Optimal stopping and embedding (Q2725312)

By using Skorokhod embedding techniques, the authors estimate the rate of convergence of the approximation of an optimal stopping problem along Brownian paths, when Brownian motion \((B_t)_{t\geq 0}\) is approximated by a normalized random walk \((S_n)_{n\in\mathbb{N}}\) which is represented by the form \(S_n= \sum^n_{k=1} X_k\). Let \({\mathcal T}\) be the set of all \(F\)-stopping times, where \(F= ({\mathcal F}_t)_{t\geq 0}\) is the natural filtration of \((B_t)_{t\geq 0}\), and set \({\mathcal T}_{0,1}= \{\tau\in {\mathcal T}\mid 0\leq \tau\leq 1\) a.s.\}, and let \({\mathcal T}^S\) be the set of all stopping times with respect to the natural filtration of \((S_n)_{n\in\mathbb{N}}\) and set \({\mathcal T}^S_{0,n}= \{\nu\in{\mathcal T}^S\mid 0\leq \nu\leq n\) a.s.\}. For a continuous bounded function \(f\) on \([0,1]\times\mathbb{R}\), define optimal values by \(P= \sup_{\tau\in{\mathcal T}_{0,1}}Ef(\tau, B_\tau)\) and \(P^{(n)}= \sup_{\nu\in{\mathcal T}^S_{0, n}} Ef(\nu/n, S_\nu/\sqrt n)\). Then the authors show that if \(E[X^4_k]< \infty\) and \(f\) satisfies some regularity conditions, then \(P^{(n)}- P= O(1/\sqrt n)\).