Difference betweem martingale property and adapted filteration

Question

What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct examples when they are the same and when they differ.

Answer 1

Let's consider a random process $X$. If $X$ is an adapted process, then we know, without any uncertainty, what its value is at the present time. This idea is formalized with measure theory.

For $X$ to be a martingale, it needs to have the following property: at any given time, our best estimate of the value at some point in the future (i.e. forecast), is the present value. In other words, the present and previous values never make us believe there's a local trend in time.

Implicit in this belief is the idea that we know what the present value is. In other words, every martingale is necessarily adapted for the concept to make sense.

Examples:

1. Let $Y_t = t$ be deterministic and increasing. It is adapted but not a martingale. This is the only real counterexample, since martingale implies adapted. But one other.
2. Let $\Omega = \{a,b\}$. let $t \in [0,1,2]$, and let $Y_0 = 0, Y_1(a) = 1, Y_2(a) = 0, Y_1(b) = -1, Y_2(b) = 0$. Let the filtration be the one generated by $Y$, so that $Y$ is by default adapted. But, $Y$ is not a martingale.

Answer 2

A random process that is adapted to a filtration is measurable (ie X_t is F_t-measurable) but not necessarily a martingale. X_t is a martingale if E(X_t | F_s) = X_s for s < t.

Answer 3

For a stochastic process $\left(X_{t}\right)$ to be adopted to a filtration $\left(\mathcal{F}_{t}\right)_{t\in T}$ the random variable $X_{t}$ must be $\mathcal{F}_{t}$-measurable for each $t\in T$. A stochastic process is a collection of random variables $X_{t}$, indexed by some set $T$. Each random variable is a mapping from a probability space into a measurable space. A measurable space is a set together with a collection of subsets of this set which fulfill certain properties. This collection is called a $\sigma$-algebra. It is nearly the same as a topology, but for a slight technical difference (countable vs finite intersection of sets belong to the collection). A topology is essentially a collection of open sets. For two topological space one can define a mapping from one into the other. If this function respects the openness of sets, i.e. the preimage of an open set is open, we say this mapping is continuous. In much the same light a $\sigma$-algebra includes (sub)sets which are measurable, i.e. it makes sense to give that subset a measure (a volume). Measurable functions are functions where the preimage of a measurable set is measurable. In Probability Theory we call a measurable function a random variable.

As a stochastic process is a collection of (different) random variables, i.e. measurable functions, a filtration is the sequence of the different $\sigma$-algebras of these random variables. So a stochastic process is adapted to a filtration, if this filtration makes every single random variable measurable according to its $\sigma$-algebra. I hope this was not too technical.

On the other hand a martingale is a stochastic process with the martingale property: $$ \textrm{for all}\space s,t\in T \textrm{ with } s\leq t\textrm{ it holds that }\mathbb{E}_{\mathbb{P}}\left[X_{t}\mid\mathcal{F}_{s}\right] = X_{s}\space\left(\mathbb{P}\textrm{-almost surely}\right) $$ This defining property depends on the probability measure $\mathbb{P}$ and the filtration $\left(\mathcal{F}_{t}\right)$. So a stochastic process is only a martingale with respect to a filtration $\left(\mathcal{F}_{t}\right)$ and a probability measure $\mathbb{P}$ if it fulfills the martingale property.

So the adapted filtration is a building block of the definition of a martingale.