63.7. DOOB MEYER DECOMPOSITION 2155
and by right continuity, it follows
limp→∞
np
∑k=1
ξ(t pk
)X(t p
k−1,tpk ](s) = ξ (s)
and so the dominated convergence theorem applies and it follows
limp→∞
np
∑k=1
ξ(t pk
)(A(t pk
)−A
(t pk−1
))=∫(0,t]
ξ (s)dA(s)
where this is a random variable. Thus
E(∫
(0,t]ξ (s)dA(s)
)=∫
Ω
(limp→∞
∫(0,t]
np
∑k=1
ξ(t pk
)X(t p
k−1,tpk ](s)dA(s)
)dP (63.7.32)
Now as mentioned above,∫(0,t]
np
∑k=1
ξ(t pk
)X(t p
k−1,tpk ](s)dA(s) =
np
∑k=1
ξ(t pk
)(A(t pk
)−A
(t pk−1
))and since A is increasing, this is bounded above by an expression of the form CA(t) , afunction in L1. Therefore, by the dominated convergence theorem, 63.7.32 reduces to
limp→∞
∫Ω
∫(0,t]
np
∑k=1
ξ(t pk
)X(t p
k−1,tpk ](s)dA(s)dP
= limp→∞
∫Ω
np
∑k=1
ξ(t pk
)(A(t pk
)−A
(t pk−1
))dP
= limp→∞
∫Ω
(np
∑k=1
ξ(t pk
)A(t pk
)−
np−1
∑k=0
ξ(t pk+1
)A(t pk
))dP
= limp→∞
np−1
∑k=1
∫Ω
(ξ(t pk
)−ξ
(t pk+1
))A(t pk
)dP+
∫Ω
ξ (t)A(t)dP. (63.7.33)
Since ξ is a martingale,∫Ω
ξ(t pk+1
)A(t pk
)dP =
∫Ω
E(
ξ(t pk+1
)A(t pk
)|Ft p
k
)dP
=∫
Ω
A(t pk
)E(
ξ(t pk+1
)|Ft p
k
)dP
=∫
Ω
A(t pk
)ξ(t pk
)dP
and so in 63.7.33 the term with the sum equals 0 and it reduces to
E (ξ (t)A(t)) .
This is sufficiently interesting to state as a lemma.