Degenerate parabolic stochastic partial differential equations

4334 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336
n−1
+ φi W (ti+1 ) − W (t i )
, g j M(ti+1 ) − M(t i ), g j
i=m
+ φ n
W (t) − W (tn ) , g j
M(t) − M(tn ), g j
F s
(A.4)
as one can neglect all the mixed terms due to the martingale property of β k , k ≥ 1, and (A.1).
Indeed, let i ∈ {m, . . . , n − 1} then
φi
(W (t i+1 ) − W (t i )), g j M(tm ) − M(s), g j |Fs
E
= E
E
k≥1
φ
k
i
βk (t i+1 ) − β k (t i ) , g j
M(tm ) − M(s), g j
F ti
F s
M(tm
= E ) − M(s), g j ⟨φi k , g j⟩ E β k (t i+1 ) − β k (t i )|F ti F s = 0,
k≥1
where the interchange of summation with scalar product and expectation, respectively, is justified
by the fact that
φi
k
k≥1
βk (t i+1 ) − β k (t i ) =
ti+1
t i
φ i dW
is convergent in L 2 (Ω; H).
As the next step, we proceed with (A.4). If i ∈ {m, . . . , n − 1} then we obtain using again the
martingale property of β k , k ≥ 1, and (A.1)
φi
E W (ti+1 ) − W (t i )
, g j M(ti+1 ) − M(t i ), g j |Fs
= E
= E
= E
E
k≥1
φ
k
i , g j
βk (t i+1 ) − β k (t i ) M(t i+1 ) − M(t i ), g j
F ti
F s
⟨φi k , g j⟩ E β k (t i+1 )
M(t i+1 ), g j − βk (t i )
M(t i ), g j |Fti F s
k≥1
⟨φi k , g j⟩
k≥1
ti+1
ti+1
= E ⟨ f k , φi ∗ g j⟩ U
k≥1
t i
ti+1
= E ⟨σ ∗ g j , φ ∗ g j ⟩ U dr
t i
t i
fk , σ ∗ g j
U dr
F s
fk , σ ∗ g j
U dr
F s
F s
.
The remaining terms being dealt with similarly. As a consequence, we see that (A.3) holds true
for simple processes and the general case follows by classical arguments using approximation.