Abstract
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
(H$\boldsymbol{h,m}$).
We make the following assumption on the coefficients m and h.
The function m is bounded: there exists a constant
The function h is continuous and satisfies the following growth property: there exists a constant
The functions h and m are Lipschitz continuous in their space variables uniformly in their time variable: there exist two constants
for all
Proposition A.1.
Suppose Assumptions (H
Moreover, w satisfies the following space regularity property:
for all
We first need the following lemma.
Lemma A.2.
Under Assumption (H
for
Proof.
Fix
for
Moreover, we have
which gives the result. ∎
Proof of Proposition A.1.
For
From [19, Theorem 1.1], we get existence and uniqueness of the solution to this BSDE for all
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
for
for
for
for
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
for
Proposition A.3.
Under Assumptions (H
Proof.
By a mollification argument, we can find regular functions
uniformly on compact sets.
Fix now
From (A.3) we get
From [20, Theorem 3.2], we have
where
From the uniqueness of solutions to Lipschitz BSDEs we get by applying Itô’s formula
Since
Therefore, we have
We then conclude using (A.4). ∎
Proposition A.4.
Under Assumptions (H
for all
Proof.
We take the same notations as in the proof of Proposition A.1. We fix
By a classical argument using (H
Then, using Proposition A.3, we have
for
From (A.2) we get
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