Doob decomposition theorem

Doob decomposition theorem 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.

Contents 1 Statement 1.1 Remark 2 Proof 2.1 Existence 2.2 Uniqueness 3 Corollary 3.1 Proof 4 Example 5 Application 6 Generalization 7 Citations 8 References Statement Let {displaystyle (Omega ,{mathcal {F}},mathbb {P} )} be a probability space, I = {0, 1, 2, ..., N} with {displaystyle Nin mathbb {N} } or {displaystyle I=mathbb {N} _{0}} a finite or an infinite index set, {displaystyle ({mathcal {F}}_{n})_{nin I}} a filtration of  {displaystyle {mathcal {F}}} , and X = (Xn)n∈I an adapted stochastic process with E[|Xn|] < ∞ for all n ∈ I. Then there exists a martingale M = (Mn)n∈I and an integrable predictable process A = (An)n∈I starting with A0 = 0 such that Xn = Mn + An for every n ∈ I. Here predictable means that An is {displaystyle {mathcal {F}}_{n-1}} -measurable for every n ∈ I {0}. This decomposition is almost surely unique.[2][3][4] Remark The theorem is valid word by word also for stochastic processes X taking values in the d-dimensional Euclidean space {displaystyle mathbb {R} ^{d}} or the complex vector space {displaystyle mathbb {C} ^{d}} . This follows from the one-dimensional version by considering the components individually. Proof Existence Using conditional expectations, define the processes A and M, for every n ∈ I, explicitly by {displaystyle A_{n}=sum _{k=1}^{n}{bigl (}mathbb {E} [X_{k},|,{mathcal {F}}_{k-1}]-X_{k-1}{bigr )}}         (1) and {displaystyle M_{n}=X_{0}+sum _{k=1}^{n}{bigl (}X_{k}-mathbb {E} [X_{k},|,{mathcal {F}}_{k-1}]{bigr )},}         (2) where the sums for n = 0 are empty and defined as zero. Here A adds up the expected increments of X, and M adds up the surprises, i.e., the part of every Xk that is not known one time step before. Due to these definitions, An+1 (if n + 1 ∈ I) and Mn are Fn-measurable because the process X is adapted, E[|An|] < ∞ and E[|Mn|] < ∞ because the process X is integrable, and the decomposition Xn = Mn + An is valid for every n ∈ I. The martingale property {displaystyle mathbb {E} [M_{n}-M_{n-1},|,{mathcal {F}}_{n-1}]=0}     a.s. also follows from the above definition (2), for every n ∈ I {0}. Uniqueness To prove uniqueness, let X = M' + A' be an additional decomposition. Then the process Y := M − M' = A' − A is a martingale, implying that {displaystyle mathbb {E} [Y_{n},|,{mathcal {F}}_{n-1}]=Y_{n-1}}     a.s., and also predictable, implying that {displaystyle mathbb {E} [Y_{n},|,{mathcal {F}}_{n-1}]=Y_{n}}     a.s. for any n ∈ I {0}. Since Y0 = A'0 − A0 = 0 by the convention about the starting point of the predictable processes, this implies iteratively that Yn = 0 almost surely for all n ∈ I, hence the decomposition is almost surely unique. Corollary A real-valued stochastic process X is a submartingale if and only if it has a Doob decomposition into a martingale M and an integrable predictable process A that is almost surely increasing.[5] It is a supermartingale, if and only if A is almost surely decreasing. Proof If X is a submartingale, then {displaystyle mathbb {E} [X_{k},|,{mathcal {F}}_{k-1}]geq X_{k-1}}     a.s. for all k ∈ I {0}, which is equivalent to saying that every term in definition (1) of A is almost surely positive, hence A is almost surely increasing. The equivalence for supermartingales is proved similarly. Example Let X = (Xn)n∈ {displaystyle mathbb {N} _{0}} be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. Fn = σ(X0, . . . , Xn) for all n ∈ {displaystyle mathbb {N} _{0}} . By (1) and (2), the Doob decomposition is given by {displaystyle A_{n}=sum _{k=1}^{n}{bigl (}mathbb {E} [X_{k}]-X_{k-1}{bigr )},quad nin mathbb {N} _{0},} and {displaystyle M_{n}=X_{0}+sum _{k=1}^{n}{bigl (}X_{k}-mathbb {E} [X_{k}]{bigr )},quad nin mathbb {N} _{0}.} If the random variables of the original sequence X have mean zero, this simplifies to {displaystyle A_{n}=-sum _{k=0}^{n-1}X_{k}}     and     {displaystyle M_{n}=sum _{k=0}^{n}X_{k},quad nin mathbb {N} _{0},} hence both processes are (possibly time-inhomogeneous) random walks. If the sequence X = (Xn)n∈ {displaystyle mathbb {N} _{0}} consists of symmetric random variables taking the values +1 and −1, then X is bounded, but the martingale M and the predictable process A are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale 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 X = (X0, X1, . . . , XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, . . . , FN), and let {displaystyle mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . . . , UN) denote the Snell envelope of X with respect to {displaystyle mathbb {Q} } . The Snell envelope is the smallest {displaystyle mathbb {Q} } -supermartingale dominating 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 U = M + A denote the Doob decomposition with respect to {displaystyle mathbb {Q} } of the Snell envelope U into a martingale M = (M0, M1, . . . , MN) and a decreasing predictable process A = (A0, A1, . . . , AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way[10][11] is {displaystyle tau _{text{max}}:={begin{cases}N&{text{if }}A_{N}=0,\min{nin {0,dots ,N-1}mid A_{n+1}<0}&{text{if }}A_{N}<0.end{cases}}} Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in Fn for every n ∈ {0, 1, . . . , N − 1}, hence 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 τmax the discounted value process U is a martingale with respect to {displaystyle mathbb {Q} } . Generalization The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.[12] Citations ^ Doob (1953), see (Doob 1990, pp. 296−298) ^ Durrett (2005) ^ (Föllmer & Schied 2011, Proposition 6.1) ^ (Williams 1991, Section 12.11, part (a) of the Theorem) ^ (Williams 1991, Section 12.11, part (b) of the Theorem) ^ (Lamberton & Lapeyre 2008, Chapter 2: Optimal stopping problem and American options) ^ (Föllmer & Schied 2011, Chapter 6: American contingent claims) ^ (Föllmer & Schied 2011, Proposition 6.10) ^ (Föllmer & Schied 2011, Theorem 6.11) ^ (Lamberton & Lapeyre 2008, Proposition 2.3.2) ^ (Föllmer & Schied 2011, Theorem 6.21) ^ (Schilling 2005, Problem 23.11) References Doob, Joseph L. (1953), Stochastic Processes, New York: Wiley, ISBN 978-0-471-21813-5, MR 0058896, Zbl 0053.26802 Doob, Joseph L. (1990), Stochastic Processes (Wiley Classics Library ed.), New York: John Wiley & Sons, Inc., ISBN 0-471-52369-0, MR 1038526, Zbl 0696.60003 Durrett, Rick (2010), Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics (4. ed.), Cambridge University Press, ISBN 978-0-521-76539-8, MR 2722836, Zbl 1202.60001 Föllmer, Hans; Schied, Alexander (2011), Stochastic Finance: An Introduction in Discrete Time, De Gruyter graduate (3. rev. and extend ed.), Berlin, New York: De Gruyter, ISBN 978-3-11-021804-6, MR 2779313, Zbl 1213.91006 Lamberton, Damien; Lapeyre, Bernard (2008), Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall/CRC financial mathematics series (2. ed.), Boca Raton, FL: Chapman & Hall/CRC, ISBN 978-1-58488-626-6, MR 2362458, Zbl 1167.60001 Schilling, René L. (2005), Measures, Integrals and Martingales, Cambridge: Cambridge University Press, ISBN 978-0-52185-015-5, MR 2200059, Zbl 1084.28001 Williams, David (1991), Probability with Martingales, Cambridge University Press, ISBN 0-521-40605-6, MR 1155402, Zbl 0722.60001 hide vte Stochastic processes Discrete time Bernoulli processBranching processChinese restaurant processGalton–Watson processIndependent and identically distributed random variablesMarkov chainMoran processRandom walk Loop-erasedSelf-avoidingBiasedMaximal entropy Continuous time Additive processBessel processBirth–death process pure birthBrownian motion BridgeExcursionFractionalGeometricMeanderCauchy processContact processContinuous-time random walkCox processDiffusion processEmpirical processFeller processFleming–Viot processGamma processGeometric processHawkes processHunt processInteracting particle systemsItô diffusionItô processJump diffusionJump processLévy processLocal timeMarkov additive processMcKean–Vlasov processOrnstein–Uhlenbeck processPoisson process CompoundNon-homogeneousSchramm–Loewner evolutionSemimartingaleSigma-martingaleStable processSuperprocessTelegraph processVariance gamma processWiener processWiener sausage Both Branching processGalves–Löcherbach modelGaussian processHidden Markov model (HMM)Markov processMartingale DifferencesLocalSub-Super-Random dynamical systemRegenerative processRenewal processStochastic chains with memory of variable lengthWhite noise Fields and other Dirichlet processGaussian random fieldGibbs measureHopfield modelIsing model Potts modelBoolean networkMarkov random fieldPercolationPitman–Yor processPoint process CoxPoissonRandom fieldRandom graph Time series models Autoregressive conditional heteroskedasticity (ARCH) modelAutoregressive integrated moving average (ARIMA) modelAutoregressive (AR) modelAutoregressive–moving-average (ARMA) modelGeneralized autoregressive conditional heteroskedasticity (GARCH) modelMoving-average (MA) model Financial models Binomial options pricing modelBlack–Derman–ToyBlack–KarasinskiBlack–ScholesChan–Karolyi–Longstaff–Sanders (CKLS)ChenConstant elasticity of variance (CEV)Cox–Ingersoll–Ross (CIR)Garman–KohlhagenHeath–Jarrow–Morton (HJM)HestonHo–LeeHull–WhiteLIBOR marketRendleman–BartterSABR volatilityVašíčekWilkie Actuarial models BühlmannCramér–LundbergRisk processSparre–Anderson Queueing models BulkFluidGeneralized queueing networkM/G/1M/M/1M/M/c Properties Càdlàg pathsContinuousContinuous pathsErgodicExchangeableFeller-continuousGauss–MarkovMarkovMixingPiecewise deterministicPredictableProgressively measurableSelf-similarStationaryTime-reversible Limit theorems Central limit theoremDonsker's theoremDoob's martingale convergence theoremsErgodic theoremFisher–Tippett–Gnedenko theoremLarge deviation principleLaw of large numbers (weak/strong)Law of the iterated logarithmMaximal ergodic theoremSanov's theoremZero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy) Inequalities Burkholder–Davis–GundyDoob's martingaleDoob's upcrossingKunita–WatanabeMarcinkiewicz–Zygmund Tools Cameron–Martin formulaConvergence of random variablesDoléans-Dade exponentialDoob decomposition theoremDoob–Meyer decomposition theoremDoob's optional stopping theoremDynkin's formulaFeynman–Kac formulaFiltrationGirsanov theoremInfinitesimal generatorItô integralItô's lemmaKarhunen–Loève theoremKolmogorov continuity theoremKolmogorov extension theoremLévy–Prokhorov metricMalliavin calculusMartingale representation theoremOptional stopping theoremProkhorov's theoremQuadratic variationReflection principleSkorokhod integralSkorokhod's representation theoremSkorokhod spaceSnell envelopeStochastic differential equation TanakaStopping timeStratonovich integralUniform integrabilityUsual hypothesesWiener space ClassicalAbstract Disciplines Actuarial mathematicsControl theoryEconometricsErgodic theoryExtreme value theory (EVT)Large deviations theoryMathematical financeMathematical statisticsProbability theoryQueueing theoryRenewal theoryRuin theorySignal processingStatisticsStochastic analysisTime series analysisMachine learning List of topicsCategory Categories: Theorems regarding stochastic processesMartingale theory