Let be a probability space, with or a finite or an infinite index set, a filtration of , and an adapted stochastic process with for all. Then there exists a martingale and an integrable predictable process starting with such that for every. Here predictable means that is -measurable for every. This decomposition is almost surely unique.
Remark
The theorem is valid word by word also for stochastic processes taking values in the -dimensional Euclidean space or the complex vector space. This follows from the one-dimensional version by considering the components individually.
Proof
Existence
Using conditional expectations, define the processes and, for every, explicitly by and where the sums for are empty and defined as zero. Here adds up the expected increments of, and adds up the surprises, i.e., the part of every that is not known one time step before. Due to these definitions, and are -measurable because the process is adapted, and because the process is integrable, and the decomposition is valid for every. The martingale property also follows from the above definition, for every.
Uniqueness
To prove uniqueness, let be an additional decomposition. Then the process is a martingale, implying that and also predictable, implying that for any. Since by the convention about the starting point of the predictable processes, this implies iteratively that almost surely for all, hence the decomposition is almost surely unique.
Corollary
A real-valued stochastic process is a submartingaleif and only if it has a Doob decomposition into a martingale and an integrable predictable process that is almost surely increasing. It is a supermartingale, if and only if is almost surely decreasing.
Proof
If is a submartingale, then for all, which is equivalent to saying that every term in definition of is almost surely positive, hence is almost surely increasing. The equivalence for supermartingales is proved similarly.
Example
Let be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. for all. By and, the Doob decomposition is given by and If the random variables of the original sequence have mean zero, this simplifies to hence both processes are random walks. If the sequence consists of symmetric random variables taking the values and , then is bounded, but the martingale and the predictable process are unbounded simple random walks, and Doob's optional stopping theorem might not be applicable to the martingale unless the stopping time has a finite expectation.
Application
In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let denote the non-negative, discounted payoffs of an American option in a -period financial market model, adapted to a filtration, and let denote an equivalent martingale measure. Let denote the Snell envelope of with respect to . The Snell envelope is the smallest -supermartingale dominating and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. Let denote the Doob decomposition with respect to of the Snell envelope into a martingale and a decreasing predictable process with. Then the largest stopping time to exercise the American option in an optimal way is Since is predictable, the event is in for every, hence is indeed a stopping time. It gives the last moment before the discounted value of the American option will drop in expectation; up to time the discounted value process is a martingale with respect to .
