Modern Stochastics: Theory and Applications

The classical definition of martingales is extended to a more general case in the space of Banach lattices by V. Troitsky [6]. In the Banach lattice framework, martingales are defined without a probability space and the famous Doob’s convergence theorem was reproduced. Moreover, under certain conditions on the Banach lattice, it was shown that the set of bounded martingales forms a Banach lattice with respect to the point-wise order. In 2011, H. Gessesse and V. Troitsky [2] produced several sufficient conditions for the space of bounded martingales on a Banach lattice to be a Banach lattice itself. They also provided examples showing that the space of bounded martingales is not necessarily a vector lattice. Several other works have been done by other authors with regard to martingales in vector lattices, such as [4, 3].

In the theory of random processes, not just the study of martingale convergence is important, but the study of convergence of martingale-like stochastic sequences and processes, and the determination of interrelation between them are also crucial. So it is natural to ask if martingale-like sequences can be defined in a vector lattice or Banach lattice framework. In this article, we define and study martingale-like sequences in Banach lattices along the same lines as martingales are defined and studied in [6].

Classically, a martingale-like sequence is defined as follows (for instance, see a paper by A. Melnikov [5]). Consider a probability space $(\varOmega ,\mathcal{F},P)$ and a filtration ${({\mathcal{F}_{n}})_{n=1}^{\infty }}$, i.e., an increasing sequence of complete sub-sigma-algebras of $\mathcal{F}$. An integrable stochastic sequence $x=({x_{n}},{\mathcal{F}_{n}})$ is an ${L^{1}}$-martingale if An integrable stochastic sequence $x=({x_{n}},{\mathcal{F}_{n}})$ is an E-martingale if

Here we extend the definition of ${L^{1}}$-martingales and E-martingales in a general Banach lattice X following the same lines as the definition of martingales in Banach lattices in [6]. First we mention some terminology and definitions from the theory of Banach lattices for the reader convenience. For more detailed exploration, we refer the reader to [1]. A vector lattice is a vector space equipped with a lattice order relation, which is compatible with the linear structure. A Banach lattice is a vector lattice with a Banach norm which is monotone, i.e., $0\leqslant x\leqslant y$ implies $\| x\| \leqslant \| y\| $, and satisfies $\| x\| =\| |x|\| $ for any two vectors x and y. A vector lattice is said to be order complete if every nonempty subset that is bounded above has a supremum. We say that a Banach lattice has order continuous norm if $\| {x_{\alpha }}\| \to 0$ for every decreasing net $({x_{\alpha }})$ with $\inf {x_{\alpha }}=0$. A Banach lattice with order continuous norm is order complete. A sublattice Y of a vector lattice is called an (order) ideal if $y\in Y$ and $|x|\leqslant |y|$ imply $x\in Y$. An ideal Y is called a band if $x={\sup _{\alpha }}{x_{\alpha }}$ implies $x\in Y$ for every positive increasing net $({x_{\alpha }})$ in Y. Two elements x and y in a vector lattice are said to be disjoint whenever $|x|\wedge |y|=0$ holds. If J is a nonempty subset of a vector lattice, then its disjoint complement ${J^{d}}$ is the set of all elements of the lattice, disjoint to every element of J. A band Y in a vector lattice X that satisfies $X=Y\otimes {Y^{d}}$ is refered to as a projection band. Every band in an order complete vector lattice is a projection band. An operator T on a vector lattice X is positive if $Tx\geqslant 0$ for every $x\geqslant 0$. A sequence of positive projections $({E_{n}})$ on a vector lattice X is called a filtration if ${E_{n}}{E_{m}}={E_{n\wedge m}}$. A sequence of positive contractive projections $({E_{n}})$ on a normed lattice X is called a contractive filtration if ${E_{n}}{E_{m}}={E_{n\wedge m}}$. A filtration $({E_{n}})$ in a normed lattice X is called dense if ${E_{n}}x\to x$ for each x in X. In many articles such as in [6], a martingale with respect to a filtration $({E_{n}})$ in a vector lattice X is defined as a sequence $({x_{n}})$ in X such that ${E_{n}}{x_{m}}={x_{n}}$ whenever $m\ge n$.

Note that Definition 2 is equivalent to saying a sequence $({x_{n}})$ is an $\mathcal{E}$-martingale if there exists $l\geqslant 1$ such that ${E_{n}}{x_{m}}={x_{n}}$ whenever $m\geqslant n\ge l$. The symbol “$\mathcal{E}$” stands for eventual so when we say $({x_{n}})$ is an $\mathcal{E}$-martingale, we are saying that after a first few finite elements of the sequence, the sequence becomes a martingale.

Sequences defined by Definition 1 and Definition 2 are collectively called martingale-like sequences. Notice that every martingale $({x_{n}})$ in a vector lattice X with respect to a filtration $({E_{n}})$ is obviously an $\mathcal{E}$-martingale with respect to the filtration $({E_{n}})$. Moreover, every $\mathcal{E}$-martingale $({x_{n}})$ in a Banach lattice X with respect to a contractive filtration $({E_{n}})$ is an X-martingale with respect to the contrative filtration $({E_{n}})$. Note that for every x in a vector lattice X and a filtration $({E_{n}})$ in X, the sequence $({E_{n}}x)$ is an $\mathcal{E}$-martingale with respect to the filtration $({E_{n}})$. If x is in a normed space X and $({E_{n}})$ is a contractive filtration, then the sequence $({E_{n}}x)$ is an X-martingale with respect to the contractive filtration $({E_{n}})$.

By considering any nonzero martingale $({x_{n}})$ in a Banach lattice X with respect to filtration $({E_{n}})$ where ${x_{1}}$ is nonzero without loss of generality, we can define a sequence $({y_{n}})$ such that ${y_{1}}=2{x_{1}}$ and ${y_{n}}={x_{n}}$ for all $n\geqslant 2$. Then one can see that $({y_{n}})$ is an $\mathcal{E}$-martingale with respect to the filtration $({E_{n}})$. However, $({y_{n}})$ is not a martingale.

Note that every sequence which converges to zero is an X-martingale with respect to any contractive filtration $({E_{n}})$ because if ${x_{n}}\to 0$ and $m>n$ then $\| {E_{n}}{x_{m}}-{x_{n}}\| \leqslant \| {x_{m}}\| +\| {x_{n}}\| \to 0$ as $n\to \infty $. So one can easily create an X-martingale $({x_{n}})$ which is not $\mathcal{E}$-martingale by setting ${x_{n}}=\frac{1}{n}x$ where x is a nonzero vector in X.

A martingale-like sequence $A=({x_{n}})$ with respect to a contractive filtration $({E_{n}})$ on a normed lattice X is said to be bounded if its norm defined by $\| A\| ={\sup _{n}}\| {x_{n}}\| $ is finite. Given a contractive filtration $({E_{n}})$ on a normed lattice X, we denote the set of all bounded X-martingales with respect to the contractive filtration $({E_{n}})$ by ${M_{X}}={M_{X}}(X,({E_{n}}))$ and the set of all bounded $\mathcal{E}$-martingales with respect to the contractive filtration $({E_{n}})$ by ${M_{E}}={M_{E}}(X,({E_{n}}))$. With the introduction of the sup norm in these spaces, one can show that ${M_{X}}$ and ${M_{E}}$ are normed spaces. Keeping the notation M of [6] for all bounded martingales with respect to the contractive filtration $({E_{n}})$ and from the preceding arguments, these spaces form a nested increasing sequence of linear subspaces $M\subset {M_{E}}\subset {M_{X}}\subset {\ell _{\infty }}(X)$, with the norm being exactly the ${\ell _{\infty }}(X)$ norm.

The following proposition confirms that for any convergent element $A=({x_{n}})$ of ${M_{X}}$ we can find a sequence in ${M_{E}}$ that converges to A.

In [6] and [2] several sufficient conditions are established where the set of bounded martingales M is a Banach lattice. In [2], counter examples are provided where M is not a Banach lattice. So, one may similarly ask when are ${M_{X}}$ and ${M_{E}}$ Banach spaces and Banach lattices? We start by showing a counter example that illustrates that ${M_{E}}$ is not necessarily a Banach space.

Given a vector (Banach) lattice X and a filtration (respectively contractive) $({E_{n}})$ on X, we can introduce order structure on the spaces ${M_{E}}$ and ${M_{X}}$ as follows. For two bounded $\mathcal{E}$-martingales (respectively X-martingales) $A=({x_{n}})$ and $B=({y_{n}})$, we write $A\geqslant B$ if ${x_{n}}\geqslant {y_{n}}$ for each n. With this order ${M_{E}}$ and ${M_{X}}$ are ordered vector spaces and the monotonicity of the norm follows from the monotonicity of the norm of X, i.e. for two $\mathcal{E}$-martingales (respectively X-martingales) with $0\leqslant A\leqslant B$, we have $\| A\| \leqslant \| B\| $. For two $\mathcal{E}$-martingales (respectively X-martingales) $A=({x_{n}})$ and $B=({y_{n}})$, one may guess that $A\vee B$ (or $A\wedge B$) can be computed by the formulas $A\vee B=({x_{n}}\vee {y_{n}})$ (or $A\wedge B=({x_{n}}\wedge {y_{n}})$). We show in the following theorem that this is in fact the case in order for ${M_{E}}$ to be a vector lattice. However, this is not obvious to show in the case of ${M_{X}}$.

Using the equivalence in Theorem 6, the following examples illustrate that ${M_{E}}$ is not always a vector lattice.

Under the pointwise order structure on ${M_{X}}$, for an X-martingale $A=({x_{n}})$, we can refer to Example 8 to show that the sequence $(|{x_{n}}|)$ is not necessarily an X-martingale. However, under certain assumptions, we can show that $(|{x_{n}}|)$ is an X-martingale for every X-martingale $A=({x_{n}})$ making ${M_{X}}$ a Banach lattice.