Doob's martingale convergence theorems

In mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph Leo Doob.

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 < +∞,

$N_{s}\geq \mathbf {E} {\big [}N_{t}{\big |}F_{s}{\big ]}.$

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

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

Then the pointwise limit

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

exists for P-almost all ω ∈ Ω.

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.

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

• 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.

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

Corollary: convergence theorem for continuous martingales

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

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

for some p > 1. Then there exists a random variable M ∈ L¹(Ω, P; R) such that M_t → M as t → +∞ both P-almost surely and in L¹(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

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

for some p > 1. Then there exists a random variable M ∈ L¹(Ω, P; R) such that M_k → M as k → +∞ both P-almost surely and in L¹(Ω, P; R)

Convergence of conditional expectations

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

$\mathbf {E} {\big [}X{\big |}F_{k}{\big ]}\to \mathbf {E} {\big [}X{\big |}F_{\infty }{\big ]}{\mbox{ as }}k\to \infty$

both P-almost surely and in L¹.

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth edition ed.). Berlin: Springer. ISBN 3-540-04758-1. {{cite book}}: |edition= has extra text (help) (See Appendix C)