2438 CHAPTER 71. STOCHASTIC O.D.E. ONE SPACE

and for such a fixed ω /∈ N, there is no change in increasing m. Hence, u is progressivelymeasurable and satisfies the integral equation 71.3.21.

It remains to verify uniqueness. Suppose there are two solutions u,v each progressivelymeasurable solutions of the given integral equation. Then let τn be a stopping time

τn = inf{t : |u(t)|+ |v(t)|> 2n}

Then a repeat of the arguments given in the above claim shows that on [0,τn∧T ] the twofunctions uτn ,vτn are equal on [0,τn∧T ] off a set of measure zero Nn. Let N be the unionof the exceptional sets. Then for ω /∈ N, u(t,ω) = v(t,ω) for all t ∈ [0,τn∧T ]. However,τn (ω) = ∞ for all n large enough because each of these functions is continuous. Hence,the two functions are equal on [0,T ] for such ω . This shows uniqueness.