Doob's martingale convergence theorems

In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob.^[1]

Statement of the theorems

In the following, (Ω, F, F_∗, P), F_∗ = (F_t)_{t ≥ 0}, will be a filtered probability space and N : [0, +∞) × Ω → R will be a right-continuous supermartingale with respect to the filtration F_∗; in other words, for all 0 ≤ s ≤ t < +∞,

N_{s}\geq \operatorname {E} {\big [}N_{t}\mid F_{s}{\big ]}.

Doob's first martingale convergence theorem

Doob's first martingale convergence theorem provides a sufficient condition for the random variables N_t to have a limit as t → +∞ in a pointwise sense, i.e. for each ω in the sample space Ω individually.

For t ≥ 0, let N_t⁻ = max(−N_t, 0) and suppose that

\sup _{t>0}\operatorname {E} {\big [}N_{t}^{-}{\big ]}<+\infty .

Then the pointwise limit

N(\omega )=\lim _{t\to +\infty }N_{t}(\omega )

exists and is finite for P-almost all ω ∈ Ω.^[2]

Doob's second martingale convergence theorem

It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any L^p space. In order to obtain convergence in L¹ (i.e., convergence in mean), one requires uniform integrability of the random variables N_t. By Chebyshev's inequality, convergence in L¹ implies convergence in probability and convergence in distribution.

The following are equivalent:

(N_t)_{t > 0} is uniformly integrable, i.e.

\lim _{C\to \infty }\sup _{t>0}\int _{\{\omega \in \Omega \,\mid \,N_{t}(\omega )>C\}}\left|N_{t}(\omega )\right|\,\mathrm {d} \mathbf {P} (\omega )=0;

there exists an integrable random variable N ∈ L¹(Ω, P; R) such that N_t → N as t → +∞ both P-almost surely and in L¹(Ω, P; R), i.e.

\operatorname {E} \left[\left|N_{t}-N\right|\right]=\int _{\Omega }\left|N_{t}(\omega )-N(\omega )\right|\,\mathrm {d} \mathbf {P} (\omega )\to 0{\text{ as }}t\to +\infty .

Corollary: convergence theorem for continuous martingales

Let M : [0, +∞) × Ω → R be a continuous martingale such that

\sup _{t>0}\operatorname {E} {\big [}{\big |}M_{t}{\big |}^{p}{\big ]}<+\infty

for some p > 1. Then there exists a random variable M ∈ L^p(Ω, P; R) such that M_t → M as t → +∞ both P-almost surely and in L^p(Ω, P; R).

Discrete-time results

Similar results can be obtained for discrete-time supermartingales and submartingales, the obvious difference being that no continuity assumptions are required. For example, the result above becomes

Let M : N × Ω → R be a discrete-time martingale such that

\sup _{k\in \mathbf {N} }\operatorname {E} {\big [}{\big |}M_{k}{\big |}^{p}{\big ]}<+\infty

for some p > 1. Then there exists a random variable M ∈ L^p(Ω, P; R) such that M_k → M as k → +∞ both P-almost surely and in L^p(Ω, P; R)

Convergence of conditional expectations: Lévy's zero–one law

Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.

Let (Ω, F, P) be a probability space and let X be a random variable in L¹. Let F_∗ = (F_k)_k∈N be any filtration of F, and define F_∞ to be the minimal σ-algebra generated by (F_k)_k∈N. Then

\operatorname {E} {\big [}X\mid F_{k}{\big ]}\to \operatorname {E} {\big [}X\mid F_{\infty }{\big ]}{\text{ as }}k\to \infty

both P-almost surely and in L¹.

This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if A is an event in F_∞, then the theorem says that $\mathbf {P} [A\mid F_{k}]\to \mathbf {1} _{A}$ almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be. This sounds almost like a tautology, but the result is still non-trivial. For instance, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have $\mathbf{P}[ A ] = \mathbf{1}_A$ almost surely, hence $\mathbf {P} [A]\in \{0,1\}$ .

Similarly we have the Levy's downwards theorem :

Let (Ω, F, P) be a probability space and let X be a random variable in L¹. Let (F_k)_k∈N be any decreasing sequence of sub-sigma algebras of F, and define F_∞ to be the intersection. Then

\operatorname {E} {\big [}X\mid F_{k}{\big ]}\to \operatorname {E} {\big [}X\mid F_{\infty }{\big ]}{\text{ as }}k\to \infty

both P-almost surely and in L¹.

Doob's upcrossing inequality

The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems.^[2]
Hypothesis.
Let $N$ be a natural number. Let $X_{n}$ , for $n=1,\ldots ,N$ , be a martingale with respect to a filtration $\mathcal{F}_n$ , for $n=1,\ldots ,N$ . Let $a$ , $b$ be two real numbers with $a<b$ .
Define the random variables $U_{n}$ , for $n=1,\ldots ,N$ , as follows: $U_{n}=m$ if and only if $m$ is the largest integer such that there exist integers $j_{1},k_{1},j_{2},k_{2},\ldots ,j_{m}$ , $k_{m}$ satisfying 1 ≤ $j_1$ < $k_{1}<j_{2}<k_{2}<\cdots <j_{m}<k_{m}\leq n$ and, for $i=1,\ldots ,m$ , for each pair $j_{i},k_{i}$ the inequalities $X_{j_{i}}<a$ and $X_{k_{i}}>b$ are satisfied. Each $U_{n}$ is called the number of upcrossings with respect to the interval $a,b$ for the martingale $X_{i}$ , $i=1,\ldots ,n$ .
Conclusion.

(b-a)\operatorname {E} [U_{n}]\leq \operatorname {E} [(X_{n}-a)^{-}]

.^[3]^[4]

References

↑ Doob, J. L. (1953). Stochastic Processes. New York: Wiley.
1 2 "Martingale Convergence Theorem" (PDF). Massachusetts Institute of Tecnnology, 6.265/15.070J Lecture 11-Additional Material, Advanced Stochastic Processes, Fall 2013, 10/9/2013.
↑ Bobrowski, Adam (2005). Functional Analysis for Probability and Stochastic Processe: An Introduction. Cambridge University Press. pp. 113–114. ISBN 9781139443883.
↑ Gushchin, A. A. (2014). "On pathwise counterparts of Doob's maximal inequalities". Proceedings of the Steklov Institute of Mathematics. 287 (287): 118–121. arXiv:1410.8264. doi:10.1134/S0081543814080070.
↑ Doob, Joseph L. (2012). Measure theory. Graduate Texts in Mathematics, Vol. 143. Springer. p. 197. ISBN 9781461208778.

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Appendix C)

Durrett, Rick (1996). Probability: theory and examples (Second ed.). Duxbury Press. ISBN 978-0-534-24318-0. ; Durrett, Rick (2010). 4th edition. ISBN 9781139491136.

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[1] Doob, J. L. (1953). Stochastic Processes. New York: Wiley.

[MITnotes-2] 1 2 "Martingale Convergence Theorem" (PDF). Massachusetts Institute of Tecnnology, 6.265/15.070J Lecture 11-Additional Material, Advanced Stochastic Processes, Fall 2013, 10/9/2013.

[Bobrowski-3] Bobrowski, Adam (2005). Functional Analysis for Probability and Stochastic Processe: An Introduction. Cambridge University Press. pp. 113–114. ISBN 9781139443883.

[4] Gushchin, A. A. (2014). "On pathwise counterparts of Doob's maximal inequalities". Proceedings of the Steklov Institute of Mathematics. 287 (287): 118–121. arXiv:1410.8264. doi:10.1134/S0081543814080070.

[5] Doob, Joseph L. (2012). Measure theory. Graduate Texts in Mathematics, Vol. 143. Springer. p. 197. ISBN 9781461208778.