Kenneth Kuttler

33.2. THE STOCHASTIC INTEGRAL WHEN f (s) ∈L2 (U,H) 897

Proof: It only remains to verify 33.23. The first equal sign is obvious from the defini-tion of

∫ t0 f dM. Both sides equal

mn−1

∑r=0

fr (M (t ∧σ ∧ tr+1)−M (t ∧σ ∧ tr))

Now consider 33.24. From 33.23, the left side is

(sup

t∈[0,T ]

∥∥∥∥∫ t

0f dMσ

∥∥∥∥2)≤

∫Ω

∫ T

0∥ f∥2

L2d [Mσ ]dP

=∫

Ω

∫ T

0∥ f∥2

L2d [M]σ dP

=∫

Ω

∫ T

0X[0,σ ] ∥ f∥2

L2d [M]dP =

∫Ω

∫ T

∥∥X[0,σ ] f∥∥2

L2d [M]dP

This is because when t > σ , [M]σ (t) = [M] (σ), a constant. Thus the contribution to theconventional integral is 0 from then on. On the other hand, [M]σ (t) = [M] (t) for t ≤ σ andso the last equation follows. ■

Now here is a nice proposition which is a summary of what has just been discussedalong with some other observations.

Proposition 33.2.6 For ∥ f (t,ω)∥ bounded by K and continuous in t for each ω havingvalues in L2 (U,H) , and for M a continuous bounded martingale with values in U, t →∫ t

0 f dM is a continuous martingale with values in H. Assume that [M] (T ) ∈ L1 (Ω). Alsothe fundamental inequality

(sup

t∈[0,T ]

∥∥∥∥∫ t

0f dM

∥∥∥∥2)≤∫

Ω

∫ T

0∥ f∥2

L2d [M]dP (33.25)

is valid for f . In addition, f →∫ t

0 f dM is linear on the linear space of bounded continuousin t adapted functions. If σ is a stopping time,∫ t∧σ

0f dM =

∫ t

0f dMσ (33.26)

Proof: Since the integral is a limit of integrals of elementary functions for ω off a setof measure zero and since this integral is linear on these functions, the integral is linear.Finally, consider the claim 33.26. From Proposition 31.7.2, and letting { fn} be elemen-tary functions approximating f as above, then by that proposition again, there is a furthersubsequence still denoted with n such that for a.e. ω,∫ t

0f dMσ = lim

n→∞

∫ t

0fndMσ = lim

n→∞

∫ t∧σ

0fndM =

∫ t∧σ

0f dM ■

Note how this does not require f to be of bounded variation. If f were of bounded vari-ation, you would get pointwise convergence of the Stieltjes sums for

∫ t0 fndM to

∫ t0 f dM

as a consequence of simple considerations involving Stieltjes integrals. Then the new in-formation is that the Stieltjes integral

∫ t0 f dM for f of bounded variation is a continuous

martingale.

33.2. THE STOCHASTIC INTEGRAL WHEN f (s) € %2(U,H) 897Proof: It only remains to verify 33.23. The first equal sign is obvious from the defini-tion of {9 fdM. Both sides equalmn—1Vf (M(tAoAt+1)-M (tAoAt,))Now consider 33.24. From 33.23, the left side is1 ° Pe pl2=E| sup < Lf fll d[M°]dP(0 | feaT2 oOd|M|" dP[ff Witatr 2=[ [ %oailsleammlar = [| %ooif\h,,almlarThis is because when t > o, [M]° (t) = [M](o), a constant. Thus the contribution to theconventional integral is 0 from then on. On the other hand, [M]° (t) = [M] (rt) for t < o andso the last equation follows.Now here is a nice proposition which is a summary of what has just been discussedalong with some other observations.[ ' fdM?Proposition 33.2.6 For || f (t,@)|| bounded by K and continuous int for each @ havingvalues in %y(U,H), and for M a continuous bounded martingale with values in U, t >Jo. fdM is a continuous martingale with values in H. Assume that (M]|(T) € L' (Q). Alsothe fundamental inequalityPi a2< [ [ \i/ll2,4[M\aP (33.25)supte[0,7]is valid for f. In addition, f > fo FfdM is linear on the linear space of bounded continuousin t adapted functions. If o is a stopping time,[ O° tdM = [ ' faM® (33.26)Proof: Since the integral is a limit of integrals of elementary functions for @ off a setof measure zero and since this integral is linear on these functions, the integral is linear.Finally, consider the claim 33.26. From Proposition 31.7.2, and letting {f,} be elemen-tary functions approximating f as above, then by that proposition again, there is a furthersubsequence still denoted with n such that for a.e. @,t\o t\o[ fam? = lim | judM? = =im/[ f,dm=[ faveneo JO 0Note how this does not require f to be of bounded variation. If f were of bounded vari-ation, you would get pointwise convergence of the Stieltjes sums for fo FtndM to fo fdMas a consequence of simple considerations involving Stieltjes integrals. Then the new in-formation is that the Stieltjes integral fp fdM for f of bounded variation is a continuousmartingale.