Lévy-Sheffer Systems and the Longstaff-Schwartz Algorithm for ...

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Results of Glasserman and Yu (2004) ◮ Simple models, but precise results ◮ Assume that there is an exact representation Cn(x) = K k=0 βnkψnk(x) = β T n ψn(x). ◮ Estimate mean square error MSE = E[|β − ˆβ| 2 ] ◮ Some simplifying assumptions ◮ In practice, the convergence behavior will be even worse
Results of Glasserman and Yu (2004) and SG (2008) ◮ What is the highest K for which MSE → 0 as N, K → ∞? ◮ Geometric Brownian motion: √ log N (Gl., Yu 2004) ◮ Brownian motion: log N (Gl., Yu 2004) ◮ Geometric Poisson process: log N/ log log N (SG 2008) ◮ Geometric Gamma process: log N/ log log N (SG 2008) ◮ Geometric Pascal process: log N/ log log N (SG 2008) ◮ Geometric Meixner process: log N/ log log N (SG 2008)
Page 1 and 2: Lévy-Sheffer Systems and the Longs
Page 3 and 4: Motivation ◮ Many financial instr
Page 5 and 6: Algorithms ◮ An optimal stopping
Page 7 and 8: Dynamic Programming ◮ Vn(x) = val
Page 9 and 10: The Longstaff-Schwartz Algorithm
Page 11 and 12: The Longstaff-Schwartz Algorithm
Page 13 and 14: Convergence Results ◮ K basis fun
Page 15: Results of Glasserman and Yu (2004)
Page 19 and 20: The Models ◮ Geometric Poisson pr
Page 21 and 22: Properties of the Meixner Process
Page 23 and 24: Lévy-Sheffer Systems ◮ Sheffer S
Page 25 and 26: Examples of Lévy-Sheffer Systems
Page 27 and 28: Convergence Results (SG 2008) ◮ N
Page 29 and 30: Proof Ingredients of the Convergenc
Page 31: References ◮ J.F. Carriere: Valua

Lévy-Sheffer Systems and the Longstaff-Schwartz Algorithm for ...

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