suppose that we are given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with a filtration $(\mathcal F_t)_{t \in \mathcal T}$. I want to consider two different cases for the index set $\mathcal T$, namely
- $\mathcal{T}=\mathbb{N}$ (discrete case), and
- $\mathcal{T}=[0, \infty)$ (continuous case)
In both cases the definition of a martingale reads like this:
A stochastic process $X=(X_t)_{t \in \mathcal T}$ is called a martingale with respect to the filtration $(\mathcal F_t)_{t \in \mathcal T}$ if $X$ is integrable, adapted to $(\mathcal F_t)_{t \in \mathcal T}$ and satisfies the martingale property: \begin{align} \tag{1} \label{1} \mathbb{E}[X_t\mid \mathcal F_s] =X_s, \quad \text{for all} \ s\leq t, \ s,t \in \mathcal{T}. \end{align}
Now it is well known that in discrete case $\mathcal{T}=\mathbb{N}$ the martingale property is equivalent to the following: \begin{align} \tag{2} \label{2} \mathbb{E}[X_{t+1}\mid \mathcal F_t] =X_t, \quad \text{for all} \ t\in \mathcal{T}. \end{align} This easily follows with the tower property of conditional expectations.
However in continuous time this does not work anymore: if $\mathcal{T}=[0,\infty)$ then $(\ref{2})$ is strictly weaker than $(\ref{1})$.
Edit: I changed the index set in (\ref{2})!! Here comes my question: do you know an example of a stochastic process $X=(X_t)_{t \in [0, \infty)}$ which is integrable, adapted to a filtration and satisfies $(\ref{2})$ but is not a martingale? I tried to, but didn't get very far. Thank you.