Kenneth Kuttler

65.14. A TECHNICAL INTEGRATION BY PARTS RESULT 2267

because then X[0,τn]R((

Z ◦ J−1)∗(X))◦ J would end up being integrable in the right way

and you could define the stochastic integral provided τn > t whenever n is large enough.However, this is problematic because t → X (t) is not known to be continuous. Therefore,some other condition must be assumed.

Lemma 65.14.2 Suppose t → X (t) is weakly continuous into H for a.e.ω, and that X isadapted. Then the τn described above is a stopping time.

Proof: Let B≡ {x ∈ H : |x|> n} . Then the complement of B is a closed convex set. Itfollows that BC is also weakly closed. Hence B must be weakly open. Now t → X (t) isadapted as a function mapping into the topological space consisting of H with the weaktopology because it is in fact adapted into the strong topolgy. Therefore, the above τn isjust the first hitting time of an open set by a continuous process so τn is a stopping time byProposition 62.7.5. Also, by the assumption that t→ X (t) is weakly continuous, it followsthat X (t) for t ∈ [0,T ] is weakly bounded. Hence, for each ω off a set of measure zero,|X (t)| is bounded for t ∈ [0,T ] . This follows from the uniform boundedness theorem. Itfollows that τn = ∞ for n large enough.

Hence the weak continuity of t → X (t) suffices to define the stochastic integral in65.14.22. It remains to verify some sort of convergence in the case that

limn→∞

[max

j≤mn−1

(tn

j+1− tnj)]

= 0

Lemma 65.14.3 Let X (s)−X lk (s)≡ ∆k (s) . Here Z ∈ L2

([0,T ]×Ω,L2

(Q1/2U,H

))and

let X ∈ L2 ([0,T ]×Ω,H) with both X and Z progressively measurable, t → X (t) beingweakly continuous into H,

limk→∞

∥∥∥X−X lk

∥∥∥L2([0,T ]×Ω,H)

= 0

Then the integral ∫ t

0R((

Z ◦ J−1)∗ (X))◦ JdW

exists as a local martingale and the following limit occurs for a suitable subsequence, stillcalled k.

limk→∞

([sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗∆k (s)

)◦ JdW (s)

∣∣∣∣≥ ε

])= 0. (65.14.23)

That is,

supt∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗(X (s)−X lk (s)

))◦ JdW (s)

∣∣∣∣converges to 0 in probability.

Proof: Let k denote a subsequence for which X lk also converges pointwise to X .

65.14. A TECHNICAL INTEGRATION BY PARTS RESULT 2267because then 2%,.,.% ((ZoJ~!)" (X)) oJ would end up being integrable in the right wayand you could define the stochastic integral provided Tt, >t whenever n is large enough.However, this is problematic because t — X (t) is not known to be continuous. Therefore,some other condition must be assumed.Lemma 65.14.2 Suppose t — X (t) is weakly continuous into H for a.e.@, and that X isadapted. Then the T, described above is a stopping time.Proof: Let B = {x € H: |x| >}. Then the complement of B is a closed convex set. Itfollows that BC is also weakly closed. Hence B must be weakly open. Now t — X (ft) isadapted as a function mapping into the topological space consisting of H with the weaktopology because it is in fact adapted into the strong topolgy. Therefore, the above T,, isjust the first hitting time of an open set by a continuous process so T,, is a stopping time byProposition 62.7.5. Also, by the assumption that t + X (t) is weakly continuous, it followsthat X (t) for t € [0,7] is weakly bounded. Hence, for each @ off a set of measure zero,|X (t)| is bounded for t € [0,7]. This follows from the uniform boundedness theorem. Itfollows that T, = co forn large enough. ffHence the weak continuity of t + X (t) suffices to define the stochastic integral in65.14.22. It remains to verify some sort of convergence in the case thatlim | max (t,,—1¢7)| =0tim | max, (6 i)Lemma 65.14.3 Let X (s) — X} (s) = Ag (s). Here Z € L? ((0,T] x Q,-4 (Q!/2U,H)) andlet X € L* ((0,T] x Q,H) with both X and Z progressively measurable, t + X (t) beingweakly continuous into H,lim |x —x!L2([0,7]xQ,H)k—y00Then the integraltR((Zos!)* (X)) osJdw[ a(@er')@) 2exists as a local martingale and the following limit occurs for a suitable subsequence, stillcalled k.jin ( ae, [B((Zo)o-)'au(s)) osaw (5) >) =0. (65.14.23)That is,ep, [ 8 (Ze)eF") (X) XL) esaw(s)converges to 0 in probability.Proof: Let k denote a subsequence for which Xx} also converges pointwise to X.