In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob.[1]
The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.
Let
(\Omega,l{F},P)
N\in\N
I=\N0
(l{F}n)n
l{F}
l{F}n-1
\Realsd
\Complexd
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, (if) and are -measurable because the process is adapted, and because the process is integrable, and the decomposition is valid for every . The martingale property
E[Mn-Mn-1|l{F}n-1]=0
also follows from the above definition, for every .
To prove uniqueness, let be an additional decomposition. Then the process is a martingale, implying that
E[Yn|l{F}n-1]=Yn-1
and also predictable, implying that
E[Yn|l{F}n-1]=Yn
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.
A real-valued stochastic process is a submartingale if 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.
If is a submartingale, then
E[Xk|l{F}k-1]\geXk-1
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.
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
An=\sum
n | |
k=1 |
l(E[Xk]-Xk-1r), n\inN0,
and
Mn=X0+\sum
n | |
k=1 |
l(Xk-E[Xk]r), n\inN0.
If the random variables of the original sequence have mean zero, this simplifies to
An=-\sum
n-1 | |
k=0 |
Xk
Mn=\sum
n | |
k=0 |
Xk, n\inN0,
hence both processes are (possibly time-inhomogeneous) 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 not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale unless the stopping time has a finite expectation.
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
Q
Q
\taumax:=\begin{cases}N&ifAN=0,\\min\{n\in\{0,...,N-1\}\midAn+1<0\}&ifAN<0.\end{cases}
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
Q
The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.