Biomedical Engineering Reference
In-Depth Information
1
N
Q [ s N (
∑
k ∈ H ( s , t )
d
)=
(
)( ·×{
} )=
ε X k ( t ) ∈M (R
) ,
t
Q N
t
s
(31)
: C k
where H
(
s
,
t
)= {
k
∈{
1
,...,
N
(
t
) }
(
t
)=
s
}.
Again from Ito's formula, we obtain
that the time evolution of the measure Q N (
t
)
,forany t
∈ [
0
,
T
]
and for any B
∈
d
B (R
)
is the following
3
c = 1
3
c = 1
f
(
x
,
c
)
Q N (
t
)(
d x
,
c
)=
f
(
x
,
c
)
Q N (
0
)(
d x
,
c
)
B
B
2
c = 1 σ
c
2 Δ x f
t
Q [ c ]
N
+
(
x
,
c
)
(
s
)(
d x
)
0
B
Q [ 2 ]
N
+
B β [ ∇
x g
(
x
) − ∇
x u
(
x
)] · ∇
x f
(
x
,
2
)
(
s
)(
d x
)
2
c = 1
j
Q [ c ]
N
+
B (
f
(
x
,
j
) −
f
(
x
,
c
))
m cj (
x
,
c
)
(
s
)(
d x
)
=
c
d s
2
c = 1
Q [ c ]
N
+
h
(
x
,
c
)
f
(
x
,
c
)
(
s
)(
d x
)
B
+
M N [
Q N ,
W
](
t
) ,
(32)
where
N B ( s )
k = 1 σ c ∇ x f ( X k
2
c = 1
t
1
N
d W k
M N [
Q N ,
W
](
t
)=
(
s
) ,
c
)
(
s
) δ c , C k
(
s
)
0
N B (
d s
N B (
)
k = 1
s
2
c = 1 h ( X k
t
1
N
X k
C k
+
f
(
(
s
) ,
(
s
))
d s
) −
(
s
) ,
c
) δ c , C k ( s ) ]
0
f
N B (
)
k = 1
s
t
1
N
X k
C k
+
(
(
s
) ,
(
s
))
0
d s
− ∑
j
X k
X k
C k
X k
C k
C k [
f
(
(
s
) ,
j
) −
f
(
(
s
) ,
(
s
))
m C k j (
(
t
) ,
(
t
))
.
(33)
=
{F
} t ∈ R + generated
It is a zero mean martingale with respect to the natural filtration
t
X k
C k
{ (
(
) ,
(
)) ,
(
) } t ∈ R + . Equation ( 32 ) is coupled with the system
by the process
t
t
N
t
of PDEs
)+ α g Q [ 2 ]
K N
∂
g
(
x
,
t
)
= −
d g g
(
x
,
t
)+
D g Δ
g
(
x
,
t
∗
(
x
)
;
(34)
N
∂
t
 
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