Doob decomposition theorem

Let $(\backslash Omega,\; \backslash mathcal\{F\},\; \backslash mathbb\{P\})$ be a probability space, {{math|1=I = {0, 1, 2, ..., N}}} with $N\; \backslash in\; \backslash N$ or $I\; =\; \backslash N\_0$ a finite or countably infinite index set, $(\backslash mathcal\{F\}\_n)\_\{n\; \backslash in\; I\}$ a filtration of $\backslash mathcal\{F\}$, and {{math|1=X = (X_n)_n∈I}} an adapted stochastic process with {{math|E[{{!}}X_n{{!}}] < ∞}} for all {{math|n ∈ I}}. Then there exist a martingale {{math|1=M = (M_n)_n∈I}} and an integrable predictable process {{math|1=A = (A_n)_n∈I}} starting with {{math|1=A₀ = 0}} such that {{math|1=X_n = M_n + A_n}} for every {{math|n ∈ I}}.

Here predictable means that {{math|A_n}} is $\backslash mathcal\{F\}\_\{n-1\}$-measurable for every {{math|n ∈ I \ {0}}}.

This decomposition is almost surely unique.{{harvtxt|Durrett|2010}}{{harv|Föllmer|Schied|2011|loc=Proposition 6.1}}{{harv|Williams|1991|loc=Section 12.11, part (a) of the Theorem}}

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

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

{{NumBlk|:|$A\_n=\backslash sum\_\{k=1\}^n\backslash bigl(\backslash mathbb\{E\}[X\_k\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{k-1\}]-X\_\{k-1\}\backslash bigr)$|{{EquationRef|1}}}}

and

{{NumBlk|:|$M\_n=X\_0+\backslash sum\_\{k=1\}^n\backslash bigl(X\_k-\backslash mathbb\{E\}[X\_k\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{k-1\}]\backslash bigr),$|{{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|X_k}} that is not known one time step before.

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

:$\backslash mathbb\{E\}[M\_n-M\_\{n-1\}\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{n-1\}]=0$    a.s.

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

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

:$\backslash mathbb\{E\}[Y\_n\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{n-1\}]=Y\_\{n-1\}$    a.s.,

and also predictable, implying that

:$\backslash mathbb\{E\}[Y\_n\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{n-1\}]=\; Y\_n$    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|Y_n {{=}} 0}} almost surely for all {{math|n ∈ I}}, hence the decomposition is almost surely unique.

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.{{harv|Williams|1991|loc=Section 12.11, part (b) of the Theorem}} It is a supermartingale, if and only if {{math|A}} is almost surely decreasing.

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

:$\backslash mathbb\{E\}[X\_k\backslash ,|\backslash ,\backslash mathcal\{F\}\_\{k-1\}]\backslash ge\; X\_\{k-1\}$    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.

Let {{math|X {{=}} (X_n)_{n∈$\backslash mathbb\{N\}\_0$}}} be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. {{math|{{mathcal|F}}_n {{=}} σ(X₀, . . . , X_n)}} for all {{math|n ∈ $\backslash mathbb\{N\}\_0$}}. By ({{EquationNote|1}}) and ({{EquationNote|2}}), the Doob decomposition is given by

:$A\_n=\backslash sum\_\{k=1\}^\{n\}\backslash bigl(\backslash mathbb\{E\}[X\_k]-X\_\{k-1\}\backslash bigr),\backslash quad\; n\backslash in\backslash mathbb\{N\}\_0,$

and

:$M\_n=X\_0+\backslash sum\_\{k=1\}^\{n\}\backslash bigl(X\_k-\backslash mathbb\{E\}[X\_k]\backslash bigr),\backslash quad\; n\backslash in\backslash mathbb\{N\}\_0.$

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

:$A\_n=-\backslash sum\_\{k=0\}^\{n-1\}X\_k$    and    $M\_n=\backslash sum\_\{k=0\}^\{n\}X\_k,\backslash quad\; n\backslash in\backslash mathbb\{N\}\_0,$

hence both processes are (possibly time-inhomogeneous) random walks. If the sequence {{math|X {{=}} (X_n)_{n∈$\backslash mathbb\{N\}\_0$}}} 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.

In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option.{{harv|Lamberton|Lapeyre|2008|loc=Chapter 2: Optimal stopping problem and American options}}{{harv|Föllmer|Schied|2011|loc=Chapter 6: American contingent claims}} Let {{math|X {{=}} (X₀, X₁, . . . , X_N)}} 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|$\backslash mathbb\{Q\}$}} denote an equivalent martingale measure. Let {{math|U {{=}} (U₀, U₁, . . . , U_N)}} denote the Snell envelope of {{math|X}} with respect to $\backslash mathbb\{Q\}$. The Snell envelope is the smallest {{math|$\backslash mathbb\{Q\}$}}-supermartingale dominating {{math|X}}{{harv|Föllmer|Schied|2011|loc=Proposition 6.10}} and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity.{{harv|Föllmer|Schied|2011|loc=Theorem 6.11}} Let {{math|U {{=}} M + A}} denote the Doob decomposition with respect to $\backslash mathbb\{Q\}$ of the Snell envelope {{math|U}} into a martingale {{math|M {{=}} (M₀, M₁, . . . , M_N)}} and a decreasing predictable process {{math|A {{=}} (A₀, A₁, . . . , A_N)}} with {{math|A₀ {{=}} 0}}. Then the largest stopping time to exercise the American option in an optimal way{{harv|Lamberton|Lapeyre|2008|loc=Proposition 2.3.2}}{{harv|Föllmer|Schied|2011|loc=Theorem 6.21}} is

:

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

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.{{harv|Schilling|2005|loc=Problem 23.11}}

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