Numerical Appromixation of the Value of a Stochastic Differential Game with Asymmetric Information

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Center for

Mathematical Economics

Working Papers

630

December 2019

Numerical Appromixation of the Value of a Stochastic Differential Game with Asymmetric Information

Lubomir Baˇ nas, Giorgio Ferrari, and Tsiry A. Randrianasolo

Center for Mathematical Economics (IMW) Bielefeld University

Universit¨atsstraße 25 D-33615 Bielefeld·Germany e-mail: imw@uni-bielefeld.de http://www.imw.uni-bielefeld.de/wp/

ISSN: 0931-6558

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NUMERICAL APPROXIMATION OF THE VALUE OF A STOCHASTIC DIFFERENTIAL GAME WITH ASYMMETRIC

INFORMATION

UBOMÍR BAAS, GIORGIO FERRARI, AND TSIRY A. RANDRIANASOLO

Abstract. We consider a convexity constrained Hamilton-Jacobi-Bellman-type obstacle problem for the value function of a zero-sum dierential game with asymmetric information. We propose a convexity-preserving probabilistic numerical scheme for the approximation of the value function which is discrete w.r.t. the time and convexity variables, and show that the scheme converges to the unique viscosity solution of the considered problem. Furthermore, we generalize the semi-discrete scheme to obtain an implementable fully discrete numerical approximation of the value function and present numerical experiments to demonstrate the properties of the proposed numerical scheme.

1. Introduction

In this paper we consider the Hamilton-Jacobi-Bellman-type obstacle problem min

"

BtV `1

2Trpσσ^Tpt, xqD²_xVq `Hpt, x, DxV, pq, λmin

ˆ p,B²V

Bp²

˙*

“0, VpT, x, pq “ xp, gy,

(1.1)

where V ”Vpt,x,pq, pt,x,pq P r0,Ts ˆR^dˆ∆pIq, ∆pIq denotes the set of probability vectors p“ pp₁,...,p_Iq P p0,1q^I that satisfy řI

i“1p_i“1 and the Hamiltonian H will be specied below. The convexity of the solution V with respect to the variable p is enforced via the obstacle term λ_min

´ p,^B_Bp²^V2

¯, which is the minimal eigenvalue of the Hessian matrix ^B_Bp²^V2 on the tangent cone to∆pIq. More precisely, for a symmetricIˆI matrix A we denote

λminpp, Aq:“ min

zPT∆pIqppqzt0u

xAz, zy

|z|² , whereT_∆pIqppq“Ť

δą0p∆pIq ´pq{δ is the tangent cone to ∆pIq at pP∆pIq, cf. [4].

(. Ba¬as) Faculty of mathematics, Bielefeld university, Universitätsstraÿe 25, D- 33615 Bielefeld

(G. Ferrari) Center for Mathematical Economics, Bielefeld university, Univer- sitätsstraÿe 25, D-33615 Bielefeld

(T.A. Randrianasolo) Faculty of mathematics, Bielefeld university, Univer- sitätsstraÿe 25, D-33615 Bielefeld

Date: December 30, 2019.

1991 Mathematics Subject Classication. 65K15,65C20,49N70,49L25,35F21,52A27,52B55.

Key words and phrases. zero-sum stochastic dierential games; asymmetric information; probabilistic numerical approximation; discrete convex envelope; convexity constrained Hamilton-Jacobi- Bellmann equation; viscosity solution.

This research was supported by the German Research Foundation as part of the Collaborative Research Center SFB 1283.

1

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Problem (1.1) describes the value of a class of zero-sum stochastic dierential games with asymmetric information, cf. [15]. Since the seminal work by Aumann and Maschler (see [1]) in the framework of repeated games, the literature on games with asymmetric information experienced an increasing interest ([12, 19, 22], among many others), recently also in continuous-time dierential settings (see, e.g., [7,8, 9, 13,16,24,25]).

As in [15], in our game both players can adjust the dynamics of a non degenerate Itô-diusion by controlling the drift via regular controls taking values in some compact subset of a nite dimensional space. However, one player has more information than the other in the following sense (cf. [1] and [7]). Before the game starts, the payos of the game are chosen randomly with some probability p from a nite collection of size I, and the information on which payos have been realized is transmitted only to one player. Since we assume that both players can observe the actions of the other one, the uninformed player infers which game is actually played through the moves of the informed one. It turns out that it is optimal for the informed player to release information to the uninformed one in a sophisticated way aiming at manipulating the beliefs of the latter player (see [7]).

The numerical analysis of our paper hinges on the theoretical results of [7]. There it is shown (in a setting actually more general than ours) that the previously described game has a value V, whenever the so-called Isaacs conditions are satised and addi- tional technical requirements on the problem's data area fullled. The value function V depends on time t, on the state variable x, and on a probability vector pP∆pIq;

this latter variable describes the initial value of the beliefs of the uninformed player about the game she is playing. Moreover, it is shown in [4], that V can be characterized as the unique continuous viscosity solution (in the dual sense) to a second-order partial dierential equation complemented by a convexity constraint with respect the parameter p.

There exist only few results on numerical approximation of dierential games with incomplete information. Numerical approximation of (deterministic) dierential games with incomplete information was rst studied in [5] and generalized to games with incomplete information on both sides in [23]. As far as we are aware the only work on numerical approximation of stochastic dierential games with incomplete information is [15]. We note that all three aforementioned works only consider semi-discretization in the time-variable and the remaining variables are kept continuous, hence, the schemes are not implementable.

In this paper we generalize the probabilistic numerical approximation of [15] to include the discretization of the convex envelope, i.e., we propose a structure preserving probabilistic numerical approximation that is discrete in time and in the variable p and preserves the convexity of the solution. We show that the proposed numerical approximation converges to the unique viscosity solution of (1.1). The discretization in the probability variablepis constructed by approximating the lower convex envelope of the semi-discrete solution in p by its nite-dimensional counterpart. The discrete lower convex envelope is computed over a nite set of values which coincide with nodes of a simplicial partition of ∆pIq. The resulting approximation is monotone and inherits the Lipschitz continuity properties of the solution. To further reduce the complexity of the numerical approximation we employ random walk instead of the usual Wiener increments to simulate the associated Itô-diusion process. Furthermore,

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we propose an implementable fully discrete numerical scheme by combining the semi- discrete probabilistic approximation in time and p with a spatial discretization that employs linear interpolation in the state variable xover a simplicial partition.

The paper is organized as follows. In Section 2 we collect basic denitions and assumptions on the considered problem. In Section3 we introduce a probabilistic numerical scheme for the approximation of (1.1) which is discrete in the time variable t and the convexity variable p and summarize the regularity properties of the numerical approximation in Section 4. Convergence of the numerical approximation to the viscositiy solution is shown in Section 5. Finally, an implementable fully discrete numerical approximation of the problem is introduced in Section6along with numerical studies which demonstrate the practicability of the proposed approach.

2. Assumptions and preliminaries

Throughout the paper, the scalar product of two vectors x“ px₁,...,x_dq and y“ py₁,...,y_dq of R^d is denoted by xx, yy:“řd

i“1x_iy_i and the`¹-norm is denoted by |x|:“ řd

i“1|x_i|; furthermore, we use | ¨ |8 and } ¨ }8 to respectively denote the `⁸-norm and the L⁸pR^dq-norm.

2.1. Description of the game. Since the aim of this paper is to provide a numerical approximation of the solution to (1.1), we only provide here a brief and informal description of the stochastic dierential game related to the problem (1.1) and simply refer to [15] for detailed discussion of the game and further references. We consider a two-player zero-sum dierential game where two players control thed-dimensional Itô process dened fortP r0, Ts, xPR^d as

(2.1) dX_s^{t, x, u, v} “bps, X_s^{t, x, u, v}, u_s, v_sqds`σps, X_s^{t, x, u, v}qdB_s sP rt, Ts, X_t^{t, x, u, v} “x.

HereB“ B_s:sP rt, Ts(

is ad-dimensional Brownian motion on a complete probability space, b and σ are suitable Borel-Measurable functions and the controls pu, vq PUˆV and U,Vare compact subsets of some nite dimensional spaces.

The game is characterized byI congurations with respective running costsp`_iqiPt1,...,Iu: r0, Ts ˆR^dˆUˆVÑR and terminal payos pg_iqiPt1,...,Iu:R^dÑR and is played as follows. Before the game starts, one congurationiP t1,...,Iu is chosen with probability p_i and the choice of i is communicated to Player 1. Player 2 only knows the probability distribution pP∆pIq of the respective congurations. Once the game has started, both players adjust their control to minimize, for the Player 1, and to maximize, for the Player 2, the expected payo, cf. [4, Section 6.3]. We assume that both players observe their opponent's control.

2.2. General assumptions. The drift term b, the diusion term σ:“ pσ_k,lq_k,l, the running cost p`_iqiPt1,...,Iu, the terminal payo g:“ pg_iqiPt1,...,Iu, and the Hamiltonian H, cf. (1.1), satisfy the following standing assumptions:

(A₁q b:r0, Ts ˆR^dˆUˆVÑR^d is bounded and continuous in all its variables and Lipschitz continuous with respect to pt, xquniformly in pu, vq PUˆV.

(A₂q For 1ďk,lďd the function σ_k,l:r0, Ts ˆR^dÑR is bounded and Lipschitz continuous with respect to pt, xq. For any pt, xq P r0, Ts ˆR^d the matrix pσ^Tq^´1, where the superscript T means transpose, is non-singular, bounded, and Lips- chitz continuous with respect to pt, xq.

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(A₃q p`_iqiPt1,...,Iu:r0, Ts ˆR^dˆUˆVÑR is bounded and continuous in all its variables and Lipschitz continuous with respect topt, xq uniformly in pu, vq.

pg_iqiPt1,...,Iu:R^dÑR is bounded and uniformly Lipschitz continuous.

(A₄q Isaacs condition: for all pt, x, z, pq P rt₀, Ts ˆR^dˆR^dˆ∆pIq Hpt, x, z, pq:“inf

uPUsup

vPV

"

xbpt, x, u, vq, zy `

I

ÿ

i“1

p_i`_ipt, x, u, vq

*

“sup

vPV uPUinf

"

xbpt, x, u, vq, zy `

I

ÿ

i“1

p_i`_ipt, x, u, vq

* .

(A₅q In addition, there exists a constant Cą0 such that for all t,t¹P r0, Ts, x,x¹P R^d, z,z¹PR^d, p,p¹P∆pIq, the following hold

(2.2) |Hpt, x, z, pq|ďCp1`|z|q,

|Hpt, x, z, pq ´Hpt¹, x¹, z¹, p¹q|ďCp1`|z|qp|x´x¹|`|t´t¹|q

`C|z´z¹|`C|p´p¹|.

(2.3)

2.3. Viscosity solution of (1.1). Under the assumptions in the previous section Cardaliaguet [4, 7] established that there exists a unique uniformly bounded viscosity solution of problem (1.1), which is convex and uniformly Lipschitz continuous in p. We recall the notion of viscosity solution as well as the corresponding notions of subsolutions and supersolutions to (1.1) below, cf. [4], [6].

Denition 2.1. We say that V is a subsolution of (1.1) if V “Vpt, x, pq is upper semicontinuous and if, for any smooth test function φ:p0, Tq ˆR^dˆ∆pIq ÑR such that V ´φ has a local maximum on r0, Ts ˆR^dˆ∆pIq at some point pst,sx,pq P r0, Ts s ˆ R^dˆ∆pIq, one has

(2.4) min

"

Btφ`1

2Trpσσ^Tpt, xqD_x²φq `Hpt, x, Dxφ, pq, λmin

ˆ p,B²φ

Bp²

˙*

ě0, at pt, x, pq “ pst,sx,pq.s

We say that V is a supersolution of (1.1) if V “Vpt, x, pq is lower semicontinuous and if, for any smooth test function φ:p0, Tq ˆR^dˆ∆pIq ÑR such that V ´φ has a local minimum onr0, Ts ˆR^dˆ∆pIqat some pointpst,sx,pq P r0, Ts s ˆR^dˆ∆pIq, one has

(2.5) min

"

Btφ`1

2Trpσσ^Tpt, xqD_x²φq `Hpt, x, D_xφ, pq, λ_min ˆ

p,B²φ Bp²

˙*

ď0, at pt, x, pq “ pst,sx,pq.s

We say that V is a viscosity solution of (1.1) if V is a sub- and a supersolution of (1.1).

Remark 2.2. For a smooth test function φ:p0, Tq ˆR^dˆ∆pIq ÑR such that V ´φ has a local maximum on r0, Ts ˆR^dˆ∆pIq at some point pst,sx,pq P r0, Ts s ˆR^dˆ∆pIq, we have that (2.4) is equivalent to

Btφ`1

2Trpσσ^Tpt, xqD²_xφq `Hpt, x, D_xφ, pq ě0 and λ_min ˆ

p,B²φ Bp²

˙ ě0;

(2.6)

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and for the smooth test function φ:p0, Tq ˆR^dˆ∆pIq ÑR such that V ´φ has a local minimum on r0, Ts ˆR^dˆ∆pIq at some point pst,x,s pq P r0, Ts s ˆR^dˆ∆pIq, (2.5) is equivalent to

Btφ`1

2Trpσσ^Tpt, xqD_x²φq `Hpt, x, D_xφ, pq ď0 or λ_min ˆ

p,B²φ Bp²

˙ ď0.

(2.7)

3. Numerical approximation

To simplify the subsequent numerical approximation, we perform a change of measure via the Girsanov transform in the spirit of, for instance, [14, Lemma 3.8] or [17], and instead of the controlled process (2.1) we consider the simpler process

(3.1) dX_s^{t, x}“σps, X_s^{t, x}qdB_s sP rt,Ts, X_t^{t, x}“x,

fortP r0, TsandxPR^d. Notice that the dynamics in (3.1) is independent of the players' controls.

For a xed NPN and a step size τ:“T{N we introduce an equidistant partition Π_τ:“ t_n(N

n“0, t_n“nτ of the time interval r0, Ts. We dene the discrete process pXs_nⁿ¹^{, x}qn¹“n,..., N as the weak Euler approximation of (3.1), that is

(3.2) Xs_nⁿ¹^{, x}“x`

n´1

ÿ

j“n¹

σpt_j,Xs_jⁿ¹^{, x}qξ_j? τ , where ξ_n?

τ“ pξ_n¹,..., ξ_n^dq?

τ, n“1,...,N is a suitable approximation of the R^d-valued Wiener increment rWpt_nq ´Wpt_n´1qs „Np0,τq. Here, we take ξ_n to be a R^d-valued binomial random walk, i.e. ξ_n¹,...,ξ_n^d are i.i.d. random variables with the law Ppξ_n^k“

˘1q “1{2, for everyk“1,...,d; the analysis below can be easily modied to cover other choices such as, e.g., a trinomial random walk or the discrete Wiener increments. In the following we abbreviateσ_npxq:“σpt_n, xq andσ_n^´Tpxq:“ pσ^Tpt_n, xqq^´1. The approximation obtained after one step of the Euler scheme (3.2) will be denoted as

(3.3) Xs_n`1^x :“Xs_n`1^{n, x} “x`σ_npxqξ_n?

τ xPR^d.

Let tM^huhą0 be a family of regular partitions of ∆pIq into open pI´1q-simplices K (i.e., line segments, triangles, tetrahedra for I“2,3,4, respectively) with mesh-size h“max_KPM^htdiampKqu such that∆pIq “ Y_KPM^hK. The set of vertices of all KPM^h is denoted by N^h:“ tp₁,...,p_Mu.

The approximation of the value function Vpt_n, x, p_mq is denoted by V_n^mpxq for t_nP Π_τ, xPR^d, p_mPN^h. The discrete numerical solution V_n^mpxq, xPR^d, n“0,...,N´1, m“1,...,M is obtained by the following algorithm.

Algorithm 3.1. ForxPR^dsetV_N^mpxq “ xp_m, gpxqyforp_mPN^h,m“1,...,M, setV_Npxq “ V_N¹pxq,...,V_N^Mpxq(

and proceed for n“N´1,...,0 as follows:

(1) Forward step: for xPR^d compute:

(3.4) Xs_n`1^x “x`σ_npxqξ_n? τ; (2) Backward step: for xPR^d and m“1,...,M set

Zs_n^mpxq “1 τE“

V_n`1^m pXs_n`1^x qσ^´T_n pxqξn

?τ‰ (3.5) ,

Ys_n^mpxq “E“

V_n`1^m pXs_n`1^x q‰

`τ H`

t_n, x,Zs_n^mpxq, p_m˘ (3.6) ;

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(3) Convexication: forxPR^dcompute the discrete lower convex envelope V_n¹pxq,...,V_n^Mpxq( of Ys_n¹pxq,...,Ys_n^Mpxq(

as (3.7) V_n^mpxq “Vex_p“

Ys_n¹pxq,...,Ys_n^Mpxq‰

pp_mq p_mPN^h, m“1,...,M.

We note that for pP∆pIq the lower convex envelope in (3.7) is the solution of the minimization problem, cf. [10],

Vex_p“

Ys_n¹pxq,...,Ys_n^Mpxq‰

ppq:“min

" I

ÿ

k“1

Ys_n^kpxqλ_k;

I

ÿ

k“1

λ_k“1, λ_kě0,

I

ÿ

k“1

λ_kp_k“p

* . (3.8)

We will discuss ecient algorithms for the computation of the discrete lower convex envelope (3.8) in Section6.1.

It is well known that the piecewise linear interpolation does not preserve the convexity of the interpolated data. Nevertheless, cf. [10, Corollary 2.3.], there exists a data de- pendent (regular) simplicial partitionM^h_n,xof∆pIqwith nodesN_n,x^h :“ tπ¹_n,x,...,π^Mn,x^n,xu Ď N^h such that the piecewise linear interpolant of the data values at the nodesN_n,x^h over the partition M^h_n,x (for a precise denition see (3.9) below) agrees with the discrete data values `

pm,V_n^mpxq˘(M

m“1,pmPN^h of the discrete lower convex envelope (3.7) for xed 0ďnďN, xPR^d. We note that the partition M^h_n,x does not necessarily coincide with the original mesh M^h.

We consider the set of piecewise linear Lagrange basis functionstψⁱ_n,x, i“1,...,M_n,xu associated with the set of nodes N_n,x^h of the partition M^h_n,x. We recall the following properties of the the Lagrange basis functions which will be frequently used throughout the paper: aqψⁱ_n,xpπ^k_n,xq “δik, where δik is the Kronecker delta and bqřMn,x

i“1 ψ_n,xⁱ ppq “1 for anypP∆pIq. We note thataqimplies that at any pointpP∆pIqthere are at mostI basis functions with non-zero value at this point, hence the sum in bqreduces tořI

i“1. We dene the convex piecewise linear interpolantV_n^hpx,¨q,xPR^dof the discrete lower convex envelope V_n¹pxq,...,V_n^Mpxq(

over the convexity preserving partitionM^h_n,x as

(3.9) V_n^hpx, pq:“

Mn,x

ÿ

i“1

V^mpπ

in,xq

n pxqψⁱ_n,xppq, where mpπ_n,xⁱ q PN is the index of πⁱ_n,x inN^h, i.e. πⁱ_n,x“p_mpπⁱ

n,xq for some p_mpπⁱ

n,xqPN^h. We note that by construction V_n^hpx, pmq “V_n^mpxq for all pmPN^h. For the analysis below we also consider the (possibly non-convex) interpolant over the xed partition M^h

(3.10) Vr_n^hpx, pq:“

M

ÿ

m“1

V_n^mpxqψ^mppq,

where tψ^m, m“1,...,Muis the linear Lagrange basis associated with the set of nodes N^h. By a slight abuse of notation in (3.8), we observe that

(3.11) V_n^hpx, pq “Vex_p“

Vr_n^hpx,¨q‰ ppq.

Furthermore, we dene the time interpolant V_τ^hpt, x, pq of (3.9) which is continuous on r0,Ts as

(3.12) V_τ^hpt, x, pq:“V_n^hpx, pq

´t_n`1´t τ

¯

`V_n`1^h px, pq

´t´t_n τ

¯

, for tP rtn, tn`1s.

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4. Regularity properties of the numerical approximation

In this section we study regularity properties of the numerical approximation obtained by Algorithm3.1. We establish uniform boundedness, almost Hölder continuity in time and Lipschitz continuity in p, and x, respectively, of the numerical solution.

Furthermore, we show that the numerical approximation satises a monotonicity property.

We recall the following properties of the discrete lower convex envelope which are a simple consequence of its denition (3.8).

Lemma 4.1. We consider the setN:“ p₁,...,p_M(

Ă∆pIqwith associated scalar values Upp_mq,Vpp_mq, such thatUpp_mq ďVpp_mq, m“1,...,M. We denoteV“ Vpp₁q,...,Vpp_Mq(

P R^M and U“ Upp₁q,...,Upp_Mq(

PR^M. The discrete lower convex envelope Vex_p satises the following properties for pP∆pIq:

i) Monotonicity: Vex_prUsppq ďVex_prVsppq,

ii) Constant preservation: Vex_prV`θsppq “Vex_prVsppq `θ for any θPR.

4.1. Lipschitz continuity in p.

Lemma 4.2. There exists a constantCą0(which only depends onAssumptions (A1q (A₄q) such that for n“0,...,N and all xPR^d the numerical solution is Lipschitz continuous in the variable p, i.e.,

(4.1) |V_n^hpx, pq ´V_n^hpx, qq|ďC|p´q| @p, qP∆pIq.

Proof. Forn“N, by linearity the functionV_N^hpx, pq:“ xp, gpxqy is Lipschitz continuous inp for any xPR^d with a Lipschitz constantL_N that only depends on g.

We proceed by induction and assume that V_n`1^h px, pq is Lipschitz continuous in p with a Lipschitz constant L_n`1 for some nďN´2. We consider p, q P∆pIq and assume without loss of generality thatV_n^hpx, qq ´V_n^hpx, pq ě0, otherwise p and q can be commuted.

LetpPK_p, whereK_p“ rπ¹_n,xppq,...,π_n,x^I ppqs ĂM^h_n,x is the simplex given by the nodes π¹_n,xppq,...,π_n,x^I ppq PN_n,x^h . Hence, p“řI

i“1πⁱ_n,xppqψⁱ_n,xppq, where ψ_n,xⁱ ppq, i“1,...,I are the linear Lagrange basis functions onrπ_n,x¹ ppq,...,π_n,x^I ppqs.

We note thatřI

i“1ψ_n,xⁱ ppq “1andψⁱ_n,xppq ě0fori“1,...,I. Hence, by [18, Lemma 8.2.], there exist vectors ω_n,x¹ ,...,ω_n,x^I (

P∆pIq(the vectors depend on p,q,M^h_n,x and are not necessarily in N^h) such that q“řI

i“1ω_n,xⁱ pqqψⁱ_n,xppq and

(4.2) |p´q|“

I

ÿ

i“1

|π_n,xⁱ ppq ´ωⁱ_n,x|ψ_n,xⁱ ppq.

By the convexity of V_n^h, since q“řI

i“1ω_n,xⁱ ψⁱ_n,xppq it directly follows that V_n^hpx, qq ď

I

ÿ

i“1

V_n^hpx, ωⁱ_n,xqψ_n,xⁱ ppq.

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Using the above inequality, (4.2), the representationV_n^hppq|_K_p“řI

i“1V_n^hpx, π_n,xⁱ qψⁱ_n,xppq and Vex_prfs ďf we get on recalling (3.6) that

0ďV_n^hpx, qq ´V_n^hpx, pq ď

I

ÿ

i“1

`V_n^hpx, ωⁱ_n,xq ´V_n^hpx, π_n,xⁱ q˘

ψⁱ_n,xppq

ď

I

ÿ

i“1

´ E“

V_n`1^h pXs_n`1^x , ω_n,xⁱ q‰

`τ H`

t_n, x, Z_n^hpx, ωⁱ_n,xq, ω_n,xⁱ ˘

´E“

V_n`1^h pXs_n`1^x , π_n,xⁱ q‰

´τ H`

t_n, x, Z_n^hpx, π_n,xⁱ q, π_n,xⁱ ˘¯

ψⁱ_n,xppq, (4.3)

where we used that V_n^hpx, πⁱ_n,xq “Vex_prY_npxqspπ_n,xⁱ q, i“1,...,I.

By the Lipschitz continuity ofV_n`1^h , it follows from (4.3) using (4.2) and [15, Lemma 3.6]

that

|V_n^hpx, qq ´V_n^hpx, pq| ďL_n`1

I

ÿ

i“1

|π_n,xⁱ ´ωⁱ_n,x|ψ_n,xⁱ ppq “L_n`1|p´q|, (4.4)

where L_n“L_n`1p1`Cτq `Cτ.

Recursively, we get that L_n“L_N`Ct_n`CτřN

i“n`1L_n, n“1,...,N´1 and by the discrete Gronwall lemma it follows L_nďL₀ď pL_N`CTqexppCTq. Hence V_n^h is uniformly Lipschitz continuous in p with a Lispchitz constant L0 which only depends on

the Assumptions (A1q(A5q.

4.2. Lipschitz continuity in x. The next lemma can be show analogically to [15, Lemma 3.3].

Lemma 4.3. Let φ:R^dÑR be a uniformly Lipschitz continuous function with a Lisp- schitz constant L. Then, there exists a constant Cą0, depending only on the data of Assumptions (A₁q(A₅q, such that the following inequality holds for n“0,...,N´1

E“

φpXs_n`1^x q‰

`τ H

´

t_n, x, 1 τE“

φpXs_n`1^x qσ_n^´Tpxqξ_n? τ‰

, p

¯

´E“

φpXs_n`1^x¹ q‰

´τ H

´

tn, x¹, 1 τE“

φpXs_n`1^x¹ qσ_n^´Tpx¹qξn

?τ‰ , p

¯

ďCτ,L|x´x¹|, where C_τ,L:“Lp1`Cτq `Cτ.

Lemma 4.4 (Lipschitz continuity in x). For n“0,...,N the interpolant V_n^h is (i) Lipschitz continuous in x:

|V_n^hpx, pq ´V_n^hpx¹, pq|ďC|x´x¹| for all x,x¹PR^d, pP∆pIq, (ii) uniformly bounded:

|V_n^hpx, pq|ďC for all xPR^d, pP∆pIq,

where Cą0 is a constant which depends only on Assumptions (A₁q(A₅q.

From the subsequent proof it follows that the non-convex interpolant Vr_n^h dened in (3.10) enjoys the same boundedness and Lipschitz continuity properties as V_n^h.

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Proof. We xpP∆pIqand considerx,x¹PR^d. For n“N we haveVr_N^hpx, pq “V_N^hpx, pq “ xp, gpxqy and piq, piiqhold since

V_N^hpx, pq ´V_N^hpx¹, pq

“

xp, gpxq ´gpx¹qy

ďL_N|x´x¹|, V_N^hpx, pq

“

xp, gpxqy ďCN,

whereL_N and C_N are positive constant which depend only on g.

We proceed by induction. We assume that V_n`1^h px, pq, Vr_n`1^h px, pq are Lipschitz continuous in x with a Lipschitz constant L_n`1 and bounded by C_n`1. We show that V_n^hpx, pq, Vr_n^hpx, pq are Lipschitz continuous with a Lipschitz constantLn and bounded by a constantCn. On recalling (3.7), (3.11) we may write

V_n^hpx,pq “Vex_p“ Vr_n^hp¨q‰

ppq “Vex_p

„ E“

Vr_n`1^h pXs_n`1^x ,¨q‰

´τ H`

t_n, x,Zr_n^hpx,¨q,¨˘

 ppq, (4.5)

whereZr_n^hpx, pq:“¹_τE“

Vr_n`1^h pXs_n`1^x , pqσ^´T_n pxqξ_n? τ‰

. Moreover, we recall that forp_mPN^h it holds by denition

Vr_n^hpp_mq “Vex_p

„ E“

Vr_n`1^h pXs_n`1^x ,¨q‰

´τ H`

t_n, x,Zr_n^hpx,¨q,¨˘

 pp_mq. (4.6)

i) Lipschitz continuity. By Lemma 4.3 we have for p_mPN^h E“

Vr_n`1^h pXs_n`1^x ,p_mq‰

`τ H`

t_n, x,Zr_n^hpx,p_mq,p_m˘ ďE“

Vr_n`1^h pXs_n`1^x¹ ,p_mq‰

´τ H`

t_n, x¹,Zr_n^hpx¹,p_mq,p_m˘

`L_n`1|x´x¹|, (4.7)

withL_n`1:“L_np1`Cτq `Cτ. On noting (4.6) it follows from (4.7) byLemma 4.1that Vr_n^hpx,p_mq ´Vr_n^hpx¹,p_mq ďL_n`1|x´x¹|

(4.8)

We recall (3.10) and obtain from (4.8) (note Vr_n^hpx,p_mq “V_n^mpxq) that for any pP∆pIq it holds

Vr_n^hpx,pq ´Vr_n^hpx¹,pq ď

M

ÿ

m“1

´

V_n^mpxq ´V_n^mpx¹q

¯

ψ^mppq ďL_n`1|x´x¹|, (4.9)

where we used thatřM

m“1ψ^m”1,ψ^mě0for any0ďnďN. Consequently by (4.5) and Lemma 4.1 it also follows that

V_n^hpx,pq ´V_n^hpx¹,pq ďL_n`1|x´x¹|. (4.10)

After commuting the role of x and x¹ and repeating the above steps we obtain for any pP∆pIq

|V_n^hpx,pq ´V_n^hpx¹,pq|ďLn`1|x´x¹|.

(4.11)

Hence, we get recursively that L_nďL_N`Ct_n`CτřN

i“n`1L_i. By the discrete Gron- wall lemma, it follows that L_nď pL_N`CTqexppCTq. Consequently, V_n^h, Vr_n^h, n“ 0,...,N are Lipschitz continuous inx, with a Lipschitz constantL:“ pL_N`CTqexppCTq depending only on the data in Assumptions (A₁q(A₅q.

ii) boundedness. LetK_p“ rπ¹_{n, x}ppq,...,π^I_{n, x}ppqsbe a simplex ofM^h_{n, x}such thatpPK_p, i.e. p“řI

i“1πⁱ_{n, x}ppqψ_{n, x}ⁱ ppq, where tψ_{n, x}ⁱ ppq:i“1,...,Iu is the Lagrange polynomial

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basis onK_p. In particular,řI

i“1ψ_{n, x}ⁱ ppq “1andψⁱ_{n, x}ppq ě0fori“1,...,I. On recalling (3.9) we may write

V_n^hpx, pq “

I

ÿ

i“1

´ E“

V_n`1^h pXs_n`1^x , π_{n, x}ⁱ ppqq‰

`τ H`

t_n, x, Z_n^hpx, π_{n, x}ⁱ ppqq, π_{n, x}ⁱ ppq˘¯

ψⁱ_{n, x}ppq, (4.12)

where Z_n^hpx, pq:“¹_τE“

V_n`1^h pXs_n`1^x , pqσ_n^´Tpxqξ_n? τ‰

. By (2.2), since V_n`1^h is bounded by C_n`1 we estimate the right hand side of (4.12)

E“

V_n`1^h pXs_n`1^x , πⁱ_{n, x}ppqq‰

`τ H`

t_n, x, Z_n^hpx, π_{n, x}ⁱ ppqq, π_{n, x}ⁱ ppq˘

ďC_n`1`τ C`

1`|Z_n^hpx, πⁱ_n,xppqq|˘ (4.13) .

Next, we show that Z_n^hpx, π_n,xⁱ ppqq is bounded. SinceV_n`1^h is uniformly Lipschitz continuous in the variable x. On recalling (3.3), by the generalized mean value theorem [11, Theorem 2.3.7 ] there exists ΘPR^d with |Θ|₈ďC such that

(4.14) V_n`1^h pXs_n`1^x , πⁱ_{n, x}ppqq “V_n`1^h px, π_n,xⁱ ppqq ` xΘ, σ_npxqξ_n? τy.

We multiply (4.14) with p1{τqσ_n^´Tpxqξ_n and take the expectation to get Z_n^hpx, π_{n, x}ⁱ ppqq “1

τE“

V_n`1^h pXs_n`1^x , πⁱ_{n, x}ppqqσ^´T_n pxqξ_n? τ‰

“1 τE“

V_n`1^h px, π_n,xⁱ ppqqσ^´T_n pxqξ_n?

τ` xΘ, σ_npxqξ_n?

τyσ_n^´Tpxqξ_n? τ‰

“1 τE“

xΘ, σ_npxqξ_n?

τyσ_n^´Tpxqξ_n? τ‰ (4.15) .

ByAssumption (A2qσn and σ^´T_n are uniformly bounded. Hence, it follows from (4.15) that

|Z_n^hpx, π_n,xⁱ ppqq|ď1

τ}σ_n}8}σ^´T_n }8E“

|Θ|₈|ξ_n? τ|²‰

ďC.

(4.16)

We substitute (4.16), (4.13) into (4.12) and get that

|V_n^hpx, pq| ď

I

ÿ

i“1

C_nψ_{n, x}ⁱ ppq “C_n,

where C_n:“C_n`1`τ C. Consequently, V_n^hpx, pq, n“0,...,N is uniformly bounded by

C:“CN`CT.

4.3. Almost Hölder continuity in t.

Lemma 4.5. For any τą0, hą0 and xPR^d, pP∆pIq the interpolant V_τ^h dened in (3.12) satises the following inequality

|V_τ^hps,x, pq ´V_τ^hpt, x, pq|ďC|s´t|^1{2`Cτ^1{2 @s,tP r0, Ts, where the constant Cą0 depends only on Assumptions (A1q(A4q.

Proof. We consider the piecewise linear Lagrange basis functions associated with Πτ as

χ_nptq “

$

’’

&

’’

%

t´t_n´1

τ , fortP rt_n´1,t_ns, t_n`1´t

τ , fortP rtn,tn`1s, 0 otherwise,

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forn“0,...,N and note that řN

n“0χ_nptq “1 for tP r0,Ts.

For s,tP r0, Ts lettP rt_n, t_n`1s and sP rt_n`n¹, t_n`n¹_`1s. Hence, we deduce from (3.12) that

V_τ^hpt, x,pq ´V_τ^hps, x,pq “

1

ÿ

k“0

V_n`k^h px,pqχ_n`kptq ´

1

ÿ

k¹“0

V_n`n^h 1`k¹px,pqχ_n`n¹_`k¹psq

“

1

ÿ

k“0 1

ÿ

k¹“0

`V_n`k^h px,pq ´V_n`n^h 1`k¹px,pq˘

χ_n`kptqχ_n`n¹_`k¹psq.

(4.17)

forxPR^d and pP∆pIq.

Since Vex_prfs ďf we get for p_mPN^h, recall (3.6), (3.7), that V_n`k^h px,p_mq ďE“

V_n`k`1^h pXs_n`k`1^{n`k, x},p_mq‰

`τ H`

t_n`k, x, Z_n`k^h px,p_mq,p_m˘ (4.18) ,

withZ_n`k^h px,p_mq:“¹_τE“

V_n`k`1^h pXs_n`k`1^{n`k, x},p_mqσ^´T_n`kξ_n`k? τ‰

. Using (4.16),Assumption (A₅q we obtain from (4.18) that

V_n`k^h px,p_mq ´V_n`n^h 1`k¹px,p_mq ďE“

V_n`k`1^h pXs_n`k`1^{n`k, x},p_mq ´V_n`n^h 1`k¹px,p_mq‰

`τ H`

t_n`k, x, Z_n`k^h px,p_mq,p_m˘ ďE“

`Cτ`

1`C|Z_n`k^h px,p_mq|˘ ďE“

`Cτ.

(4.19)

Recursively, we estimate the rst term on the right-hand side above using the corresponding analogue of (4.18) as

V_n`k^h pXs_n`k`1^{n`k, x},p_mq ´V_n`n^h 1`k¹px,p_mq ďE

”

V_n`k`2^h pXs^n`k`1,^X^s

n`k, x n`k`1

n`k`2 ,p_mq ´V_n`n^h 1`k¹px,p_mq ı

`Cτ.

(4.20)

We substitute (4.20) into (4.19) and obtain on noting Xs_n`k`2^{n`k, x} “Xs^n`k`1,^X^s

n`k, x n`k`1

n`k`2 (cf.

(3.2)) that

V_n`k^h px,p_mq ´V_n`n^h 1`k¹px,p_mq ďE

”

V_n`k`2^h pXs_n`k`2^{n`k, x},p_mq ´V_n`n^h 1`k¹px,p_mq ı

`C2τ.

Consequently, we get afterpn¹`k¹´k´2q recursive steps that V_n`k^h px,p_mq ´V_n`n^h 1`k¹px,p_mq

ď E“

V_n`n^h 1`k¹pXs_n`n^{n`k, x}1`k¹,p_mq ´V_n`n^h 1`k¹px,p_mq‰

` pn¹`k¹´kqCτ.

(4.21)

ByLemma 4.4and Assumption (A₂qwe estimate the rst term on the right hand side of (4.21) as

E“

V_n`n^h 1`k¹px,p_mq ´V_n`n^h 1`k¹pXs_n`n^n`k,x1`k¹,p_mq‰

ďC

E“

x´Xs_n`n^n`k,x1`k¹

‰

ďC

„n`n¹`k¹´1

ÿ

j“n`k

E

σ_jpXs_j^{n`k, x}q

2

τ

1{2

ďC|t_n`k´t_n`n¹_`k¹|^1{2, and get

V_n`k^h px,p_mq ´V_n`n^h 1`k¹px,p_mq ďC|t_n`k´t_n`n¹_`k¹|^1{2`Cpt_n`n¹_`k¹´t_n`kq.

SincetP rt_n, t_n`1s and sP rt_n`n¹, t_n`n¹_`1s it follows for k, k¹“0,1 that (4.22) V_n`k^h px,p_mq ´V_n`n^h 1`k¹px,p_mq ďCT^1{2|t´s|^1{2`Cτ^1{2,

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for p_mPN^h.

On recalling that V_n^hpx,pmq “Vr_n^hpx,pmqfor pmPN^h, V_n^hpx,pq “Vexp

“

Vr_n^hpx,¨q‰ ppqfor pP∆pIq, we deduce analogically as in the proof of Lemma 4.4 (cf. (4.9), (4.10)), that the inequality (4.22) holds for any pP∆pIq. Hence, substituting (4.22) into (4.17) for pP∆pIq implies the inequality

(4.23) V_τ^hpt,x,pq ´V_τ^hps,x,pq ďC|t´s|^1{2`Cτ^1{2.

After reverting the role of s and t and repeating the above steps we get the statement

of the lemma.

4.4. Monotonicity.

Lemma 4.6. Let φ₁, φ₂:R^dÑR be two uniformly Lipschitz continuous functions that satisfy 0ď pφ₁´φ₂q ďC. Then for any xPR^d, pP∆pIq, τą0, n“0,...,N´1 it holds that

E“

φ₁pXs_n`1^x q‰

`τ H

´

t_n, x,1 τE“

φ₁pXs_n`1^x qσ_n^´Tpxqξ_n? τ‰

, p

¯

ěE“

φ₂pXs_n`1^x q‰

`τ H

´

t_n, x, 1 τE“

φ₂pXs_n`1^x qσ_n^´Tpxqξ_n? τ‰

, p

¯

´Cτ? τ , where Cą0 is a constant which depends only on Assumptions (A₁q(A₅q.

Proof. We set H:“E“

pφ₁´φ₂qpXs_n`1^x q‰

`τ H

´

t_n, x,1 τE“

φ₁pXs_n`1^x qσ^´T_n pxqξ_n? τ‰

, p

¯

´τ H

´

tn, x, 1 τE“

φ2pXs_n`1^x qσ_n^´Tpxqξn

?τ‰ , p

¯ .

By Assumption (A₅q H is uniformly Lipschitz continuous in the third variable, hence using the generalized mean value theorem [11, Theorem 2.3.7 ] there exists a ΘPR^d,

|Θ|₈ďC such that H“E“

pφ₁´φ₂qpXs_n`1^x q` 1`@

Θ,σ_n^´Tpxqξ_n? τD˘ı

“E“

1 _Ckσ´1k₈|ξn|? τě1

(pφ₁´φ₂qpXs_n`1^x q` 1`@

Θ, σ_n^´Tpxqξ_n? τD˘ı

`E“

1 _Ckσ´1k8|ξn|?

τă1(pφ₁´φ₂qpXs_n`1^x q` 1`@

. (4.24)

Next, we show that the second term of the right hand side of (4.24) is positive. Since pφ₁´φ₂q ě0, it remains to examine the term p1`@

Θ, σ_n^´Tpxqξ_n? τD˘

. We note that 1`@

Θ, σ^´T_n pxqξ_n? τD

ě1´ }Θσ^´1_n }8|ξ_n|?

τě1´Ckσ^´1_n k₈|ξ_n|? τ . For Ckσ^´1k₈|ξn|?

τă1 it holds` 1`@

Θ, σ^´T_n pxqξn

?τD˘

ą0 and hence (4.25) E“

1 _Ckσ´1k₈|ξ_n|?

τă1(pφ₁´φ₂qpXs_n`1^x q` 1`@

ě0.

Using(4.25) we deduce from (4.24) that HěE

” 1 _|ξ

n

?τ|²ě1{R(

@Θ,pφ₁´φ₂qpXs_n`1^x qσ^´T_n pxqξ_n? τDı

, where R:“C²kσ^´1k²₈.

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On noting that |ξ_n|“|ξ_n¹|`...`|ξ_n^d|“ p1`...`1q “d we deduce E

” 1 _|ξ

n|²τě1{R(|ξ_n|? τ

ı

“1 _d2τě1{R(d?

τ“1 _d2τě1{R(R1 Rd?

τď1 _d2τě1{R(Rτ d³?

τďRτ d³? τ . Since ´pφ₁´φ₂q ě ´C we conclude

Hě ´E

” 1 _|ξ

n

?τ|²ě1{R(pφ₁´φ₂qpXs_n`1^x qkΘσ^´1k₈ ξ_n?

τ ı

ě ´CE

” 1 _|ξ

n

?τ|²ě1{R(|ξ_n? τ|ı

“ ´CRd³τ? τ .

5. Convergence of the numerical approximation

In this section we prove the convergence of numerical approximation, see Theo- rem 5.1below, in several steps. First, we show that, up to a subsequence, the sequence tV_τ^huh, τą0 admits a limit denoted byw. We then prove the viscosity super/sub-solution property of every accumulation pointw. Hence, by the uniqueness property of the viscosity solution, see [4], we may conclude that the whole sequence tV_τ^huh, τą0 converges to the viscosity solution.

Theorem 5.1. UnderAssumptions (A₁q(A₅q the numerical solutionV_τ^h converges to the viscosity solution of (1.1) (uniformly on compact subsets of r0, Ts ˆR^dˆ∆pIq) in the sense that for all pt¹, x¹, p¹q Ñ pt, x, pq it holds that

τ,hÑ0lim V_τ^hpt¹,x¹,p¹q “Vpt, x, pq,

where V is the unique uniformly bounded and continuous viscosity solution to (1.1) which is convex and uniformly Lipschitz continuous in p.

5.1. Existence of a limit.

Lemma 5.2. The sequencetV_τ^huτ, hą0 admits a subsequence which converges uniformly on every compact subset of r0, Ts ˆR^dˆ∆pIq to a uniformly bounded and continuous function w which is convex and uniformly Lipschitz continuous in p.

Proof. The proof is a consequence of a slight modication of the the ArzelàAscoli theorem, [26, Section III.3]. The equi-boundedness is granted by Lemma 4.4 and the equi-continuity is granted byLemmas 4.2, 4.4 and 4.5.

5.2. Viscosity solution properties and uniqueness of the limit. Below, we show that every accumulation point w of tV_τ^huh, τą0 from Lemma 5.2 is a viscosity sub- and super-solution to (1.1) which by uniqueness of the viscosity solution V implies Theorem 5.1.

5.2.1. Viscosity subsolution property ofw.

Proposition 5.3. Every accumulation point wof the sequence tV_τ^huτ, hą0 is a viscosity subsolution of (1.1) on r0, Ts ˆR^dˆ∆pIq.