Definition 61.4.1 Let X be a stochastic process defined on an interval, I with values in
a separable Banach space, E. It is called integrable if E
< ∞ for each t ∈ I.
Also let ℱt be a filtration. An integrable and adapted stochastic process X is called a
martingale if for s ≤ t
Recalling the definition of conditional expectation this says that for F ∈ℱs
for all F ∈ℱs. A real valued stochastic process is called a submartingale if whenever
s ≤ t,
and a supermartingale if
Example 61.4.2 Let ℱt be a filtration and let Z be in L1
. Then let
This works because for s < t, E
Proposition 61.4.3 The following statements hold for a stochastic process defined
Ω having values in a real separable Banach space, E.
- If X
is a martingale then
is a submartingale.
- If g is an increasing convex function from [0,∞) to [0,∞) and
< ∞ for all t ∈
then then g
Proof:Let s ≤ t
Now by Theorem 60.1.1
on Page 6683
is a submartingale as claimed.
Consider the second claim. Recall Jensen’s inequality for submartingales, Theorem
59.1.6 on Page 6554. From the first part
and so from Jensen’s inequality,
showing that g
is also a submartingale. This proves the proposition.