In this post I’ll give a proof of Rao’s decomposition for quasimartingales. That is, every quasimartingale decomposes as the sum of a submartingale and a supermartingale. Equivalently, every quasimartingale is a difference of two submartingales, or alternatively, of two supermartingales. This was originally proven by Rao (Quasi-martingales, 1969), and is an important result in the general theory of continuous-time stochastic processes.
As always, we work with respect to a filtered probability space . It is not required that the filtration satisfies either of the usual conditions — the filtration need not be complete or right-continuous. The methods used in this post are elementary, requiring only basic measure theory along with the definitions and first properties of martingales, submartingales and supermartingales. Other than referring to the definitions of quasimartingales and mean variation given in the previous post, there is no dependency on any of the general theory of semimartingales, nor on stochastic integration other than for elementary integrands.
Recall that, for an adapted integrable process X, the mean variation on an interval is
where the supremum is taken over all elementary processes with . Then, X is a quasimartingale if and only if is finite for all positive reals t. It was shown that all supermartingales are quasimartingales with mean variation given by
Rao’s decomposition can be stated in several different ways, depending on what conditions are required to be satisfied by the quasimartingale X. As the definition of quasimartingales does differ between texts, there are different versions of Rao’s theorem around although, up to martingale terms, they are equivalent. In this post, I’ll give three different statements with increasingly stronger conditions for X. First, the following statement applies to all quasimartingales as defined in these notes. Theorem 1 can be compared to the Jordan decomposition, which says that any function with finite variation on bounded intervals can be decomposed as the difference of increasing functions or, equivalently, of decreasing functions. Replacing finite variation functions by quasimartingales and decreasing functions by supermartingales gives the following.
Theorem 1 (Rao) A process X is a quasimartingale if and only if it decomposes as
for supermartingales Y and Z. Furthermore,
- this decomposition can be done in a minimal sense, so that if is any other such decomposition then is a supermartingale.
- the inequality
holds, with equality for all if and only if the decomposition is minimal.
- the minimal decomposition is unique up to a martingale. That is, if are two such minimal decompositions, then is a martingale.