Chapter 71

Stochastic O.D.E. One Space71.1 Adapted Solutions With Uniqueness

Instead of a single σ algebra F , one can generalize to the case of a normal filtrationFt and obtain adapted solutions to finite dimensional theorems, provided one also knowspath uniqueness of the solutions. Recall that a filtration is normal includes the followingcondition which is what we will use.

Ft = ∩s>tFs (71.1.1)

Theorem 71.1.1 Suppose N(t,u,v,w,ω) ∈ Rd for u,v,w ∈ Rd , t ∈ [0,T ] and

(t,u,v,w,ω)→ N(t,u,v,w,ω)

is progressively measurable with respect to a normal filtration or more generally one whichsatisfies 71.1.1. Also suppose (t,u,v,w)→N(t,u,v,w,ω) is continuous. Suppose for eachω, there exists an estimate for any solution u(·,ω) to the integral equation

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, (71.1.2)

which is of the formsup

t∈[0,T ]|u(t,ω)| ≤C (ω)< ∞

Also let f be progressively measurable and f(·,ω) ∈ L1([0,T ] ;Rd

). Here u0 has values in

Rd and is F0 measurable and u(s−h,ω)≡ u0 (ω) whenever s−h≤ 0 and

w(t,ω)≡ w0 (ω)+∫ t

0u(s,ω)ds

where w0 is a given F0 measurable function. Also assume that for each ω there is at mostone solution to the integral equation 71.1.2. Then for h > 0, there exists a progressivelymeasurable solution u to the integral equation 71.1.2.

Proof: Let 0 = t0 < t1 < · · ·< tn = T . From Theorem 70.3.3, there exists a solution tothe integral equation u which has the property that u(t ∧ t j) is Ft j measurable. One simplyapplies this theorem to the succession of intervals determined by the given partition. Nowsuppose Pn consists of the points k2−nT ≡ tn

j so that these satisfy Pn ⊆Pn+1 and thelengths of the sub-intervals decreases to 0 with increasing n. Let un denote the solution justdescribed corresponding to Pn such that un

(t ∧ tn

j

)is Ftn

jmeasurable. As before, using

the estimate, these un (·,ω) for a fixed ω are uniformly bounded and equicontinuous. Thisis because it is a solution to the integral equation for each ω and so by assumption, thereis an estimate. Therefore, for fixed ω, there exists u(·,ω) and a subsequence, denotedas un (·,ω) which converges uniformly to u(·,ω) on [0,T ]. Therefore, u(·,ω) will be a

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