Brownian optimal stopping and random walks
The author considers the optimal stopping problem \[ P=\sup_{\tau\in {\mathcal T}_{0,T}} { E}(e^{-r\tau}f(\mu\tau+B_\tau)), \] where \((B_t)_{0\leq t\leq T}\) is Brownian motion on a bounded interval \([0,T]\), \(f\) a bounded continuous function, \(r\) and \(\mu\) real constants, and \({\mathcal T}_{0,T}\) denotes the set of all stopping times of the natural filtration of \((B_t)_{t\geq 0}\), with values in \([0,T]\). One way to compute \(P\) is to approximate the Brownian motion \((B_t)_{0\leq t\leq T}\) by a random walk \[ B^{(n)}_t=\sqrt{{T}\over{n}} \sum_{k=1}^{[nt/T]} X_k, \qquad 0\leq t\leq T, \] where \((X_n)_{n\geq 1}\) are i.i.d. random variables with \({E}(X^2_n)=1\), \({E}(X_n)=0\), and to apply dynamic programming to compute \[ P^{(n)}=\sup_{\tau\in {\mathcal T}^{(n)}_{0,T}} { E}(e^{-r\tau}f(\mu\tau+B^{(n)}_\tau)). \] The author derives error estimates for this type of approximation under various assumptions on the differentiability of \(f\) and the distribution of the \(X_n\).
scientific article; zbMATH DE number 1761092