71.3. STOCHASTIC DIFFERENTIAL EQUATIONS 2437

and so, by Gronwall’s inequality followed by the Burkholder-Davis-Gundy inequality,

E

(sup

s∈[0,t]|u(s∧ τ)− v(s∧ τ)|2H

)≤

CE

((∫ t

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

)1/2)

≤ 12

E

(sup

s∈[0,t]|u(s∧ τ)− v(s∧ τ)|2H

)+CE

(∫ t

0X[0,τ] ∥σ (s,u)−σ (s,v)∥2 ds

)and so, adjusting the constant again,

E

(sup

s∈[0,t]|u(s∧ τ)− v(s∧ τ)|2H

)

≤ CE(∫ t

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

)

≤CE(∫ t

0X[0,τ]K |u(s∧ τ)− v(s∧ τ)|2 ds

)

≤C∫ t

0E

(sup

r∈[0,s]|u(r∧ τ)− v(r∧ τ)|2

)ds

and so, Gronwall’s inequality shows that for every t,

E

(sup

s∈[0,t]|u(s∧ τ)− v(s∧ τ)|2H

)= 0

In particular, for t = T this holds. Hence

E

(sup

s∈[0,T ]|u(s∧ τ)− v(s∧ τ)|2H

)= 0

It follows that

E

(sup

s∈[0,τ∧T ]|u(s)− v(s)|2H

)= 0

so that off a set of measure zero, u(s) = v(s) for all s ∈ [0,τ]. This proves the claim.Now let the set of measure zero N be given by N ≡ ∪m<nNmn ∪ N̂ where N̂ is the set

of measure zero off which τm = ∞ for all m large enough. Then for ω /∈ N, it follows thatuτn

n (s) = uτmm (s) on [0,τm∧T ] and, for all m large enough, τm = ∞. Hence for all m large

enough, and such ω, un (s,ω) = um (s,ω) for all s∈ [0,T ] . Thus, for ω off N, it follows thatlimm→∞ uτm

m (s,ω)≡ u(s,ω) exists, this for each s ∈ [0,T ] and ω off a fixed set of measurezero. In fact, this convergence is uniform on [0,T ] because for all n sufficiently large