The Longstaff--Schwartz algorithm for L\'{e}vy models: Results on fast and slow convergence

We investigate the Longstaff--Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090--2119] and Stentoft [Manag. Sci. 50 (2004) 1193--1203] to several L\'{e}vy models, in particular the geometric Meixner model. A convenient setting to analyze this convergence problem is provided by the L\'{e}vy--Sheffer systems introduced by Schoutens and Teugels.