The setting :
Let $(\Omega,\mathcal G, P)$ be a filtered space.
Let $X$ be a (continuous) stochastic process and $\mathbb F$ the filtration generated by $X$.
For every $n\geq 0$, let $\pi^n$ a mesh on $[0,T]$ and $X^n$ a discretization of $X$ (assume something like that $X^n_t=\sum_{t_k\in\pi^n} X_{t_k} 1_{t\in[t^n_k,t^n_{k+1}[}$).
Then I build an increasing sequence of filtrations $(\{\mathcal F^n_t\}_{t\geq0})_{n\geq 0}$ each one generated by $X^n$.
Now, I have two $\mathbb F$-adapted processes $L$ and $U$ and their versions on $\mathbb F^n$ given by $L^n_t=E(L_t\mid\mathcal F^n_t)$ and $U^n_t=E(U_t\mid\mathcal F^n_t)$ for all $t\in [0,T]$ a.s.
My questions are :
- Does it exists an increasing sequence $(\pi^n)_n$ such that $L^n_t\leq L^{n+1}_t$ and $U^n_t\leq U^{n+1}_t$ for all $t\in [0,T]$ a.s.?
- or a subsequence $\psi^n\rightarrow \infty$ such that we have the aformentionned property on $\pi^{\psi^n}$?
- If no, if we assume $L^n$ converges (assume $S^2$) to $L$ and $U^n$ to $U$, could we find two alternative sequences $\tilde L^n,\tilde U^n$ increasing (extracted from $L^n$ and $U^n$), converging to $L$ and $U$ respectively ?
- If no, are there some conditions to add to get it ? (weak convergence etc)