Journal ArticleDOI
A sparse-grid method for multi-dimensional backward stochastic differential equations *
TLDR
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme and the conditional mathematical expectations derived from the original equation are approximated using sparse-Grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse- grid interpolation.Abstract:
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our sread more
Citations
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Journal ArticleDOI
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations
TL;DR: This work proposes a new method for solving high-dimensional fully nonlinear second-order PDEs and shows the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 200-dimensional G-Brownian motion.
Journal ArticleDOI
Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
TL;DR: In this paper, the authors proposed a new method for solving high-dimensional fully nonlinear second-order partial differential equations (PDE), which can be used to sample from highdimensional nonlinear expectations.
Journal ArticleDOI
A primal–dual algorithm for bsdes
TL;DR: In this paper, the primal-dual methodology is generalized to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs).
Journal ArticleDOI
Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs
TL;DR: In this paper, the error analysis of the time discretization of systems of Forward Backward Stochastic Dierential Equations with drivers having polynomial growth and that are also monotone in the state variable is performed.
Journal ArticleDOI
Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs
Yu Fu,Weidong Zhao,Tao Zhou +2 more
TL;DR: Zhao et al. as discussed by the authors proposed the sparse grid spatial discretization, where the Gaussian-Hermite quadrature rule is used to approximate the conditional expectations and for the associated high-dimensional interpolations, they adopt a spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use sparse grid approximations to recover the expansion coefficients.
References
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Journal ArticleDOI
Adapted solution of a backward stochastic differential equation
TL;DR: In this paper, the authors considered the problem of finding an adapted pair of processes with values in Rd and Rd×k, respectively, which solves an equation of the form: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∪ t 1 [g(m, x, s, g(m)) + y(m)] dW s = X.
Journal ArticleDOI
Backward Stochastic Differential Equations in Finance
TL;DR: In this article, different properties of backward stochastic differential equations and their applications to finance are discussed. But the main focus of this paper is on the theory of contingent claim valuation, especially cases with constraints.
Journal ArticleDOI
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
TL;DR: A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data.
Journal ArticleDOI
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
TL;DR: This work demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates, indicating for which problems the sparse grid stochastic collocation method is more efficient than Monte Carlo.
Journal ArticleDOI
Numerical integration using sparse grids
Thomas Gerstner,Michael Griebel +1 more
TL;DR: The usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction is suggested and their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules is shown.
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