64.13 Taking Out A Linear Transformation
It is assumed L ∈ℒ
is another separable real Hilbert space. First of all,
here is a lemma which shows ∫
at least makes sense.
Proposition 64.13.1 Suppose Φ is ℒ2
Then the same is true of LΦ. Furthermore, for each t ∈
Proof: First note that if Φ ∈ℒ2
and that the
Φ is continuous. It follows L
Φ is ℒ2
progressively measurable. All
that remains is to check the appropriate integral.
and so this proves LΦ satisfies the same conditions as Φ, being stochastically square
It follows one can consider
Assume to begin with that Φ ∈ L2
Next recall the
situation in which the definition of the integral is considered.
be an approximating sequence of elementary functions satisfying
it is also the case that
By the definition of the integral, for each t
The second equality is obvious for elementary functions.
Now consider the case where Φ is only stochastically square integrable so that all is
known is that
Then define τn as above
This sequence of stopping times works for LΦ also. Recall there were two conditions the
sequence of stopping times needed to satisfy. The first is obvious. Here is why the second
Then let t
be given and pick n
such that τn
Then from the first part, for that ω,