185 research outputs found
Numerical stability analysis of the Euler scheme for BSDEs
In this paper, we study the qualitative behaviour of approximation schemes
for Backward Stochastic Differential Equations (BSDEs) by introducing a new
notion of numerical stability. For the Euler scheme, we provide sufficient
conditions in the one-dimensional and multidimensional case to guarantee the
numerical stability. We then perform a classical Von Neumann stability analysis
in the case of a linear driver and exhibit necessary conditions to get
stability in this case. Finally, we illustrate our results with numerical
applications
A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations
The implementation of the convolution method for the numerical solution of
backward stochastic differential equations (BSDEs) introduced in [19] uses a
uniform space grid. Locally, this approach produces a truncation error, a space
discretization error, and an additional extrapolation error. Even if the
extrapolation error is convergent in time, the resulting absolute error may be
high at the boundaries of the uniform space grid. In order to solve this
problem, we propose a tree-like grid for the space discretization which
suppresses the extrapolation error leading to a globally convergent numerical
solution for the (F)BSDE. On this alternative grid the conditional expectations
involved in the BSDE time discretization are computed using Fourier analysis
and the fast Fourier transform (FFT) algorithm as in the initial
implementation. The method is then extended to higher-order time
discretizations of FBSDEs. Numerical results demonstrating convergence are also
presented.Comment: 28 pages, 8 figures; Previously titled 'Global convergence and
stability of a convolution method for numerical solution of BSDEs'
(1410.8595v1
Discrete-time approximation of multidimensional BSDEs with oblique reflections
In this paper, we study the discrete-time approximation of multidimensional
reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory
Related Fields 147 (2010) 89-121] and generalized by Hamad\`ene and Zhang
[Stochastic Process. Appl. 120 (2010) 403-426]. In comparison to the penalizing
approach followed by Hamad\`{e}ne and Jeanblanc [Math. Oper. Res. 32 (2007)
182-192] or Elie and Kharroubi [Statist. Probab. Lett. 80 (2010) 1388-1396], we
study a more natural scheme based on oblique projections. We provide a control
on the error of the algorithm by introducing and studying the notion of
multidimensional discretely reflected BSDE. In the particular case where the
driver does not depend on the variable , the error on the grid points is of
order , .Comment: Published in at http://dx.doi.org/10.1214/11-AAP771 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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