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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

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

(A.1){-tw(t,x)-w(t,x)-h(t,x,w(t,x),σ(t,x)Dw(t,x))=0,(t,x)[t¯,t¯)×d,w(t¯,x)=m(x),xd.

(H$\boldsymbol{h,m}$).

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

    |m(x)-m(x)|Lm|x-x|,
    |h(t,x,y,z)-h(t,x,y,z)|Lh(|x-x|+|y-y|+|z-z|)

    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

|w(t,x)|C(1+|x|p),(t,x)[t¯,t¯]×d.

Moreover, w satisfies the following space regularity property:

|w(t,x)-w(t,x)|e(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12|x-x|

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:

(A.2)sups[tt,t¯]𝔼[|Xst,x-Xst,x|2]e(2Lb,σ+Lb,σ2)(t¯-t¯)(1+(t¯-t¯))(|x-x|+Mb,σ|t-t|)2

for t,t[t¯,t¯] and x,xRd.

Proof.

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

𝔼[|Xst,x-Xst,x|2]𝔼[|Xtt,x-Xtt,x|2]+(2Lb,σ+Lb,σ2)ts𝔼[|Xut,x-Xut,x|2]𝑑u

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

sups[t,t¯]𝔼[|Xst,x-Xst,x|2]𝔼[|Xtt,x-Xtt,x|2]e(2Lb,σ+Lb,σ2)(t¯-t¯).

Moreover, we have

𝔼[|Xtt,x-Xtt,x|2]=𝔼[|x-x-ttb(s,Xst,x)ds-ttσ(s,Xst,x)dBs|2]
|x-x|2+Mb,σ2|t-t|2+Mb,σ2|t-t|+2Mb,σ|x-x||t-t|
(|x-x|+Mb,σ|t-t|)2(1+(t¯-t¯))),

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

𝒴ut,x=m(Xt¯t,x)+ut¯h(s,Xst,x,𝒴st,x,𝒵st,x)𝑑s-ut¯𝒵st,x𝑑Bs,u[t,t¯].

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

w(t,x)=𝒴tt,x,(t,x)[t¯,t¯]×d,

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

|𝒴st,x-𝒴st,x|2=|m(Xt¯t,x)-m(Xt¯t,x)|2+st¯(h(u,Xut,x,𝒴ut,x,𝒵ut,x)-h(u,Xut,x,𝒴ut,x,𝒵ut,x))(𝒴ut,x-𝒴ut,x)𝑑u
-st¯|𝒵ut,x-𝒵ut,x|2𝑑u-st¯(𝒴ut,x-𝒴ut,x)(𝒵ut,x-𝒵ut,x).dBu

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

𝔼[|𝒴st,x-𝒴st,x|2]Lm2𝔼[|Xt¯t,x-Xt¯t,x|2]+Lh2st¯𝔼[|Xut,x-Xut,x|2]𝑑u+(Lh24+Lh+1)st¯𝔼[|𝒴ut,x-𝒴ut,x|2]𝑑u

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

𝔼[|𝒴st,x-𝒴st,x|2]Lm2𝔼[|Xt¯t,x-Xt¯t,x|2]+(Lh2)2st¯𝔼[|Xut,x-Xut,x|2]𝑑u+(Lh2)2st¯𝔼[|𝒴ut,x-𝒴ut,x|2]𝑑u

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

𝔼[|𝒴st,x-𝒴st,x|2]e(2Lb,σ+Lb,σ2)(t¯-t¯)(1+(t¯-t¯))(Lm2+(t¯-t¯)(Lh2)2)|x-x|2+(Lh2)2st¯𝔼[|𝒴ut,x-𝒴ut,x|2]𝑑u

for s[t,t¯]. Since w has polynomial growth, we can apply Gronwall’s Lemma and we get

𝔼[|𝒴tt,x-𝒴tt,x|2]e(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))(Lm2+(t¯-t¯)(Lh2)2)|x-x|2.

Therefore, we get

|w(t,x)-w(t,x)|e(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12|x-x|.

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

𝒴ut,x=m(Xt¯t,x)+ut¯h(s,Xst,x,𝒴st,x,𝒵st,x)𝑑s-ut¯𝒵st,x𝑑Bs,u[t,t¯],

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¯].

Proof.

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

(A.3)(bn,σn,hn,mn)n+(b,σ,h,m)

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

Xut,x,n=x+tubn(s,Xst,x,n)𝑑s+tuσn(s,Xst,x,n)𝑑Bs,u[t,t¯],𝒴ut,x,n=mn(Xt¯t,x,n)+ut¯hn(s,Xst,x,n,𝒴st,x,n,𝒵st,x,n)𝑑s-ut¯𝒵st,x,n𝑑Bs,u[t,t¯].

From (A.3) we get

(A.4)Yt,x-Yt,x,n𝐒[t,t¯]2+Zt,x-Zt,x,n𝐇[t,t¯]2n+0.

From [20, Theorem 3.2], we have

Yst,x,n=wn(s,Xt,x,n),s[t,t¯],

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

Zst,x,n=(σnDwn)(s,Xst,x),s[t,t¯].

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

sup[t¯,t¯]×d|Dwn|e(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12.

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:

|w(t,x)-w(t,x)|e(3Lb,σ+2Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))32(Lm2+(t¯-t¯)(Lh2)2)12Mb,σt-t
+Mh(Mm+Mh(t¯-t¯))eMh(t¯-t¯)(t-t)
+MhMb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12(t-t)

for all t,t[t¯,t¯] and xRd.

Proof.

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

|w(t,x)-w(t,x)|=|𝒴tt,x-𝒴tt,x|
=|𝒴tt,x-𝔼[𝒴tt,Xtt,x+ttf(s,Xst,x,𝒴st,x,𝒵st,x)𝑑s]|
𝔼[|𝒴tt,x-𝒴tt,Xtt,x|]+Mhtt(1+𝔼[|𝒴st,x|]+𝔼[|𝒵st,x|])𝑑s.

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

sups[t¯,t¯]𝔼[|𝒴st,x|2]Mm2+e4(Mh+Mh2)(t¯-t¯).

Then, using Proposition A.3, we have

𝔼[|𝒵st,x|]Mb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12

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

|w(t,x)-w(t,x)|e(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12𝔼[|Xtt,x-Xtt,x|]
+Mh(Mm2+e4(Mh+Mh2)(t¯-t¯)+1)(t-t)
+MhMb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12(t-t).

From (A.2) we get

|w(t,x)-w(t,x)|e(3Lb,σ+2Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))(Lm2+(t¯-t¯)(Lh2)2)12Mb,σt-t
+Mh(Mm2+e4(Mh+Mh2)(t¯-t¯)+1)(t-t)
+MhMb,σe(2Lb,σ+Lb,σ2+(Lh2)2)(t¯-t¯)(1+(t¯-t¯))12(Lm2+(t¯-t¯)(Lh2)2)12(t-t).

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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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