds dx .
2
0
0
0 0
1
2
2
0
4
0
0
1
2
2