Convergence of submartingales to an increasing process under discretization of filtrations (Q2722492)

Let \(Y=(Y_t\), \(t\in [0,T])\) be some right continuous and admitting left limits process, \(Y^n\) be its discretization under the sequence of refining partitions \(\pi_n\) of \([0,T]\), \(\mathcal{F}_t=\sigma(Y_s, s\leq t)\), \(\mathcal{F}_t^n=\sigma(Y_s^n, s\leq t)\). The paper states the following result: let \(X\) be an \(\mathcal{F}\)-adapted, continuous, increasing process, \(X_0=0\) and \(X_T\) be square-integrable. Denote \(X^n= E[X_{.}\mid \mathcal{F}^n_{.}]\). Then (a) \(X^n\) is a positive submartingale, with the canonical decomposition \(X^n=M^n+A^n\), where \(M^n\) is a square-integrable \(\mathcal{F}^n\)-martingale of pure jumps, and \(A^n\) is a continuous increasing \(\mathcal{F}^n\)-adapted process, such that \(A^n_T\) is square integrable. (b) The following convergences hold for the uniform topology in \(\mathbb{D}\): \(X^n \rightarrow X, M^n \rightarrow 0\), \(A^n \rightarrow X\). The tightness of the family \(M^n\) is proved by using Aldous criterion.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].