Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$.
- $\xi_i$ be a real-valued random variable on $(\Omega,\mathcal A)$ and $$H_t:=\sum_{i=1}^n\xi_{i-1}1_{(t_{i-1},t_i]}(t)\;\;\;\text{for }t\ge 0$$ for some $0=t_0<\cdots<t_n$
- $M=(M_t)_{t\ge 0}$ be a $\mathbb F$-martingale and $$(H\cdot M)_t:=\sum_{i=1}^n\xi_{i-1}\left(M_{t_i\wedge t}-M_{t_{i-1}\wedge t}\right);\;\;\text{for }t\ge 0$$
I want $(H\cdot M)_{t\ge 0}$ to be a $\mathbb F$-martingale.
Let's take a look which further assumptions we need: If $t\in\left(t_{j-1},t_j\right]$, then $$(H\cdot M)_t=\sum_{i=1}^{j-1}\xi_{i-1}\left(M_{t_i}-M_{t_{i-1}}\right)+\xi_{j-1}\left(M_t-M_{t_{j-1}}\right)\;.\tag 1$$ So, it seems like we need $\xi_{i-1}$ to be $\mathcal F_{t_i}$-adapted. That's interesting, cause my textbook states, that we even need to assume $\mathcal F_{t_{i-1}}$-adaptedness. Why? I don't think that we need this here.
Which further assumptions on the $\xi_i$ do we need to make such that $(H\cdot M)_t$ is integrable?