Numerical Stability Analysis of the Euler Scheme for BSDEs

In this paper, we study the qualitative behavior 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 cases to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver $f$ and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications.

[1]  Ying Hu,et al.  Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs , 1998 .

[2]  Dan Crisan,et al.  Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing , 2010, SIAM J. Financial Math..

[3]  G. Pagès,et al.  A quantization algorithm for solving multidimensional discrete-time optimal stopping problems , 2003 .

[4]  M. Kobylanski.  Backward stochastic differential equations and partial differential equations with quadratic growth , 2000 .

[5]  A. Richou.  Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition , 2011, 1111.5137.

[6]  B. Delyon,et al.  On the robustness of backward stochastic differential equations , 2002 .

[7]  Emmanuel Gobet,et al.  Error expansion for the discretization of backward stochastic differential equations , 2006, math/0602503.

[8]  D. Brigo,et al.  Nonlinear Valuation Under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes , 2014, 1404.7314.

[9]  Jianfeng Zhang.  A numerical scheme for BSDEs , 2004 .

[10]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[11]  M. Royer,et al.  BSDEs with a random terminal time driven by a monotone generator and their links with PDEs , 2004 .

[12]  Dan Crisan,et al.  RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS , 2014 .

[13]  Jean-François Chassagneux,et al.  Linear Multistep Schemes for BSDEs , 2014, SIAM J. Numer. Anal..

[14]  D. Crisan,et al.  Second order discretization of backward SDEs and simulation with the cubature method , 2014 .

[15]  CCP Cleared or Bilateral CSA Trades with Initial/Variation Margins Under Credit, Funding and Wrong-Way Risks: A Unified Valuation Approach , 2014, 1401.3994.

[16]  Arnaud Lionnet,et al.  Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs , 2013, 1309.2865.

[17]  Stéphane Crépey.  Bilateral Counterparty Risk Under Funding Constraints — Part I: Pricing , 2015 .

[18]  Dan Crisan,et al.  Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations , 2012 .

[19]  J. Chassagneux,et al.  Numerical simulation of quadratic BSDEs , 2013, 1307.5741.

[20]  Bernard Delyon,et al.  L p solutions of Backward Stochastic Dierential Equations , 2003 .

[21]  B. Bouchard,et al.  Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , 2004 .

[22]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[23]  É. Pardoux.  BSDEs, weak convergence and homogenization of semilinear PDEs , 1999 .

[24]  Stéphane Crépey,et al.  BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART II: CVA , 2015 .

[25]  S. Peng,et al.  Backward stochastic differential equations and quasilinear parabolic partial differential equations , 1992 .

[26]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[27]  Jin Ma.  Forward-backward stochastic differential equations and their applications in finance , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[28]  J. Chassagneux.  Linear multi-step schemes for BSDEs , 2013, 1306.5548.

[29]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[30]  J. Lepeltier,et al.  Existence for BSDE with superlinear–quadratic coefficient , 1998 .