We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2005, Vol. 15, No. 3, 2172-2202. This reprint differs from the original in pagination and typographic detail. 1 2 E. GOBET, J.-P. LEMOR AND X. WARINApplications of BSDEs. Such equations, first studied by Pardoux and Peng [26] in a general form, are important tools in mathematical finance. We mention some applications and refer the reader to [10,12] for numerous references. In a complete market, for the usual valuation of a contingent claim with payoff Φ(S), Y is the value of the replicating portfolio and Z is related to the hedging strategy. In that case, the driver f is linear w.r.t. Y and Z. Some market imperfections can also be incorporated, such as higher interest rate for borrowing [4]: then, the driver is only Lipschitz continuous w.r.t. Y and Z. Related numerical experiments are developed in Section 6. In incomplete markets, the Föllmer-Schweizer strategy [14] is given by the solution of a BSDE. When trading constraints on some assets are imposed, the super-replication price [13] is obtained as the limit of nonlinear BSDEs. Connections with recursive utilities of Duffie and Epstein [11] are also available. Peng has introduced the notion of g-expectation (here g is the driver) as a nonlinear pricing rule [28]. Recently he has shown [27] the deep connection between BSDEs and dynamic risk measures, proving that any dynamic risk measure (E t ) 0≤t≤T (satisfying some axiomatic conditions) is necessarily associated to a BSDE (Y t ) 0≤t≤T (the converse being known for years). The least we can say is that BSDEs are now inevitable tools in mathematical finance. Another indirect application may concern variance reduction techniques for the Monte Carlo computations of expectations, say E(Φ) taking f ≡ 0. Indeed, T 0 Z s dW s is the so-called martingale control variate (see [24], for instance). Finally, for applications to semi-linear PDEs, we refer to [25], among others.The mathematical analysis of BSDE is now well understood (see [23] for recent references) and its numerical resolution has made recent progresses. However, even if several numerical methods have been proposed, they suffer of a high complexity in terms of computational time or are very costly in terms of computer memory. Thus, their uses in practice on real problems are difficult. Hence, it is still topical to devise more efficient algorithms. This article contributes in this direction by developing a simple approach, based on Monte Carlo regression on function bases. It is in the vein of the general regressi...

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [12], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics. * We are grateful to Mete Soner, Bruno Bouchard, Denis Talay, and Frédéric Bonnans for fruitful comments and suggestions. †

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in [EHJ17] when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minima as it can be the case with the algorithm designed in [EHJ17], and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

The aim of this paper is to discuss efficient algorithms for the pricing of American options by two recently proposed Monte-Carlo type methods, namely the Malliavian calculus and the regression based approaches. We explain how both technics can be exploded with improved complexity and efficiency. We also discuss several technics for the estimation of the corresponding hedging strategies. Numerical tests and comparisons, including the quantization approach, are performed.

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We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrödinger equations.

We propose a new numerical scheme for Backward Stochastic Differential Equations based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm.

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