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词条 Doob decomposition theorem
释义

  1. Statement of the theorem

     Corollary  Remark 

  2. Proofs

     Proof of the theorem  Existence  Uniqueness  Proof of the corollary 

  3. Example

  4. Application

  5. Generalization

  6. Citations

  7. References

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 of the theorem

Let {{math|(Ω, {{mathcal|F}}, ℙ)}} be a probability space, {{math|I {{=}} {0, 1, 2, . . . , N}}} with {{math|N ∈ ℕ}} or {{math|I {{=}} ℕ0}} a finite or an infinite index set, {{math|({{mathcal|F}}n)n∈I}} a filtration of {{mathcal|F}}, and {{math|X {{=}} (Xn)n∈I}} an adapted stochastic process with {{math|E[{{!}}Xn{{!}}] < ∞}} for all {{math|n ∈ I}}. Then there exists a martingale {{math|M {{=}} (Mn)n∈I}} and an integrable predictable process {{math|A {{=}} (An)n∈I}} starting with {{math|A0 {{=}} 0}} such that {{math|Xn {{=}} Mn + An}} for every {{math|n ∈ I}}.

Here predictable means that {{math|An}} is {{math|{{mathcal|F}}n−1}}-measurable for every {{math|n ∈ I \\ {0}}}.

This decomposition is almost surely unique.[2][3][4]

Corollary

A real-valued stochastic process {{math|X}} is a submartingale if and only if it has a Doob decomposition into a martingale {{math|M}} and an integrable predictable process {{math|A}} that is almost surely increasing.[5] It is a supermartingale, if and only if {{math|A}} is almost surely decreasing.

Remark

The theorem is valid word by word also for stochastic processes {{math|X}} taking values in the {{math|d}}-dimensional Euclidean space {{math|ℝd}} or the complex vector space {{math|ℂd}}. This follows from the one-dimensional version by considering the components individually.

Proofs

Proof of the theorem

Existence

Using conditional expectations, define the processes {{math|A}} and {{math|M}}, for every {{math|n ∈ I}}, explicitly by

{{NumBlk|:||{{EquationRef|1}}}}

and

{{NumBlk|:||{{EquationRef|2}}}}

where the sums for {{math|n {{=}} 0}} are empty and defined as zero. Here {{math|A}} adds up the expected increments of {{math|X}}, and {{math|M}} adds up the surprises, i.e., the part of every {{math|Xk}} that is not known one time step before.

Due to these definitions, {{math|An+1}} (if {{math|n + 1 ∈ I}}) and {{math|Mn}} are {{math|{{mathcal|F}}n}}-measurable because the process {{math|X}} is adapted, {{math|E[{{!}}An{{!}}] < ∞}} and {{math|E[{{!}}Mn{{!}}] < ∞}} because the process {{math|X}} is integrable, and the decomposition {{math|Xn {{=}} Mn + An}} is valid for every {{math|n ∈ I}}. The martingale property

    a.s.

also follows from the above definition ({{EquationNote|2}}), for every {{math|n ∈ I \\ {0}}}.

Uniqueness

To prove uniqueness, let {{math|X {{=}} M{{'}} + A{{'}}}} be an additional decomposition. Then the process {{math|Y :{{=}} M − M{{'}} {{=}} A{{'}} − A}} is a martingale, implying that

    a.s.,

and also predictable, implying that

    a.s.

for any {{math|n ∈ I \\ {0}}}. Since {{math|Y0 {{=}} A{{'}}0 − A0 {{=}} 0}} by the convention about the starting point of the predictable processes, this implies iteratively that {{math|Yn {{=}} 0}} almost surely for all {{math|n ∈ I}}, hence the decomposition is almost surely unique.

Proof of the corollary

If {{math|X}} is a submartingale, then

    a.s.

for all {{math|k ∈ I \\ {0}}}, which is equivalent to saying that every term in definition ({{EquationNote|1}}) of {{math|A}} is almost surely positive, hence {{math|A}} is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let {{math|X {{=}} (Xn)n∈ℕ0}} be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. {{math|{{mathcal|F}}n {{=}} σ(X0, . . . , Xn)}} for all {{math|n ∈ ℕ0}}. By ({{EquationNote|1}}) and ({{EquationNote|2}}), the Doob decomposition is given by

and

If the random variables of the original sequence {{math|X}} have mean zero, this simplifies to

    and    

hence both processes are (possibly time-inhomogenious) random walks. If the sequence {{math|X {{=}} (Xn)n∈ℕ0}} consists of symmetric random variables taking the values {{math|+1}} and {{math|−1}}, then {{math|X}} is bounded, but the martingale {{math|M}} and the predictable process {{math|A}} are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale {{math|M}} 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 {{math|X {{=}} (X0, X1, . . . , XN)}} denote the non-negative, discounted payoffs of an American option in a {{math|N}}-period financial market model, adapted to a filtration {{math|

({{mathcal|F}}0, {{mathcal|F}}1, . . . , {{mathcal|F}}N)}}, and let {{math|ℚ}} denote an equivalent martingale measure. Let {{math|U {{=}} (U0, U1, . . . , UN)}} denote the Snell envelope of {{math|X}} with respect to {{math|ℚ}}. The Snell envelope is the smallest {{math|ℚ}}-supermartingale dominating {{math|X}}[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 {{math|U {{=}} M + A}} denote the Doob decomposition with respect to {{math|ℚ}} of the Snell envelope {{math|U}} into a martingale {{math|M {{=}} (M0, M1, . . . , MN)}} and a decreasing predictable process {{math|A {{=}} (A0, A1, . . . , AN)}} with {{math|A0 {{=}} 0}}. Then the largest stopping time to exercise the American option in an optimal way[10][11] is

Since {{math|A}} is predictable, the event {{math|{τmax {{=}} n} {{=}} {An {{=}} 0, An+1 < 0}}} is in {{math|{{mathcal|F}}n}} for every {{math|n ∈ {0, 1, . . . , N − 1}}}, hence {{math|τmax}} 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 {{math|τmax}} the discounted value process {{math|U}} is a martingale with respect to {{math|ℚ}}.

Generalization

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

Citations

1. ^{{harvtxt|Doob|1953}}, see {{harv|Doob|1990|pp=296−298}}
2. ^{{harvtxt|Durrett|2005}}
3. ^{{harv|Föllmer|Schied|2011|loc=Proposition 6.1}}
4. ^{{harv|Williams|1991|loc=Section 12.11, part (a) of the Theorem}}
5. ^{{harv|Williams|1991|loc=Section 12.11, part (b) of the Theorem}}
6. ^{{harv|Lamberton|Lapeyre|2008|loc=Chapter 2: Optimal stopping problem and American options}}
7. ^{{harv|Föllmer|Schied|2011|loc=Chapter 6: American contingent claims}}
8. ^{{harv|Föllmer|Schied|2011|loc=Proposition 6.10}}
9. ^{{harv|Föllmer|Schied|2011|loc=Theorem 6.11}}
10. ^{{harv|Lamberton|Lapeyre|2008|loc=Proposition 2.3.2}}
11. ^{{harv|Föllmer|Schied|2011|loc=Theorem 6.21}}
12. ^{{harv|Schilling|2005|loc=Problem 23.11}}

References

  • {{Citation | last=Doob | first=Joseph L. | author-link=Joseph L. Doob | year=1953 | title=Stochastic Processes | publisher=Wiley | place=New York | isbn=978-0-471-21813-5 | mr=0058896 | zbl=0053.26802 }}
  • {{Citation | last=Doob | first=Joseph L. | author-link=Joseph L. Doob | year=1990 | title=Stochastic Processes | publisher=John Wiley & Sons, Inc. | place=New York | edition= Wiley Classics Library | isbn=0-471-52369-0 | mr=1038526 | zbl=0696.60003 }}
  • {{Citation | last=Durrett | first=Rick | year=2010 | title=Probability: Theory and Examples | publisher=Cambridge University Press | edition=4. | series=Cambridge Series in Statistical and Probabilistic Mathematics | isbn=978-0-521-76539-8 | mr=2722836 | zbl=1202.60001 }}
  • {{Citation | last=Föllmer | first=Hans | last2=Schied | first2=Alexander | title=Stochastic Finance: An Introduction in Discrete Time | edition=3. rev. and extend | place=Berlin, New York | publisher=De Gruyter | year=2011 | series=De Gruyter graduate | isbn=978-3-11-021804-6 | mr=2779313 | zbl=1213.91006 }}
  • {{Citation | last=Lamberton | first=Damien | last2=Lapeyre | first2=Bernard | title=Introduction to Stochastic Calculus Applied to Finance | edition=2. | place=Boca Raton, FL | publisher=Chapman & Hall/CRC | year=2008 | series=Chapman & Hall/CRC financial mathematics series | isbn=978-1-58488-626-6 | mr=2362458 | zbl=1167.60001 }}
  • {{Citation | last=Schilling | first=René L. | title=Measures, Integrals and Martingales | place=Cambridge | publisher=Cambridge University Press | year=2005 | isbn=978-0-52185-015-5 | mr=2200059 | zbl=1084.28001 }}
  • {{Citation| author-link=David Williams (mathematician) | first=David | last=Williams | title=Probability with Martingales | publisher= Cambridge University Press | year=1991 | isbn =0-521-40605-6 | mr=1155402 | zbl=0722.60001 }}
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3 : Theorems regarding stochastic processes|Martingale theory|Articles containing proofs
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