A few days ago, I wrote a post on some decomposition theorems for discrete-time martingales. Today, I want to talk a little bit more about decomposition theorems. In particular, I want to show that the Krickeberg decomposition theorem is not (almost surely) unique, unlike the Doob decomposition. I also want to cover another result known as the Riesz decomposition, which makes use of the celebrated martingale convergence theorem.
In what follows, all random variables are taken to defined on a complete probability space and all filtrations are assumed to be complete.
As we will need the definition in the statement of the Riesz decomposition theorem, we first introduce the stochastic idea of potential. I have not studied enough probability theory and PDE to say for sure, but I imagine that the term “potential” is motivated by the connections between (sub-,super-)martingales and (sub-,super-)harmonic functions.
Definition 1. We say that a nonnegative supermartingale is a potential if as .
Recall that a supermartingale has the defining property , so by the tower property of conditional expectation, the sequence is decreasing. In the case of a potential, we have that .
To see that the Krickeberg decomposition is not (a.s.) unique, we need the martingale convergence theorem, which tells us that any nonnegative martingale converges almost surely to an random variable as . I claim that is a nonnegative martingale. Nonnegativity is immediate from a.s., and for any ,
Applying the Krickeberg decomposition, we can write
where are nonnegative martingales. Since the sum of nonnegative martingales is again a nonnegative martingale, we see that
is another Krickeberg decomposition for .
We’re now ready to give the proof of the Riesz decomposition, which is similar to that of the Krickeberg decomposition.
Theorem 2. Any supermartingale satisfying can be uniquely written
where is a martingale and is a potential.
Proof. We first prove uniqueness. Suppose are two Riesz decompositions. Then
The monotone convergence theorem implies that
since are potentials. Hence, for all .
We now prove the existence of the Riesz decomposition. Since is a submartingale, we have for ,
By induction, we see that
By the monotone convergence theorem, is a -measurable random variable (in the extended sense). Since by hypothesis, we by the monotone convergence theorem that is a.s. finite and
To see that is a martingale, observe that by the conditional monotone convergence theorem,
We complete the proof by showing that is a potential. It is evident from above that and
for all . That is a supermartingale is immediate from the martingale property of and supermartingale property of .