Stability of conditional median under discretization of filtrations (Q1852831)

Let \(X\) be a random vector in \(\mathbb{R}^k\) and \((Y_s)_{s\leq 1}\) be a stochastic process. The best approximations of \(X\) knowing \((Y_s)_{s\leq t}\) for the \(L_p\)-criterion are the conditional \(L_p\)-medians of \(X\) given \((Y_s)_{s \leq t}\), i.e. the (random) points in \(\mathbb{R}^k\) that minimize the function \[ \alpha \mapsto\varphi (\alpha,t) (\omega)=\int \|x-\alpha\|^p P_{X\mid {\mathcal F}_t} (dx,\omega), \] where \(({\mathcal F}_s)_{s\leq 1}\) is the filtration generated by \((Y_s)_{s\leq 1}\). Now suppose that \((Y_s)_{s\leq 1}\) is only known on a finite set of times. It is proven that, under the right assumptions, if the number of these observation times tends to infinity, one rediscovers assymptotically the set of best approximations for the \(L_p\)-minimizing criterion.