Degenerate parabolic stochastic partial differential equations

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4333
where σ is an (F t )-progressively measurable L 2 (U; H)-valued stochastically integrable
process, i.e.
T
E ∥σ ∥ 2 L 2 (U;H) dr < ∞,
(A.2)
0
then
M(t) =
t
0
σ dW,
∀t ∈ D, P-a.s.
In particular, M can be defined for all t ∈ [0, T ] such that it has a modification which is a
continuous (F t )-martingale.
Proof. The crucial point to be shown here is the following: for any (F t )-progressively
measurable L 2 (U; H)-valued process φ satisfying (A.2) and any s, t ∈ D, s ≤ t, j ≥ 1, it
holds, P-a.s.,
M(t) t
t
E − M(s), g j φ dW, g j − ⟨σ ∗ g j , φ ∗ g j ⟩ U dr
F s = 0. (A.3)
s
We consider simple processes first. Let φ be an (F t )-adapted simple process with values in
finite-dimensional operators of L(U; H) that satisfies (A.2), i.e.
φ(t) = φ 0 1 {0} (t) +
I
φ i 1 (ti ,t i+1 ](t), t ∈ [0, T ],
i=0
where {0 = t 0 < t 1 < · · · < t I = T } is a division of [0, T ] such that t i ∈ D, i = 0, . . . , I . Then
the stochastic integral in (A.3) is given by
t
s
s
φ dW = φ m−1 W (tm ) − W (s) n−1
+ φ i W (ti+1 ) − W (t i )
+ φ n W (t) − W (tn )
i=m
=
φm−1 k βk (t m ) − β k (s) n−1
+
k≥1
i=m
+ φn
k
βk (t) − β k (t n )
φ k i
βk (t i+1 ) − β k (t i )
provided t m−1 ≤ s < t m , t n ≤ t < t n+1 , φi k = φ i f k . Next, we write
M(t) − M(s) = M(t m ) − M(s) n−1
+ M(ti+1 ) − M(t i ) + M(t) − M(t n )
i=m
and conclude
M(t) t
E − M(s), g j φ dW, g j F s
s
φm−1
= E W (tm ) − W (s)
, g j M(tm ) − M(s), g j