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 $f$ and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications

A comparison principle for PDEs arising in approximate hedging problems: application to Bermudan options

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard, Elie and Touzi in [1] and is known to solve an Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in [2]. [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et al. Bsdes with weak terminal condition. The Annals of Probability, 43(2):572-604,2015

Discrete-time approximation of doubly reflected BSDEs

We study the discrete time approximation of doubly reflected BSDEs in a multidimensional setting. As in Ma and Zhang (2005) or Bouchard and Chassagneux (2006), we introduce the discretely reflected counterpart of these equations. We then provide representation formulae which allow us to obtain new regularity results. We also propose an Euler scheme's type approximation and give new convergence results for both discretely and continuously reflected BSDEs

A backward dual representation for the quantile hedging of Bermudan options

Within a Markovian complete financial market, we consider the problem of hedging a Bermudan option with a given probability. Using stochastic target and duality arguments, we derive a backward numerical scheme for the Fenchel transform of the pricing function. This algorithm is similar to the usual American backward induction, except that it requires two additional Fenchel transformations at each exercise date. We provide numerical illustrations

An optimal transport approach for the multiple quantile hedging problem

We consider the multiple quantile hedging problem, which is a class of partial hedging problems containing as special examples the quantile hedging problem (F{\"o}llmer \& Leukert 1999) and the PnL matching problem (introduced in Bouchard \& Vu 2012). In complete non-linear markets, we show that the problem can be reformulated as a kind of Monge optimal transport problem. Using this observation, we introduce a Kantorovitch version of the problem and prove that the value of both problems coincide. In the linear case, we thus obtain that the multiple quantile hedging problem can be seen as a semi-discrete optimal transport problem, for which we further introduce the dual problem. We then prove that there is no duality gap, allowing us to design a numerical method based on SGA algorithms to compute the multiple quantile hedging price