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

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Choice of Basis Functions ◮ Choice of basis functions? ◮ Examples in <strong>the</strong> literature: Laguerre polynomials, decreasing exponential functions, etc. ◮ Numerical study by Stentoft (2004) ◮ Few rigorous results on good choices ◮ Glasserman, Yu (2004): How many basis functions should one use?
Results of Glasserman <strong>and</strong> Yu (2004) ◮ N Monte Carlo paths, K basis functions ◮ Underlying process St is (geometric) Brownian motion ◮ Basis functions are Hermite polynomials (Brownian motion) H1(x), . . . , HK (x) or monomials (geometric Brownian motion) x 1 , . . . , x K . ◮ Investigate convergence of mean square error as N, K → ∞.
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: Convergence Results ◮ K basis fun
Page 17 and 18: 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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