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A Nonintrusive Stratified Resampler for Regression Monte Carlo: Application to Solving Nonlinear Equations

Emmanuel Gobet, Gang Liu, Jorge P. Zubelli

2018
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SIAM Journal on Numerical Analysis
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Our goal is to solve certain dynamic programming equations associated to a given 5 Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that 6 the model for X is not known in full detail and only a root sample X 1 , . . . , X M of such process 7 is available. By a stratification of the space and a suitable choice of a probability measure ν, we 8 design a new resampling scheme that allows to compute local regressions (on basis functions) in each 9 stratum.
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... in each 9 stratum. The combination of the stratification and the resampling allows to compute the solution to 10 the dynamic programming equation (possibly in large dimensions) using only a relatively small set 11 of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates 12 in L 2 (ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root 13 paths. 14 Key words. discrete Dynamic Programming Equations, empirical regression scheme, resam-15 pling methods, small-size sample 16 AMS subject classifications. 62G08, 62G09, 93Exx 17 2 . 32 33 This Regression Monte Carlo methodology has been investigated in [9] to solve Back-34 ward Stochastic Differential Equations associated to semi-linear partial differential 35 equations (PDEs) [16], with some tight error estimates. Generally speaking, it is well 36 known that the number of simulations M has to be much larger than the dimension 37 of the vector space L and thus the number of coefficients we are seeking. 38 * This work is part of the Chair Financial Risks of the Risk Foundation, the Finance for Energy Market Research Centre and the ANR project CAESARS (ANR-15-CE05-0024). 79 stratum H k , and averaged over all the strata. The statistical error is bounded 80 by C/M with a constant C which does not depend on the number of strata: 81 only relatively small M is necessary to get low statistical errors. This is in 82 agreement with the motivation that the root sample has a relatively small 83 size. The interdependency error is an unusual issue, it is related to the strong 84 dependency between regression problems (because they all use the same root 85 sample). The analysis as well as the framework are original. The error es-86 timates take different forms according to the problem at hand (Section 3 or 87 This manuscript is for review purposes only.

doi:10.1137/16m1066865
fatcat:uvmlx5dxdfa6dal56znc3lbwzm