Английская Википедия:Doob decomposition theorem

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Шаблон:Short description 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.

Statement

Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a probability space, Шаблон:Math with <math>N \in \N</math> or <math>I = \N_0</math> a finite or countably infinite index set, <math>(\mathcal{F}_n)_{n \in I}</math> a filtration of <math>\mathcal{F}</math>, and Шаблон:Math an adapted stochastic process with Шаблон:Math for all Шаблон:Math. Then there exist a martingale Шаблон:Math and an integrable predictable process Шаблон:Math starting with Шаблон:Math such that Шаблон:Math for every Шаблон:Math. Here predictable means that Шаблон:Math is <math>\mathcal{F}_{n-1}</math>-measurable for every Шаблон:Math. This decomposition is almost surely unique.[2][3][4]

Remark

The theorem is valid word for word also for stochastic processes Шаблон:Math taking values in the Шаблон:Math-dimensional Euclidean space <math>\Reals^d</math> or the complex vector space <math>\Complex^d</math>. This follows from the one-dimensional version by considering the components individually.

Proof

Existence

Using conditional expectations, define the processes Шаблон:Math and Шаблон:Math, for every Шаблон:Math, explicitly by

Шаблон:NumBlk

and

Шаблон:NumBlk

where the sums for Шаблон:Math are empty and defined as zero. Here Шаблон:Math adds up the expected increments of Шаблон:Math, and Шаблон:Math adds up the surprises, i.e., the part of every Шаблон:Math that is not known one time step before. Due to these definitions, Шаблон:Math (if Шаблон:Math) and Шаблон:Math are Шаблон:Math-measurable because the process Шаблон:Math is adapted, Шаблон:Math and Шаблон:Math because the process Шаблон:Math is integrable, and the decomposition Шаблон:Math is valid for every Шаблон:Math. The martingale property

<math>\mathbb{E}[M_n-M_{n-1}\,|\,\mathcal{F}_{n-1}]=0</math>    a.s.

also follows from the above definition (Шаблон:EquationNote), for every Шаблон:Math}.

Uniqueness

To prove uniqueness, let Шаблон:Math be an additional decomposition. Then the process Шаблон:Math is a martingale, implying that

<math>\mathbb{E}[Y_n\,|\,\mathcal{F}_{n-1}]=Y_{n-1}</math>    a.s.,

and also predictable, implying that

<math>\mathbb{E}[Y_n\,|\,\mathcal{F}_{n-1}]= Y_n</math>    a.s.

for any Шаблон:Math}. Since Шаблон:Math by the convention about the starting point of the predictable processes, this implies iteratively that Шаблон:Math almost surely for all Шаблон:Math, hence the decomposition is almost surely unique.

Corollary

A real-valued stochastic process Шаблон:Math is a submartingale if and only if it has a Doob decomposition into a martingale Шаблон:Math and an integrable predictable process Шаблон:Math that is almost surely increasing.[5] It is a supermartingale, if and only if Шаблон:Math is almost surely decreasing.

Proof

If Шаблон:Math is a submartingale, then

<math>\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]\ge X_{k-1}</math>    a.s.

for all Шаблон:Math}, which is equivalent to saying that every term in definition (Шаблон:EquationNote) of Шаблон:Math is almost surely positive, hence Шаблон:Math is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let Шаблон:Math be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. Шаблон:Math for all Шаблон:Math. By (Шаблон:EquationNote) and (Шаблон:EquationNote), the Doob decomposition is given by

<math>A_n=\sum_{k=1}^{n}\bigl(\mathbb{E}[X_k]-X_{k-1}\bigr),\quad n\in\mathbb{N}_0,</math>

and

<math>M_n=X_0+\sum_{k=1}^{n}\bigl(X_k-\mathbb{E}[X_k]\bigr),\quad n\in\mathbb{N}_0.</math>

If the random variables of the original sequence Шаблон:Math have mean zero, this simplifies to

<math>A_n=-\sum_{k=0}^{n-1}X_k</math>    and    <math>M_n=\sum_{k=0}^{n}X_k,\quad n\in\mathbb{N}_0,</math>

hence both processes are (possibly time-inhomogeneous) random walks. If the sequence Шаблон:Math consists of symmetric random variables taking the values Шаблон:Math and Шаблон:Math, then Шаблон:Math is bounded, but the martingale Шаблон:Math and the predictable process Шаблон:Math are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale Шаблон:Math 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.[6][7] Let Шаблон:Math denote the non-negative, discounted payoffs of an American option in a Шаблон:Math-period financial market model, adapted to a filtration Шаблон:Math, and let Шаблон:Math denote an equivalent martingale measure. Let Шаблон:Math denote the Snell envelope of Шаблон:Math with respect to <math>\mathbb{Q}</math>. The Snell envelope is the smallest Шаблон:Math-supermartingale dominating Шаблон:Math[8] and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity.[9] Let Шаблон:Math denote the Doob decomposition with respect to <math>\mathbb{Q}</math> of the Snell envelope Шаблон:Math into a martingale Шаблон:Math and a decreasing predictable process Шаблон:Math with Шаблон:Math. Then the largest stopping time to exercise the American option in an optimal way[10][11] is

<math>\tau_{\text{max}}:=\begin{cases}N&\text{if }A_N=0,\\\min\{n\in\{0,\dots,N-1\}\mid A_{n+1}<0\}&\text{if } A_N<0.\end{cases}</math>

Since Шаблон:Math is predictable, the event Шаблон:Math} is in Шаблон:Math for every Шаблон:Math}, hence Шаблон:Math 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 Шаблон:Math the discounted value process Шаблон:Math is a martingale with respect to <math>\mathbb{Q}</math>.

Generalization

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.[12]

Citations

Шаблон:Reflist

References

Шаблон:Stochastic processes

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  1. ↑ Шаблон:Harvtxt, see Шаблон:Harv
  2. ↑ Шаблон:Harvtxt
  3. ↑ Шаблон:Harv
  4. ↑ Шаблон:Harv
  5. ↑ Шаблон:Harv
  6. ↑ Шаблон:Harv
  7. ↑ Шаблон:Harv
  8. ↑ Шаблон:Harv
  9. ↑ Шаблон:Harv
  10. ↑ Шаблон:Harv
  11. ↑ Шаблон:Harv
  12. ↑ Шаблон:Harv
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Doob_decomposition_theorem&oldid=14362364