2428 CHAPTER 71. STOCHASTIC O.D.E. ONE SPACE

where u(t) ∈ H a Hilbert space. Consider F (u) = 12 |u|

2 . Also let R denote the Riesz mapfrom H → H ′ such that ⟨Rx,y⟩ ≡ (x,y)H . Then proceding formally, to see what the Itoformula says,

dF = DF (u)du+12

D2F (u)(du,du)+O(du3)

Recall then that du =−Ndt + f dt +ΦdW and so recalling (dW,dW ) = dt,

R(u)(−Ndt + f dt +ΦdW )+12∥Φ∥2 dt

Hence

12|u(t)|2H −

12|u0|2H +

∫ t

0(N,u)H ds− 1

2

∫ t

0∥Φ∥2

L2ds =

∫ t

0( f ,u)ds+

∫ t

0Ru(Φ)dW

The last term is a martingale or local martingale M whose quadratic variation is given by

[M] (t) =∫ t

0∥Φ∥2

L2|u|2 ds

This is all that is of importance in what follows. Therefore, this martingale may be simplydenoted as M (t) in what follows.

Under the assumption 71.2.5 you can include instead of the term∫ t

0 ΦdW, the moregeneral term

∫ t0 σ (s,u,ω)dW . This will be shown by doing the argument and indicating

what extra assumptions are needed as this is done. Let z be progressively measurable andin L2 (Ω;C ([0,T ] ;Rn)). Also assume that σ has linear growth. That is

∥σ (s,u,ω)∥L2≤ a+b |u|Rn (71.2.6)

Then from the above theorem, there exists a unique progressively measurable solution u to

u(t,ω)−u0(ω)+∫ t

0N(s,u(s,ω),u(s−h,ω) ,w(s,ω) ,ω)ds

=∫ t

0f(s,ω)ds+

∫ t

0σ (s,z)dW, (71.2.7)

This holds for all ω . There is no exceptional set needed. Now assume

u0 ∈ L2 (Ω) (71.2.8)

and also a Lipschitz condition

∥σ (s,u,ω)−σ (s, û,ω)∥L2≤ K |u− û| (71.2.9)

Then let u coincide with z and û come from ẑ. Then applying the Ito formula, one canobtain the following for a constant C which does not depend on u, û.

12|u(t)− û(t)|2−C

∫ t

0|u(s)− û(s)|2 ds−K

∫ t

0|u(s)−û(s)|2 ds = M (t)