We study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments.
A Regularity estimates on solutions to parabolic semi-linear PDEs
We recall in this appendix an existence and uniqueness results for viscosity solution to semi-linear PDEs. We also give a regularity property with an explicit form for the Lipschitz and Hölder constants. Although, this regularity is classical in PDE theory, we choose to provide such a result as we did not find any explicit mention of the dependence of the regularity coefficient in the literature.
We fix and we consider a PDE of the form
We make the following assumption on the coefficients m and h.
The function m is bounded: there exists a constant such that
The function h is continuous and satisfies the following growth property: there exists a constant such that
The functions h and m are Lipschitz continuous in their space variables uniformly in their time variable: there exist two constants and such that
for all , , and .
Suppose Assumptions (H) and (H) hold. The PDE (A.1) admits a unique viscosity solution w with polynomial growth: there exist an integer and a constant C such that
Moreover, w satisfies the following space regularity property:
for all and .
We first need the following lemma.
Under Assumption (H), we have the following estimate:
for and .
Fix such that and . From Itô’s formula and (H), we have
for . By Gronwall’s Lemma we get
Moreover, we have
which gives the result. ∎
Proof of Proposition A.1.
For , we introduce the following BSDE: find such that
is continuous and is the unique viscosity solution to (A.1) with polynomial growth.
We now turn to the regularity estimate. We first check the regularity with respect to the variable x.
Fix and . By Itô’s formula we have
for . Using the Lipschitz properties of h and m and the Young inequality, we get
for . Since for , we get
for . Then, using (A.2), we get
for . Since w has polynomial growth, we can apply Gronwall’s Lemma and we get
Therefore, we get
In this last result we prove that under our assumptions the Z component of a solution to a BSDE is bounded. We recall that denotes the solution to
Under Assumptions (H) and (H), the process satisfies
By a mollification argument, we can find regular functions and satisfying (H), with same constants as b, σ, and satisfying (H) with same constants as h and m for such that
uniformly on compact sets. Fix now and denote by the solution to
From (A.3) we get
From [20, Theorem 3.2], we have
where is a regular solution to
From the uniqueness of solutions to Lipschitz BSDEs we get by applying Itô’s formula
Since , and satisfy (H), we get from Proposition A.1
Therefore, we have
We then conclude using (A.4). ∎
Under Assumptions (H) and (H) the unique viscosity solution with linear growth w given in (A.1) satisfies the following time regularity property:
for all and .
We take the same notations as in the proof of Proposition A.1. We fix such that and . We have
By a classical argument using (H), Young’s inequality and Gronwall’s Lemma we have
Then, using Proposition A.3, we have
for . From the regularity with respect to the variable x given in Proposition A.1 we get
From (A.2) we get
 G. Barles, Solutions de Viscosité des Équations de Hamilton–Jacobi, Math. Appl. (Berlin) 17, Springer, Paris, 1994. Search in Google Scholar
 S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, J. Mach. Learn. Res. 20 (2019), Paper No. 74. Search in Google Scholar
 J.-F. Chassagneux, R. Elie and K. Idris, A numerical probabilistic scheme for super-replication with convex constraints on the Delta, in preparation. Search in Google Scholar
 M. G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. 10.1090/S0273-0979-1992-00266-5Search in Google Scholar
 J. Cvitanić, I. Karatzas and H. M. Soner, Backward stochastic differential equations with constraints on the gains-process, Ann. Probab. 26 (1998), no. 4, 1522–1551. Search in Google Scholar
 C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Math. Stud. 29, North-Holland, Amsterdam, 1978. Search in Google Scholar
 A. Géron, Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems, O’Reilly Media, Sebastopol, 2019. Search in Google Scholar
 K. Hornik, M. Stinchcombe and H. White, Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks, Neural Networks 3 (1990), 551–560. 10.1016/0893-6080(90)90005-6Search in Google Scholar
 E. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic Analysis and Related Topics, VI (Geilo 1996), Progr. Probab. 42, Birkhäuser, Boston (1998), 79–127. 10.1007/978-1-4612-2022-0_2Search in Google Scholar
 E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic Partial Differential Equations and Their Applications (Charlotte 1991), Lect. Notes Control Inf. Sci. 176, Springer, Berlin (1992), 200–217. 10.1007/BFb0007334Search in Google Scholar
 E. Pardoux, F. Pradeilles and Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 467–490. 10.1016/S0246-0203(97)80101-XSearch in Google Scholar
 S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type, Probab. Theory Related Fields 113 (1999), no. 4, 473–499. 10.1007/s004400050214Search in Google Scholar
 H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Stoch. Model. Appl. Probab. 61, Springer, Berlin, 2009. 10.1007/978-3-540-89500-8Search in Google Scholar
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