Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Abstract

Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

Kriging metamodels and experimental design for Bermudan option pricing

“…This could be combined with an importance sampling strategy to implement the pathgeneration approach of standard LSM while generating a more efficient design. See [13,23,14] for various aspects of importance sampling for optimal stopping. Another variant would be a "double importance sampling" procedure of first generating a non-adaptive (say density-based) design, and then adding (non-sequentially) design points/paths in the vicinity of estimated ∂Ŝ t .…”

Three ways to solve partial differential equations with neural networks — A review

“…The authors of [6] applied their method to a number of examples including the heat equation, the Black‐Scholes option pricing equation and others with particular emphasis on the accurate and fast solution in high dimensions . Classical numerical approximation schemes for Kolmogorov partial differential equations are numerous, and include finite difference approximations [25,97,98], finite element methods [21,27,55,130], numerical schemes based on Monte‐Carlo methods [56,59,60,64], as well as approximations based on a discretization of the underlying stochastic differential equations (SDEs) [76,96]. Establishing a link of the proposed method to the classical approaches, which might be highly accurate and efficient in up to three dimensions, it shares also similarity to Monte‐Carlo methods since it relies on the connection between PDEs and SDEs in the form of the Feynman–Kac theorem and uses a discrete approximation of the SDE associated with equation ().…”

Variational Characterization of Free Energy: Theory and Algorithms

“…Since we can simulate the forward process and we know the functional dependence of (Y s , Z s ) on X s , the idea here is again to use the representation Equation (42) of γ in terms of a finite basis. It turns out that the coefficient vector α ∈ R N can be computed by solving a least-squares problem in every time step of the time-discretized backward SDE, which is why methods for solving an FBSDE like Equation (62) are termed least squares Monte Carlo; for the general approach we refer to [40,41]; details for the situation at hand will be addressed in a forthcoming paper [42].…”