143 research outputs found
Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection
We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result
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
Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles
We study a discrete time approximation scheme for the solution of a doubly
reflected Backward Stochastic Differential Equation (DBBSDE in short) with
jumps, driven by a Brownian motion and an independent compensated Poisson
process. Moreover, we suppose that the obstacles are right continuous and left
limited (RCLL) processes with predictable and totally inaccessible jumps and
satisfy Mokobodski's condition. Our main contribution consists in the
construction of an implementable numerical sheme, based on two random binomial
trees and the penalization method, which is shown to converge to the solution
of the DBBSDE. Finally, we illustrate the theoretical results with some
numerical examples in the case of general jumps
Minimal supersolutions of convex BSDEs
We study the nonlinear operator of mapping the terminal value to the
corresponding minimal supersolution of a backward stochastic differential
equation with the generator being monotone in , convex in , jointly lower
semicontinuous and bounded below by an affine function of the control variable
. We show existence, uniqueness, monotone convergence, Fatou's lemma and
lower semicontinuity of this operator. We provide a comparison principle for
minimal supersolutions of BSDEs.Comment: Published in at http://dx.doi.org/10.1214/13-AOP834 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Reflected Backward Stochastic Difference Equations and Optimal Stopping Problems under g-expectation
In this paper, we study reflected backward stochastic difference equations
(RBSDEs for short) with finitely many states in discrete time. The general
existence and uniqueness result, as well as comparison theorems for the
solutions, are established under mild assumptions. The connections between
RBSDEs and optimal stopping problems are also given. Then we apply the obtained
results to explore optimal stopping problems under -expectation. Finally, we
study the pricing of American contingent claims in our context.Comment: 29 page
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'
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