A proof and an application of the continuous parameter martingale convergence theorem

Abstract

A proof of the continuous parameter martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale. A probabilistic proof of Liouville’s theorem is also presented applying the continuous parameter martingale convergence theorem.

Keywords:

• Martingale inequality
• martingale convergence
• super-martingale
• Liouville’s theorem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 60G42
• 60G44
• 60J45
• 60J65

1 Introduction

Unless otherwise stated, in this article all stochastic processes are defined on a filtered probability space $(\Omega ,\mathcal{F},{({\mathcal{F}}_r)}_{r\in \mathbb{R}},\mathbb{P})$ and “almost surely” is abbreviated “a.s.”

The continuous parameter martingale convergence theorem states the following:

Theorem 1.1.

Let ${(X_r)}_{r\in \mathbb{R}}$ be a right-continuous integrable sub-martingale.

1. If $\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|X_r\right|\right]<+\infty ,$ then there exists an integrable random variable X such that $\underset{r\to +\infty }{\lim }{X}_r=X$ a.s.

2. The limit $\underset{r\to -\infty }{\lim }{X}_r$ exists a.s. in $\left[-\infty ,+\infty \right[.$

Over the years, the martingale convergence theorem has received much interest in the probability literature.

Doob (1953) gave an original proof of the continuous parameter martingale convergence theorem applying the up-crossing lemma (see also Dellacherie and Meyer 1982 and Karatzas and Shreve 1998 for more details). Helms and Loeb (1982) presented another proof of Theorem 1.1 (1) employing non-standard analysis. Kruglov (2009) deduced Theorem 1.1 from convergence theorems of separable sub-martingales.

Various interesting proofs of the discrete parameter martingale convergence theorem are also discovered periodically. Doob (1940) first presented a proof without up-crossings. Isaac (1965) provided a proof using classical martingale inequalities and ingenious reductions. Garsia (1970) gave a proof due to E. Bishop applying technical Jensen-like inequalities. Dinges (1970) obtained a proof from a special point-wise criterion of convergence. Meyer (1972) offered a proof due to S. D. Chatterji using convergence theorems of closed martingales. Lamb (1973) gave a functional analytic proof. Austin, Edgar, and Tulcea (1974) provided a proof applying an approximation lemma. Chacon (1974) and Chen (1976) offered proofs using a version of Fatou’s lemma. Chen (1981) presented a proof employing elementary martingale theorems. Al-Hussaini (1981) obtained a proof from a projective limit and Banach’s principle. Petersen (1983) provided a proof due to J. Horowitz using a simple decomposition and an elementary optional sampling theorem. Malliavin (1995) offered a proof applying a sample function property of martingales. Kruglov (2001) derived a proof by approximating random variables. Garling (2007) gave a proof applying Banach-Alaoglu theorem.

Motivated by the historical interest in this problem, the purpose of this article is to present a new elementary proof of Theorem 1.1, avoiding the usual up-crossing inequality and using the classical martingale inequality:(1) $$\delta \mathbb{P}\left(\underset{r\in \mathbb{Q}\cap \left[u,u+v\right]}{\sup }\left|Y_r\right|>\delta \right)\le \mathbb{E}\left[\left|Y_u\right|\right]+2\mathbb{E}\left[\left|Y_{u+v}\right|\right],$$(1) where $(u,v,\delta )\in \mathbb{R}\times {({\mathbb{R}}_{+})}^2$ and ${(Y_r)}_{r\in \mathbb{R}}$ is an integrable super-martingale.

The discrete parameter martingale convergence theorem is also needed along with the following decomposition theorem.

Theorem 1.2

(Krickerberg decomposition). Let ${(Y_r)}_{r\in {\mathbb{R}}_{+}}$ be an integrable sub-martingale for which $\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|Y_r\right|\right]<+\infty$. Then, there exist a non-negative martingale ${(U_r)}_{r\in {\mathbb{R}}_{+}}$ and a non-negative super-martingale ${(W_r)}_{r\in {\mathbb{R}}_{+}}$ such that $\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|U_r\right|\right]<+\infty ,\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|W_r\right|\right]<+\infty ,$ and for all $r\in {\mathbb{R}}_{+},Y_r=U_r-W_r.$

The reader is referred to Dellacherie and Meyer (1982), Karatzas and Shreve (1998), Krickeberg (1965), and Revuz and Yor (1999) for a proof of inequality (1) and Theorem 1.2.

Lastly, we note that our approach gives a new proof of the theorem on the convergence of an integrable reversed discrete parameter sub-martingale (see Chow and Teicher 1997 and Neveu 1975 for a general statement and a proof using the up-crossing lemma).

The article is organized as follows. In Section 2 we prove convergence theorems for non-negative super-martingales. In Section 3 we prove Theorem 1.1. Finally, in Section 4 we apply Theorem 1.1 to give a probabilistic proof of Liouville’s theorem.

2 Preliminary tools

In this section we present the necessary tools for our proof of Theorem 1.1.

The following lemma is crucial for the proof of Theorem 2.2 below.

Lemma 2.1.

If Y is an integrable random variable, then ${(\mathbb{E}\left[Y|{\mathcal{F}}_{-k}\right])}_{k\in \mathbb{N}}$ converges to $\mathbb{E}\left[Y|\underset{k\in \mathbb{N}}{\cap }{\mathcal{F}}_{-k}\right]$ in $L^1.$

Proof.

For every $k\in \mathbb{N},$ let $H_k:=\mathbb{E}\left[Y|{\mathcal{F}}_{-k}\right].$ We fix ${\mathcal{F}}_{\infty }:=\underset{q\in \mathbb{N}}{\cap }{\mathcal{F}}_{-q}.$

We consider first the case where $Y\in L^2.$ We have$$\forall (q,k)\in {\mathbb{N}}^2,\mathbb{E}\left[{\left(H_{k+q}-H_q\right)}^2\right]=\mathbb{E}\left[H_q^2\right]-\mathbb{E}\left[H_{k+q}^2\right].$$

We note that the sequence ${(\mathbb{E}\left[H_k^2\right])}_{k\in \mathbb{N}}$ is non-increasing and bounded below, hence it converges in $\mathbb{R}.$

Consequently, there exists $H\in L^2$ such that $\underset{k\to +\infty }{\lim }$ $\mathbb{E}\left[{(H_k-H)}^2\right]=0$ and H is ${\mathcal{F}}_{\infty }$-measurable.

Since$$\forall G\in {\mathcal{F}}_{\infty },{\int }_G\mathrm{}H \mathrm{d}\mathbb{P}=\underset{q\to +\infty }{\lim }{\int }_G\mathrm{}{H}_q \mathrm{d}\mathbb{P}={\int }_G\mathrm{}Y \mathrm{d}\mathbb{P},$$it follows that $H=\mathbb{E}\left[Y|{\mathcal{F}}_{\infty }\right]$ a.s.

Turning to the general case, we have for all $(k,q)\in {\mathbb{N}}^2,$ $$\begin{array}{c}\mathbb{E}\left[\left|H_k-\mathbb{E}\left[Y|{\mathcal{F}}_{\infty }\right]\right|\right]\le 2\mathbb{E}\left[\left|Y-Y{\mathbb{1}}_{\left\{\left|Y\right|\le q\right\}}\right|\right]\\ +\mathbb{E}[|\mathbb{E}\left[Y{\mathbb{1}}_{\left\{\left|Y\right|\le q\right\}}|{\mathcal{F}}_{-k}\right]\\ -\mathbb{E}\left[Y{\mathbb{1}}_{\left\{\left|Y\right|\le q\right\}}|{\mathcal{F}}_{\infty }\right]|].\end{array}$$

Considering $k\to +\infty$ and from the special case which we already proved, we deduce that for every $q\in \mathbb{N},$ $$\underset{k\to +\infty }{\lim \sup }\mathbb{E}\left[\left|H_k-\mathbb{E}\left[Y|{\mathcal{F}}_{\infty }\right]\right|\right]\le 2\mathbb{E}\left[\left|Y-Y{\mathbb{1}}_{\left\{\left|Y\right|\le q\right\}}\right|\right].$$

Applying the dominated convergence theorem, we conclude the proof. □

We provide next a theorem on the almost sure convergence of a uniformly bounded non-negative super-martingale.

Theorem 2.2.

If ${(Y_r)}_{r\in \mathbb{R}}$ is an integrable super-martingale such that$$\exists c\in {\mathbb{R}}_{+},\forall r\in \mathbb{R},0\le Y_r\le c,$$then there exist integrable random variables Y and $Y\prime$ such that $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{Y}_r=Y$ a.s. and $\underset{r\to -\infty ,r\in \mathbb{Q}}{\lim }{Y}_r=Y\prime$ a.s.

Proof.

• First, we will prove the almost sure convergence at $+\infty .$

  Since $\underset{k\in \mathbb{N}}{\sup }\mathbb{E}\left[\left|Y_k\right|\right]\le c<+\infty ,$ it follows from the discrete parameter martingale convergence theorem that ${(Y_k)}_{k\in \mathbb{N}}$ converges a.s. to a random variable Y such that $0\le Y\le c$ a.s.

  To prove the result in its generality, we begin by fixing $k\in {\mathbb{N}}^{*}$ and $\delta \in {\mathbb{R}}_{+}^{*}.$

  Noticing that ${(Y_{r+k}-Y_k)}_{r\in {\mathbb{R}}_{+}}$ is a super-martingale relative to ${({\mathcal{F}}_{r+k})}_{r\in {\mathbb{R}}_{+}}$ and applying inequality (1), we obtain that for all $q\in {\mathbb{N}}^{*},$

  $\begin{array}{c}\delta \mathbb{P}(\underset{r\in \mathbb{Q}\cap \left[0,q\right]}{\sup }\left|Y_{r+k}-Y\right|>\delta )\\ \le \delta \mathbb{P}\left(\underset{r\in \mathbb{Q}\cap \left[0,q\right]}{\sup }\left|Y_{r+k}-Y_k\right|>\frac{\delta }{2}\right)+\delta \mathbb{P}\left(\left|Y_k-Y\right|>\frac{\delta }{2}\right)\\ \le 4\mathbb{E}\left[\left|Y_{q+k}-Y_k\right|\right]+2\mathbb{E}\left[\left|Y_k-Y\right|\right]\\ \le 4\mathbb{E}\left[\left|Y_{q+k}-Y\right|\right]+6\mathbb{E}\left[\left|Y_k-Y\right|\right].\end{array}$

  We note that $\underset{q\to +\infty }{\lim }\mathbb{E}\left[\left|Y_{q+k}-Y\right|\right]=0$ from the dominated convergence theorem.

  Hence for any $k\in {\mathbb{N}}^{*}$ and all $\delta \in {\mathbb{R}}_{+}^{*},$

  $\begin{array}{c}\delta \mathbb{P}(\underset{r\in \mathbb{Q}\cap \left[k,+\infty \right[}{\sup }\left|Y_r-Y\right|>\delta )\\ \le \delta \mathbb{P}(\underset{q\in {\mathbb{N}}^{*}}{\cup }\left\{\underset{r\in \mathbb{Q}\cap \left[0,q\right]}{\sup }\left|Y_{r+k}-Y\right|>\delta \right\})\\ \le \delta \underset{q\to +\infty }{\lim \inf }\mathbb{P}(\underset{r\in \mathbb{Q}\cap [0,q]}{\sup }\left|Y_{r+k}-Y\right|>\delta )\\ \le 6\mathbb{E}\left[\left|Y_k-Y\right|\right].\end{array}$

  Taking $k\to +\infty$ and applying the dominated convergence theorem, we conclude that for all $\delta \in {\mathbb{R}}_{+}^{*},$

  $\underset{k\to +\infty }{\lim }\mathbb{P}(\underset{r\in \mathbb{Q}\cap \left[k,+\infty \right[}{\sup }\left|Y_r-Y\right|>\delta )=0,$

  in other words $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{Y}_r=Y$ a.s.

• The proof of the almost sure convergence at $-\infty$ is essentially the same as before, therefore it’s sufficient to show that the sequence ${(Y_{-k})}_{k\in \mathbb{N}}$ converges a.s.

  We will provide an elegant proof that doesn’t use the up-crossing inequality.

  For every $k\in \mathbb{N},$ let ${\xi }_k:=Y_{-k-1}-\mathbb{E}\left[Y_{-k}|{\mathcal{F}}_{-k-1}\right]$ and $V_k:=\underset{q\in \mathbb{N}}{\sup }({\sum }_{n=0}^q{\xi }_{k+n}).$

  We note that the sequence ${(\mathbb{E}\left[Y_{-k}\right])}_{k\in \mathbb{N}}$ is non-decreasing and bounded above, hence it has a finite limit denoted by l.

  Also for every $k\in \mathbb{N},0\le V_k\le V_{k+1}$ a.s., therefore by monotone convergence theorem we have

  $\forall k\in \mathbb{N},\mathbb{E}\left[V_k\right]=\sum _{q\in \mathbb{N}}\mathbb{E}\left[{\xi }_{q+k}\right]=l-\mathbb{E}\left[Y_{-k}\right]<+\infty .$

  So ${(V_k)}_{k\in \mathbb{N}}$ is a sequence of integrable random variable such that $\underset{k\to +\infty }{\lim }\mathbb{E}\left[V_k\right]=0.$

  Next, we check that ${(Y_k+V_{-k})}_{k\in {\mathbb{Z}}_{-}}$ is an integrable martingale relative to ${({\mathcal{F}}_k)}_{k\in {\mathbb{Z}}_{-}}.$

  We fix $k\in {\mathbb{Z}}_{-}$.

  $Y_k+V_{-k}$ is ${\mathcal{F}}_k$-measurable and integrable. We also have

  $\begin{array}{c}\forall G\in {\mathcal{F}}_{k-1},{\int }_G\mathrm{}{Y}_k \mathrm{d}\mathbb{P}+{\int }_G\mathrm{}{V}_{-k} \mathrm{d}\mathbb{P}\\ ={\int }_G\mathrm{}\mathbb{E}\left[Y_k|{\mathcal{F}}_{k-1}\right] \mathrm{d}\mathbb{P}+\sum _{q\in \mathbb{N}}{\int }_G\mathrm{}{\xi }_{q-k} \mathrm{d}\mathbb{P}\\ ={\int }_G\mathrm{}{Y}_{k-1} \mathrm{d}\mathbb{P}-{\int }_G\mathrm{}{\xi }_{-k} \mathrm{d}\mathbb{P}+\sum _{q\in \mathbb{N}}{\int }_G\mathrm{}{\xi }_{q-k} \mathrm{d}\mathbb{P}\\ ={\int }_G\mathrm{}{Y}_{k-1} \mathrm{d}\mathbb{P}+\sum _{q\in \mathbb{N}}{\int }_G\mathrm{}{\xi }_{q+1-k} \mathrm{d}\mathbb{P}\\ ={\int }_G\mathrm{}{Y}_{k-1} \mathrm{d}\mathbb{P}+{\int }_G\mathrm{}{V}_{1-k} \mathrm{d}\mathbb{P}.\end{array}$

  Consequently, ${(Y_k+V_{-k})}_{k\in {\mathbb{Z}}_{-}}$ is a martingale and hence for all $k\in \mathbb{N},Y_{-k}=\mathbb{E}\left[Y_0+V_0|{\mathcal{F}}_{-k}\right]-V_k$ a.s.

  It follows from Lemma 2.1 that ${(Y_{-k})}_{k\in \mathbb{N}}$ converges to $Y\prime :=\mathbb{E}\left[Y_0+V_0|\underset{k\in \mathbb{N}}{\cap }{\mathcal{F}}_{-k}\right]$ in $L^1.$

  Lastly, noticing that ${(Y_k-Y\prime )}_{k\in {\mathbb{Z}}_{-}}$ is a super-martingale relative to ${({\mathcal{F}}_k)}_{k\in {\mathbb{Z}}_{-}}$ and using the discrete version of inequality (1), we have for every $\delta \in {\mathbb{R}}_{+}^{*}$ and every $(q,n)\in {\mathbb{N}}^2,$

  $\begin{array}{c}\delta \mathbb{P}(\underset{0\le k\le q}{\max }\left|Y_{-k-n}-Y\prime \right|>\delta )\le \mathbb{E}\left[\left|Y_{-q-n}-Y\prime \right|\right]\\ +2\mathbb{E}\left[\left|Y_{-n}-Y\prime \right|\right].\end{array}$

  Letting $q\to +\infty$ we obtain that for any $\delta \in {\mathbb{R}}_{+}^{*}$ and all $n\in \mathbb{N},$

  $\mathbb{P}(\underset{k\in \mathbb{N}}{\sup }\left|Y_{-k-n}-Y\prime \right|>\delta )\le \frac{2}{\delta }\mathbb{E}\left[\left|Y_{-n}-Y\prime \right|\right].$

  So for every $\delta \in {\mathbb{R}}_{+}^{*},\underset{n\to +\infty }{\lim }\mathbb{P}(\underset{k\in \mathbb{N}}{\sup }\left|Y_{-k-n}-Y\prime \right|>\delta )$ $=0,$ concluding the proof. □

Remark 2.3.

The proof in Theorem 2.2 of the almost sure convergence at $-\infty$ relies on Doob decomposition of an integrable reversed discrete parameter sub-martingale (we refer to Dudley 2002 for a general statement and a proof).

We end this section by stating and proving a general version of Theorem 2.2.

Theorem 2.4.

Let ${(Y_r)}_{r\in \mathbb{R}}$ is a non-negative super-martingale. Then, the limits $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{Y}_r$ and $\underset{r\to -\infty ,r\in \mathbb{Q}}{\lim }{Y}_r$ exist a.s. in ${\overline{\mathbb{R}}}_{+}.$ Further, if the sample paths of ${(Y_r)}_{r\in \mathbb{R}}$ are right-continuous, then $\underset{r\to +\infty }{\lim }{Y}_r$ and $\underset{r\to -\infty }{\lim }{Y}_r$ exist a.s. in ${\overline{\mathbb{R}}}_{+}.$

Proof.

We will only prove the almost sure existence of the limit at $+\infty ,$ the proof is analogous at $-\infty$.

The idea is to truncate properly so that the super-martingale property is preserved.

We fix $q\in \mathbb{N}.$

${(\min (q,Y_r))}_{r\in \mathbb{R}}$ is a non-negative super-martingale uniformly bounded by q. We deduce from Theorem 2.2 that $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \sup }$ $\min$ $(q,Y_r)=$ $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \inf }$ $\min$ (q, $Y_r$) a.s.

We also have the following relations:$$\begin{array}{c}\min \left(q,\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \sup }{Y}_r\right)=\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \sup }\min \left(q,Y_r\right),\\ \min \left(q,\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \inf }{Y}_r\right)=\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \inf }\min \left(q,Y_r\right).\end{array}$$

So for every $q\in \mathbb{N},\min$ $(q,\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \sup }{Y}_r)=$ $\min (q,\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \inf }$ $Y_r)$ a.s. Hence $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \sup }{Y}_r=\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim \inf }{Y}_r$ a.s., yielding that $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{Y}_r$ exists a.s. in ${\overline{\mathbb{R}}}_{+}.$

If ${(Y_r)}_{r\in \mathbb{R}}$ is right-continuous, then $\underset{r\to +\infty }{\lim }{Y}_r=$ $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{Y}_r$ a.s. □

Remark 2.5.

Theorem 2.4 holds true if $\mathbb{Q}$ is replaced by a countable dense subset D of $\mathbb{R}.$

3 Proof of the continuous parameter martingale convergence theorem

Finally, we are ready to prove our theorem.

Proof of Theorem 1.1.

Since the sample paths of ${(X_r)}_{r\in \mathbb{R}}$ are right-continuous, it’s sufficient to prove that the limits of ${(X_r)}_{r\in \mathbb{Q}}$ at $+\infty$ and $-\infty$ exist almost surely.

1. Applying Theorem 1.2, there exist a non-negative martingale ${(U_r)}_{r\in {\mathbb{R}}_{+}}$ and a non-negative super-martingale ${(W_r)}_{r\in {\mathbb{R}}_{+}}$ such that $\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|U_r\right|\right]<+\infty ,\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|W_r\right|\right]<+\infty ,$ and for every $r\in {\mathbb{R}}_{+},X_r=U_r-W_r.$

   Theorem 2.4 yields that $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{U}_r$ and $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }$ W_r exist a.s. in ${\overline{\mathbb{R}}}_{+},$ we denote these almost sure limits by U and W, respectively.

   It follows by Fatou’s lemma that U and W are integrable, in particular they are finite a.s. and hence $\underset{r\to +\infty ,r\in \mathbb{Q}}{\lim }{X}_r=U-W$ a.s.

2. To verify the result, we need to write X_r suitably.

   We note that ${(\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right])}_{r\in {\mathbb{R}}_{-}}$ is an integrable martingale such that for all $r\in {\mathbb{R}}_{-},\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right]\ge \mathbb{E}\left[X_0|{\mathcal{F}}_r\right]\ge X_r$ a.s., so ${(\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right]-X_r)}_{r\in {\mathbb{R}}_{-}}$ is a non-negative super-martingale.

   Applying again Theorem 2.4 and since $X_0\in L^1,$ the limits $\underset{r\to -\infty ,r\in \mathbb{Q}}{\lim }\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right]$ and $\underset{r\to -\infty ,r\in \mathbb{Q}}{\lim }(\mathbb{E}[\left|X_0\right||{\mathcal{F}}_r]-$ $X_r)$ exist a.s. in $\mathbb{R}$ and ${\overline{\mathbb{R}}}_{+},$ respectively.

   By writing for every $r\in {\mathbb{R}}_{-},X_r=\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right]-(\mathbb{E}\left[\left|X_0\right||{\mathcal{F}}_r\right]-X_r),$ we conclude that $\underset{r\to -\infty ,r\in \mathbb{Q}}{\lim }{X}_r$ exists a.s. in $\left[-\infty ,+\infty \right[.$

   □

Remark 3.1.

Applying the procedure used in proving Theorem 1.1 we can show that if D is a countable dense subset of $\mathbb{R}$ and ${(Y_r)}_{r\in \mathbb{R}}$ is an integrable sub-martingale, then there exist integrable stochastic processes ${(V_r)}_{r\in \mathbb{R}}$ and ${(V{\prime }_r)}_{r\in \mathbb{R}}$ such that for every $u\in \mathbb{R},\underset{r\downarrow u,r\in D}{\lim }{Y}_r=V_u$ a.s., $\underset{r\uparrow u,r\in D}{\lim }{Y}_r=V{\prime }_u$ a.s., $\mathbb{E}\left[V_u|{\mathcal{F}}_u\right]\ge Y_u$ a.s., and $\mathbb{E}\left[Y_u|\sigma (\underset{r\in ]-\infty ,u[}{\cup }{\mathcal{F}}_r)\right]\ge V{\prime }_u$ a.s. by the following sequence of arguments:

1. For all $(u,t)\in {\mathbb{R}}^2,Y_t=\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_t\right]-(\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_t\right]-Y_t)$ and ${(\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_r\right]-Y_r)}_{r\in [u-1,u+1]}$ is a non-negative integrable super-martingale.

2. For every $u\in \mathbb{R},\underset{r\downarrow u,r\in D}{\lim }$ $\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_r\right]=\underset{k\to +\infty }{\lim }\mathbb{E}$ $\left[\left|Y_{u+1}\right||{\mathcal{F}}_{u+\frac{1}{k}}\right]=$ $\mathbb{E}\left[\left|Y_{u+1}\right||\underset{r\in ]u,+\infty [}{\cap }{\mathcal{F}}_r\right]$ a.s. and $\underset{r\uparrow u,r\in D}{\lim }$ $\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_r\right]=$ $\underset{k\to +\infty }{\lim }\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_{u-\frac{1}{k}}\right]=$ $\mathbb{E}$ $\left[\left|Y_{u+1}\right||\sigma (\underset{r\in ]-\infty ,u[}{\cup }{\mathcal{F}}_r)\right]$ a.s.

3. There exist integrable stochastic processes ${(Z_r)}_{r\in \mathbb{R}}$ and ${(Z{\prime }_r)}_{r\in \mathbb{R}}$ such that for all $u\in \mathbb{R},\underset{r\downarrow u,r\in D}{\lim }$ $(\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_r\right]-Y_r)=$ $\underset{k\to +\infty }{\lim }(\mathbb{E}[\left|Y_{u+1}\right||$ ${\mathcal{F}}_{u+\frac{1}{k}}]-$ $Y_{u+\frac{1}{k}})=Z_u$ a.s., $\underset{r\uparrow u,r\in D}{\lim }$ $(\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_r\right]-Y_r)=$ $\underset{k\to +\infty }{\lim }$ $\left(\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_{u-\frac{1}{k}}\right]-Y_{u-\frac{1}{k}}\right)=Z{\prime }_u$ a.s., $\mathbb{E}\left[Z_u|{\mathcal{F}}_u\right]$ $\le \mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_u\right]-Y_u$ a.s., and $\mathbb{E}\left[\mathbb{E}\left[\left|Y_{u+1}\right||{\mathcal{F}}_u\right]-Y_u|\sigma (\underset{r\in ]-\infty ,u[}{\cup }{\mathcal{F}}_r)\right]\le Z{\prime }_u$ a.s.

We refer to Dellacherie and Meyer (1982) for another proof using the up-crossing inequality.

4 Application to Liouville’s theorem

In this section we apply Theorem 1.1 to prove the following classical theorem from harmonic analysis.

Theorem 4.1

(Liouville’s theorem). Let $d\in {\mathbb{N}}^{*}.$ If $f:{\mathbb{R}}^d\to \mathbb{R}$ is a harmonic function such that$$\exists c\in {\mathbb{R}}_{+},\forall x\in {\mathbb{R}}^d,|f(x)|\le c,$$

then$$\exists \alpha \in \mathbb{R},\forall x\in {\mathbb{R}}^d,f(x)=\alpha .$$

Proof.

Let $({\mathbb{R}}^d,\Vert ·\Vert )$ be the d-dimensional Euclidean space, ${(B_r)}_{r\in {\mathbb{R}}_{+}}$ be the d-dimensional Brownian motion starting from $0,{\lambda }_d$ be the Lebesgue measure on $({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d)),$ and $$\begin{array}{c}h:{\mathbb{R}}_{+}^{*}\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}\\ (r,x,y)\mapsto h(r,x,y)={\left(2\pi r\right)}^{-\frac{d}{2}}{\mathrm{e}}^{-\frac{1}{2r}{\Vert y-x\Vert }^2}\end{array}$$

For all $u\in {\mathbb{R}}_{+},$ let ${\mathcal{G}}_u:=\sigma (\underset{r\in [0,u]}{\cup }\sigma (B_r)).$

Since $D_{1}h=\frac{1}{2}{\sum }_{q=d+2}^{2d+1}{D}_{qq}h,$ we obtain by integration by parts that $$\begin{array}{c}\forall (x,u)\in {\mathbb{R}}^d\times {\mathbb{R}}_{+}^{*},0=\frac{1}{2}\mathbb{E}\left[\Delta f\left(x+B_u\right)\right]\\ =\frac{1}{2}{\int }_{{\mathbb{R}}^d}\Delta f(x+y)h(u,0,y)\mathrm{ d}y\\ =\frac{1}{2}{\int }_{{\mathbb{R}}^d}\Delta f(y)h(u,x,y)\mathrm{ d}y\\ ={\int }_{{\mathbb{R}}^d}f(y)D_{1}h(u,x,y)\mathrm{ d}y.\end{array}$$

It follows by Fubini’s theorem that $$\begin{array}{c}\forall (x,u,k)\in {\mathbb{R}}^d\times {\mathbb{R}}_{+}^{*}\times {\mathbb{N}}^{*},\mathbb{E}\left[f\left(x+B_u\right)\right]\\ ={\int }_{{\mathbb{R}}^d}f(x+y)h(u,0,y)\mathrm{ d}y\\ ={\int }_{{\mathbb{R}}^d}f(y)h(u,x,y)\mathrm{ d}y\\ ={\int }_{{\mathbb{R}}^d}f\left(y\right)h\left(\frac{u}{k},x,y\right)\mathrm{ d}y\\ ={\int }_{{\mathbb{R}}^d}f(x+y)h\left(\frac{u}{k},0,y\right)\mathrm{ d}y\\ =\mathbb{E}\left[f\left(x+B_{\frac{u}{k}}\right)\right].\end{array}$$

Taking $k\to +\infty$ and applying the dominated convergence theorem, we find that$$\forall (x,u)\in {\mathbb{R}}^d\times {\mathbb{R}}_{+},f(x)=\mathbb{E}\left[f\left(x+B_u\right)\right].$$

Next, we check that ${(f(B_r))}_{r\in {\mathbb{R}}_{+}}$ is an integrable martingale relative to ${({\mathcal{G}}_r)}_{r\in {\mathbb{R}}_{+}.}$

For every $u\in {\mathbb{R}}_{+},f(B_u)$ is ${\mathcal{G}}_u$-measurable and integrable. We also have$$\begin{array}{c}\forall (r,u)\in {\left({\mathbb{R}}_{+}\right)}^2,r\le u\Rightarrow \mathbb{E}\left[f\left(B_u\right)|{\mathcal{G}}_r\right]\\ =\mathbb{E}\left[f\left(B_u-B_r+B_r\right)|{\mathcal{G}}_r\right]\\ ={\int }_{{\mathbb{R}}^d}f\left(y+B_r\right)\mathrm{d}{\mathbb{P}}_{B_u-B_r}(y)\\ ={\int }_{{\mathbb{R}}^d}f\left(y+B_r\right)\mathrm{d}{\mathbb{P}}_{B_{u-r}}(y)\\ =f(B_r)\mathrm{a}.\mathrm{s}.\end{array}$$

Consequently, ${(f(B_r))}_{r\in {\mathbb{R}}_{+}}$ is a continuous martingale relative to ${({\mathcal{G}}_r)}_{r\in {\mathbb{R}}_{+}}$ such that $\underset{r\in {\mathbb{R}}_{+}}{\sup }\mathbb{E}\left[\left|f(B_r)\right|\right]\le c<+\infty .$

It follows from Theorem 1.1 that there exists an integrable random variable ζ such that $\underset{r\to +\infty }{\lim }f(B_r)=\zeta$ a.s.

Noticing that for all $k\in \mathbb{N},$ $$\begin{array}{c}\mathbb{E}\left[\left|f(B_{k+1})-f\left(B_k\right)\right|\right]\\ \le \mathbb{E}\left[\left|f\left(B_{k+1}-B_k+B_k\right)-f\left(B_k\right)\right|\right]\\ \le {\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\left|f(x+y)-f(y)\right|\mathrm{d}{\mathbb{P}}_{\left(B_{k+1}-B_k,B_k\right)}(x,y)\\ \le {\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\left|{\int }_{{\mathbb{R}}^d}\left(f(x+y+v)-f(y+v)\right)\mathrm{\hspace{1em}d}{\mathbb{P}}_{B_1}(v)\right|\\ \mathrm{\hspace{1em}\hspace{1em}d}{\mathbb{P}}_{\left(B_{k+2}-B_{k+1},B_k\right)}(x,y)\\ \le {\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d\times {\mathbb{R}}^d}\left|f\left(x+y+v\right)-f\left(y+v\right)\right|\\ \text{\hspace{1em}\hspace{1em}d}{\mathbb{P}}_{\left(B_{k+1}-B_k,B_{k+2}-B_{k+1},B_k\right)}(v,x,y)\\ \le \mathbb{E}\left[|f\left(B_{k+2}\right)-f\left(B_{k+1}\right)|\right],\end{array}$$ we get that $$\forall k\in \mathbb{N},\mathbb{E}\left[\left|f\left(B_1\right)-f(0)\right|\right]\le \mathbb{E}\left[\left|f\left(B_{k+1}\right)-f\left(B_k\right)\right|\right].$$

Considering $k\to +\infty$ and using the dominated convergence theorem, we obtain that ${\int }_{{\mathbb{R}}^d}\left|f(y)-f(0)\right|h(1,0,y)\mathrm{ d}y=0$ and hence ${\lambda }_d(\left\{f\ne f(0)\right\})=0.$

Therefore $$\begin{array}{c}\forall (x,k)\in {\mathbb{R}}^d\times {\mathbb{N}}^{*},{\int }_{{\mathbb{R}}^d}\left|f\left(x+\frac{y}{k}\right)-f(0)\right|h(1,0,y)\mathrm{ d}y\\ ={\int }_{{\mathbb{R}}^d}\left|f(y)-f(0)\right|h\left(\frac{1}{k},x,y\right)\mathrm{ d}y=0.\end{array}$$

Applying the dominated convergence theorem, we conclude that $$\forall x\in {\mathbb{R}}^d,f(x)=f(0).$$

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Remark 4.2.

An alternative approach to Theorem 4.1 is to use Itô’s formula, Theorem 1.1, and Blumenthal’s 0-1 law (see Chung and Williams 2014 for more details).

Disclosure Statement

No potential conflict of interest was reported by the author(s).