Martingale convergence theorems #
The martingale convergence theorems are a collection of theorems characterizing the convergence of a martingale provided it satisfies some boundedness conditions. This file contains the almost everywhere martingale convergence theorem which provides an almost everywhere limit to an L¹ bounded submartingale. It also contains the L¹ martingale convergence theorem which provides an L¹ limit to a uniformly integrable submartingale. Finally, it also contains the Lévy upwards theorems.
Main results #
MeasureTheory.Submartingale.ae_tendsto_limitProcess: the almost everywhere martingale convergence theorem: an L¹-bounded submartingale adapted to the filtrationℱconverges almost everywhere to its limit process.MeasureTheory.Submartingale.memℒp_limitProcess: the limit process of an Lᵖ-bounded submartingale is Lᵖ.MeasureTheory.Submartingale.tendsto_snorm_one_limitProcess: part a of the L¹ martingale convergence theorem: a uniformly integrable submartingale adapted to the filtrationℱconverges almost everywhere and in L¹ to an integrable function which is measurable with respect to the σ-algebra⨆ n, ℱ n.MeasureTheory.Martingale.ae_eq_condexp_limitProcess: part b the L¹ martingale convergence theorem: iffis a uniformly integrable martingale adapted to the filtrationℱ, thenf nequals𝔼[g | ℱ n]almost everywhere wheregis the limiting process off.MeasureTheory.Integrable.tendsto_ae_condexp: part c the L¹ martingale convergence theorem: given a⨆ n, ℱ n-measurable functiongwhereℱis a filtration,𝔼[g | ℱ n]converges almost everywhere tog.MeasureTheory.Integrable.tendsto_snorm_condexp: part c the L¹ martingale convergence theorem: given a⨆ n, ℱ n-measurable functiongwhereℱis a filtration,𝔼[g | ℱ n]converges in L¹ tog.
Almost everywhere martingale convergence theorem #
We will now prove the almost everywhere martingale convergence theorem.
The a.e. martingale convergence theorem states: if f is an L¹-bounded ℱ-submartingale, then
it converges almost everywhere to an integrable function which is measurable with respect to
the σ-algebra ℱ∞ := ⨆ n, ℱ n.
Mathematically, we proceed by first noting that a real sequence $(x_n)$ converges if (a) $\limsup_{n \to \infty} |x_n| < \infty$, (b) for all $a < b \in \mathbb{Q}$ we have the number of upcrossings of $(x_n)$ from below $a$ to above $b$ is finite. Thus, for all $\omega$ satisfying $\limsup_{n \to \infty} |f_n(\omega)| < \infty$ and the number of upcrossings of $(f_n(\omega))$ from below $a$ to above $b$ is finite for all $a < b \in \mathbb{Q}$, we have $(f_n(\omega))$ is convergent.
Hence, assuming $(f_n)$ is L¹-bounded, using Fatou's lemma, we have $$ \mathbb{E} \limsup_{n \to \infty} |f_n| \le \limsup_{n \to \infty} \mathbb{E}|f_n| < \infty $$ implying $\limsup_{n \to \infty} |f_n| < \infty$ a.e. Furthermore, by the upcrossing estimate, the number of upcrossings is finite almost everywhere implying $f$ converges pointwise almost everywhere.
Thus, denoting $g$ the a.e. limit of $(f_n)$, $g$ is $\mathcal{F}_\infty$-measurable as for all $n$, $f_n$ is $\mathcal{F}_n$-measurable and $\mathcal{F}_n \le \mathcal{F}_\infty$. Finally, $g$ is integrable as $|g| \le \liminf_{n \to \infty} |f_n|$ so $$ \mathbb{E}|g| \le \mathbb{E} \limsup_{n \to \infty} |f_n| \le \limsup_{n \to \infty} \mathbb{E}|f_n| < \infty $$ as required.
Implementationwise, we have tendsto_of_no_upcrossings which shows that
a bounded sequence converges if it does not visit below $a$ and above $b$ infinitely often
for all $a, b ∈ s$ for some dense set $s$. So, we may skip the first step provided we can prove
that the realizations are bounded almost everywhere. Indeed, suppose $|f_n(\omega)|$ is not
bounded, then either $f_n(\omega) \to \pm \infty$ or one of $\limsup f_n(\omega)$ or
$\liminf f_n(\omega)$ equals $\pm \infty$ while the other is finite. But the first case
contradicts $\liminf |f_n(\omega)| < \infty$ while the second case contradicts finite upcrossings.
Furthermore, we introduce Filtration.limitProcess which chooses the limiting random variable
of a stochastic process if it exists, otherwise returning 0. Hence, instead of showing an
existence statement, we phrase the a.e. martingale convergence theorem by showing that a
submartingale converges to its limitProcess almost everywhere.
If a stochastic process has bounded upcrossing from below a to above b,
then it does not frequently visit both below a and above b.
A stochastic process that frequently visits below a and above b has infinite upcrossings.
A realization of a stochastic process with bounded upcrossings and bounded liminfs is convergent.
We use the spelling < ∞ instead of the standard ≠ ∞ in the assumptions since it is not as easy
to change < to ≠ under binders.
An L¹-bounded submartingale has bounded upcrossings almost everywhere.
An L¹-bounded submartingale converges almost everywhere.
Almost everywhere martingale convergence theorem: An L¹-bounded submartingale converges
almost everywhere to a ⨆ n, ℱ n-measurable function.
The limiting process of an Lᵖ-bounded submartingale is Lᵖ.
L¹ martingale convergence theorem #
We will now prove the L¹ martingale convergence theorems.
The L¹ martingale convergence theorem states that:
(a) if f is a uniformly integrable (in the probability sense) submartingale adapted to the
filtration ℱ, it converges in L¹ to an integrable function g which is measurable with
respect to ℱ∞ := ⨆ n, ℱ n and
(b) if f is actually a martingale, f n = 𝔼[g | ℱ n] almost everywhere.
(c) Finally, if h is integrable and measurable with respect to ℱ∞, (𝔼[h | ℱ n])ₙ is a
uniformly integrable martingale which converges to h almost everywhere and in L¹.
The proof is quite simple. (a) follows directly from the a.e. martingale convergence theorem
and the Vitali convergence theorem as our definition of uniform integrability (in the probability
sense) directly implies L¹-uniform boundedness. We note that our definition of uniform
integrability is slightly non-standard but is equivalent to the usual literary definition. This
equivalence is provided by MeasureTheory.uniformIntegrable_iff.
(b) follows since given $n$, we have for all $m \ge n$, $$ \|f_n - \mathbb{E}[g \mid \mathcal{F}_n]\|_1 = \|\mathbb{E}[f_m - g \mid \mathcal{F}_n]\|_1 \le \|\|f_m - g\|_1. $$ Thus, taking $m \to \infty$ provides the almost everywhere equality.
Finally, to prove (c), we define $f_n := \mathbb{E}[h \mid \mathcal{F}_n]$. It is clear that $(f_n)_n$ is a martingale by the tower property for conditional expectations. Furthermore, $(f_n)_n$ is uniformly integrable in the probability sense. Indeed, as a single function is uniformly integrable in the measure theory sense, for all $\epsilon > 0$, there exists some $\delta > 0$ such that for all measurable set $A$ with $\mu(A) < δ$, we have $\mathbb{E}|h|\mathbf{1}_A < \epsilon$. So, since for sufficently large $\lambda$, by the Markov inequality, we have for all $n$, $$ \mu(|f_n| \ge \lambda) \le \lambda^{-1}\mathbb{E}|f_n| \le \lambda^{-1}\mathbb|g| < \delta, $$ we have for sufficently large $\lambda$, for all $n$, $$ \mathbb{E}|f_n|\mathbf{1}_{|f_n| \ge \lambda} \le \mathbb|g|\mathbf{1}_{|f_n| \ge \lambda} < \epsilon, $$ implying $(f_n)_n$ is uniformly integrable. Now, to prove $f_n \to h$ almost everywhere and in L¹, it suffices to show that $h = g$ almost everywhere where $g$ is the almost everywhere and L¹ limit of $(f_n)_n$ from part (b) of the theorem. By noting that, for all $s \in \mathcal{F}_n$, we have $$ \mathbb{E}g\mathbf{1}_s = \mathbb{E}[\mathbb{E}[g \mid \mathcal{F}_n]\mathbf{1}_s] = \mathbb{E}[\mathbb{E}[h \mid \mathcal{F}_n]\mathbf{1}_s] = \mathbb{E}h\mathbf{1}_s $$ where $\mathbb{E}[g \mid \mathcal{F}_n = \mathbb{E}[h \mid \mathcal{F}_n]$ almost everywhere by part (b); the equality also holds for all $s \in \mathcal{F}_\infty$ by Dynkin's theorem. Thus, as both $h$ and $g$ are $\mathcal{F}_\infty$-measurable, $h = g$ almost everywhere as required.
Similar to the a.e. martingale convergence theorem, rather than showing the existence of the
limiting process, we phrase the L¹-martingale convergence theorem by proving that a submartingale
does converge in L¹ to its limitProcess. However, in contrast to the a.e. martingale convergence
theorem, we do not need to introduce an L¹ version of Filtration.limitProcess as the L¹ limit
and the a.e. limit of a submartingale coincide.
Part a of the L¹ martingale convergence theorem: a uniformly integrable submartingale
adapted to the filtration ℱ converges a.e. and in L¹ to an integrable function which is
measurable with respect to the σ-algebra ⨆ n, ℱ n.
If a martingale f adapted to ℱ converges in L¹ to g, then for all n, f n is almost
everywhere equal to 𝔼[g | ℱ n].
Part b of the L¹ martingale convergence theorem: if f is a uniformly integrable martingale
adapted to the filtration ℱ, then for all n, f n is almost everywhere equal to the conditional
expectation of its limiting process wrt. ℱ n.
Part c of the L¹ martingale convergence theorem: Given an integrable function g which
is measurable with respect to ⨆ n, ℱ n where ℱ is a filtration, the martingale defined by
𝔼[g | ℱ n] converges almost everywhere to g.
This martingale also converges to g in L¹ and this result is provided by
MeasureTheory.Integrable.tendsto_snorm_condexp
Part c of the L¹ martingale convergence theorem: Given an integrable function g which
is measurable with respect to ⨆ n, ℱ n where ℱ is a filtration, the martingale defined by
𝔼[g | ℱ n] converges in L¹ to g.
This martingale also converges to g almost everywhere and this result is provided by
MeasureTheory.Integrable.tendsto_ae_condexp
Lévy's upward theorem, almost everywhere version: given a function g and a filtration
ℱ, the sequence defined by 𝔼[g | ℱ n] converges almost everywhere to 𝔼[g | ⨆ n, ℱ n].
Lévy's upward theorem, L¹ version: given a function g and a filtration ℱ, the
sequence defined by 𝔼[g | ℱ n] converges in L¹ to 𝔼[g | ⨆ n, ℱ n].