In different books/handnotes I read different types of definition for simple processes.
The first one (that I've learned and intuitive for me) is the following: For $0\leq t_{0}<t_{1}<\ldots<t_{n}$
$$\varphi^{\left(1\right)}\left(t\right)=\sum_{i=0}^{n-1}\varphi_{t_{i}}\cdot\chi_{\left[t_{i},t_{i+1}\right)}\left(t\right),$$
which is a right continous process (with some additional adaptation criteria).
The second definition that I come across a lot of time is: For $0\leq t_{0}<t_{1}<\ldots<t_{n}$
$$\varphi^{\left(2\right)}\left(t\right)=\sum_{i=0}^{n-1}\varphi_{t_{i}}\cdot\chi_{\left(t_{i},t_{i+1}\right]}\left(t\right),$$
which is a left continous process.
The $I$ integral of $\varphi^{\left(1\right)}$ is usually defined as
$$I\left(t\right)=\sum_{i=0}^{n-1}\varphi_{t_{i}}\left(W_{t_{i+1}\wedge t}-W_{t_{i}\wedge t}\right),$$
where $W$ is an appropriate Wiener process. I find this definition understandable.
However, for the second case I don't understand how we can define the $I$ integral of $\varphi^{\left(2\right)}$ with the same way:
$$I\left(t\right)=\sum_{i=0}^{n-1}\varphi_{t_{i}}\left(W_{t_{i+1}\wedge t}-W_{t_{i}\wedge t}\right).$$
This is not intuitive for me at all. For example $\left[0,T\right]=\left[0,2\right]$. For $t\in\left[0,1\right]$ $\varphi^{\left(2\right)}\left(t\right)=0$ and for $t\in\left(1,2\right]$ $\varphi^{\left(2\right)}\left(t\right)=1$. In this case $\varphi^{\left(2\right)}$ indeed a simple process as defined according to the second definition of simple process and the integral at $t=1.5$ is
$$I\left(1.5\right)=0\cdot\left(W_{1}-W_{0}\right)+0\cdot\left(W_{1.5}-W_{1}\right)=0.$$
I would expect the integral to be
$$I\left(1.5\right)=1\cdot\left(W_{1.5}-W_{1}\right).$$
So when the integrand is defined as $\varphi^{\left(2\right)}$, then why don't we define the integral as
$$I\left(t\right)=\sum_{i=0}^{n-1}\varphi_{t_{i+1}}\left(W_{t_{i+1}\wedge t}-W_{t_{i}\wedge t}\right)?$$
I can't see any adaptation problem here, since at $t\in\left(t_{i},t_{i+1}\right]$ we already know what $\varphi_{t_{i+1}}$ is.