2436 CHAPTER 71. STOCHASTIC O.D.E. ONE SPACE

Then

P

(sup

s∈[0,T ]|uτn

n (s)|2H >

(32

)n)≤ E (C (u0,w0, f ))

(23

)n

Now an application of the Borel Cantelli lemma shows that there exists a set of measurezero N̂ such that for ω /∈ N̂, it follows that for all n large enough,

sups∈[0,T ]

|uτnn (s)|2H < (3/2)n

and so τn = ∞ for all n large enough.Claim: For m< n, there is a set of measure zero Nmn such that if ω /∈Nmn, then uτn

n (s)=uτm

m (s) on [0,T ∧ τm].Proof of the claim: Note that τm ≤ τn. Therefore, these are both progressively mea-

surable solutions to the integral equation

u(t ∧ τm,ω)−u0(ω)+∫ t

0X[0,τm]N (s,u(s,ω),u(s−h,ω) ,wu (s,ω) ,ω)ds

=∫ t

0X[0,τm] f (s,ω)ds+

∫ t

0X[0,τm]σ (s,u,ω)dW. (71.3.24)

wherewu (t) = w0 +

∫ t

0u(s)ds.

To save notation, refer to these functions as u,v and let τm = τ . Subtract and use the Itoformula to obtain

12|u(t ∧ τ)− v(t ∧ τ)|2H ≤

∫ t

0X[0,τm] (N (s,u(s),u(s−h) ,wu (s))−N (s,v(s),v(s−h) ,wu−v (s)) ,u− v)ds

+ sups∈[0,t]

|M (t)|

where the quadratic variation of the martingale M (t) is dominated by∫ t

0X[0,τ] ∥σ (s,u,ω)−σ (s,v,ω)∥2 |u− v|2 ds

Then from the assumption that N is locally Lipschitz and routine manipulations,

12|u(t ∧ τ)− v(t ∧ τ)|2H ≤Cm

∫ t

0X[0,τ] |u− v|2 ds+ sup

s∈[0,t]|M (s)|

and so, adjusting the constants yields

sups∈[0,t]

|u(s∧ τ)− v(s∧ τ)|2H

≤ Cm

∫ t

0X[0,τ] sup

r∈[0,s]|u(r∧ τ)− v(r∧ τ)|2 ds+ sup

s∈[0,t]|M (s)|