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Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

Idris Kharroubi, Thomas Lim and Xavier Warin


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 t¯,t¯[0,T] and we consider a PDE of the form



We make the following assumption on the coefficients m and h.

  1. (i)

    The function m is bounded: there exists a constant Mm such that

    |m(x)|Mmfor all xd.
  2. (ii)

    The function h is continuous and satisfies the following growth property: there exists a constant Mh such that

    |h(t,x,y,z))|Mh(1+|y|+|z|)for all t[0,T]xdy and zd.
  3. (iii)

    The functions h and m are Lipschitz continuous in their space variables uniformly in their time variable: there exist two constants Lh and Lm such that


    for all t[0,T], x,xd, y,y and z,zd.

Proposition A.1.

Suppose Assumptions (Hb,σ) and (Hh,m) hold. The PDE (A.1) admits a unique viscosity solution w with polynomial growth: there exist an integer p1 and a constant C such that


Moreover, w satisfies the following space regularity property:


for all t[t¯,t¯] and x,xRd.

We first need the following lemma.

Lemma A.2.

Under Assumption (Hb,σ), we have the following estimate:


for t,t[t¯,t¯] and x,xRd.


Fix t,t[t¯,t¯] such that tt and x,xd. From Itô’s formula and (Hb,σ), we have


for s[t,t¯]. By Gronwall’s Lemma we get


Moreover, we have


which gives the result. ∎

Proof of Proposition A.1.

For (t,x)[t¯,t¯]×d, we introduce the following BSDE: find (𝒴t,x,𝒵t,x)𝐒[t,t¯]2×𝐇[t,t¯]2 such that


From [19, Theorem 1.1], we get existence and uniqueness of the solution to this BSDE for all (t,x)[t¯,t¯]×d. From [19, Theorem 2.2] and [21, Theorem 5.1], the function w defined by


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 t[t¯,t¯] and x,xd. By Itô’s formula we have


for s[t,t¯]. Using the Lipschitz properties of h and m and the Young inequality, we get


for s[t,t¯]. Since (x24+x+1)x2 for x2, we get


for s[t,t¯]. Then, using (A.2), we get


for s[t,t¯]. 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 (𝒴t,x,𝒵t,x)𝐒[t,t¯]2×𝐇[t,t¯]2 denotes the solution to


for (t,x)[t¯,t¯]×d.

Proposition A.3.

Under Assumptions (Hb,σ) and (Hh,m), the process Zt,x satisfies

|𝒵t,x|Mb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12ddt-a.e.on Ω×[t,t¯].


By a mollification argument, we can find regular functions bn and σn satisfying (Hb,σ), with same constants as b, σ, hn and mn satisfying (Hh,m) with same constants as h and m for n1 such that


uniformly on compact sets. Fix now (t,x)[t¯,t¯]×d and denote by (Xt,x,n,𝒴t,x,n,𝒵t,x,n)𝐒[t,t¯]2×𝐒[t,t¯]2×𝐇[t,t¯]2 the solution to


From (A.3) we get


From [20, Theorem 3.2], we have


where wn is a regular solution to

{-twn-wn-hn(,wn,σnDwn)=0on [t¯,t¯)×d,wn(t¯,)=mnon d.

From the uniqueness of solutions to Lipschitz BSDEs we get by applying Itô’s formula


Since σn, mn and hn satisfy (Hh,m), we get from Proposition A.1


Therefore, we have

|𝒵st,x,n|Mb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12ddt-a.e.on Ω×[t,t¯].

We then conclude using (A.4). ∎

Proposition A.4.

Under Assumptions (Hb,σ) and (Hh,m) the unique viscosity solution with linear growth w given in (A.1) satisfies the following time regularity property:


for all t,t[t¯,t¯] and xRd.


We take the same notations as in the proof of Proposition A.1. We fix t,t[t¯,t¯] such that tt and xd. We have


By a classical argument using (Hh,m), Young’s inequality and Gronwall’s Lemma we have


Then, using Proposition A.3, we have


for s[t,t]. From the regularity with respect to the variable x given in Proposition A.1 we get


From (A.2) we get



[1] G. Barles, Solutions de Viscosité des Équations de Hamilton–Jacobi, Math. Appl. (Berlin) 17, Springer, Paris, 1994. Search in Google Scholar

[2] S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, J. Mach. Learn. Res. 20 (2019), Paper No. 74. Search in Google Scholar

[3] Y. Z. Bergman, Option pricing with differential interest rates, Rev. Financ. Stud. 8 (1995), 475–500. 10.1093/rfs/8.2.475Search in Google Scholar

[4] B. Bouchard, R. Elie and L. Moreau, Regularity of BSDEs with a convex constraint on the gains-process, Bernoulli 24 (2018), no. 3, 1613–1635. 10.3150/16-BEJ806Search in Google Scholar

[5] M. Broadie, J. Cvitanić and H. M. Soner, Optimal replication of contingent claims under portfolio constraints, Rev. Financ. Stud. 11 (1998), 59–79. 10.1093/rfs/11.1.59Search in Google Scholar

[6] Q. Chan-Wai-Nam, J. Mikael and X. Warin, Machine learning for semi-linear PDEs, J. Sci. Comput. 79 (2019), no. 3, 1667–1712. 10.1007/s10915-019-00908-3Search in Google Scholar

[7] 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

[8] 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

[9] 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

[10] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Math. Stud. 29, North-Holland, Amsterdam, 1978. Search in Google Scholar

[11] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1–71. 10.1111/1467-9965.00022Search in Google Scholar

[12] 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

[13] E. Gobet and J.-P. Lemor, Numerical simulation of BSDEs using empirical regression methods: Theory and practice, preprint (2008), Search in Google Scholar

[14] K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989), 359–366. 10.1016/0893-6080(89)90020-8Search in Google Scholar

[15] 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

[16] C. Huré, H. Pham and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp. 89 (2020), no. 324, 1547–1579. 10.1090/mcom/3514Search in Google Scholar

[17] C. Huré, H. Pham and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp. 89 (2020), no. 324, 1547–1579. 10.1090/mcom/3514Search in Google Scholar

[18] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, preprint (2014), Search in Google Scholar

[19] 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

[20] 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

[21] 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

[22] 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

[23] 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

[24] R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, 1970. 10.1515/9781400873173Search in Google Scholar

Received: 2020-08-20
Revised: 2020-11-25
Accepted: 2020-12-21
Published Online: 2021-01-15
Published in Print: 2021-03-01

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