# Partially Observed Nonzero-Sum Differential Game of BSDEs with Delay and Applications

Authors: Qiguang An; Qingfeng Zhu
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

A class of partially observed nonzero-sum differential games for backward stochastic differential equations with time delays is studied, in which both game system and cost functional involve the time delays of state variables and control variables under each participant with different observation equations. A necessary condition (maximum principle) for the Nash equilibrium point to this kind of partially observed game is established, and a sufficient condition (verification theorem) for the Nash equilibrium point is given. A partially observed linear quadratic game is taken as an example to illustrate the application of the maximum principle.


---

## Body

## 1. Introduction

Game theory has penetrated into many fields of economics and attracted more and more attention. A series of studies on game theory was given by [1–9]. There have been many papers about the differential games driven by backward stochastic differential equations (BSDEs), such as [10, 11]. The prospective progress of many systems relies as much on their past history as on their current state. The optimal control problem of stochastic systems with time delays is studied by [12–22]. The game problem of stochastic systems with the time-delayed generator is discussed by [23–25].Nevertheless, in the aforementioned control and game problems, it is assumed that the information is fully observed. This does not make sense in real life. In general, only partial information is available for controllers in most cases. The latest research studies on the partially observed optimal control issues of stochastic differential systems were given by [26–32]. The partially observed game issues of stochastic systems were studied by [33–36].As far as we know, the results in regard to partially observed differential games corresponding to backward stochastic systems with time delays (BSDDE) are few. This problem will be investigated in this paper. Comparing the above results, our work differs in several aspects. Firstly, we research such a kind of partially observed differential game problem corresponding to the BSDDE, which enriches the game theory of backward stochastic systems. Secondly, under the circumstance of different observation equations for each participant, our controlled systems and utility functions include the delays of state variables and control variables. Thirdly, we study a class of linear quadratic (LQ) game corresponding to backward stochastic systems with the time-delayed generator and give the specific expression of the Nash equilibrium point.The outline of this article is as follows. We present the main hypotheses and the partially observed differential game problem of BSDDE in Section2. In Section 3, we obtain the necessary optimality conditions of the partially observed game of BSDDE. Section 4 is devoted to the sufficient maximum principle. In Section 5, we take a partially observed LQ game as an example to illustrate the application of our maximum principle. Section 6 is the conclusion of this paper.


## 2. Statement of the Problems

Throughout our article,Ω,ℱ,ℱtt≥0,ℙ is a complete filtered probability space, on which three mutually independent one-dimensional standard Brownian motions Wt,Y1t, and Y2t are defined. Let ℱtW,ℱt1, and ℱt2 be the natural filtrations generated by W⋅,Y1⋅, and Y2⋅, respectively. We set ℱt=ℱtW⊗ℱt1⊗ℱt2. For all t∈−δ,0, ℱt≡ℱ0, which is the trivial σ-field, and ℱ:=ℱT+δ. Set ℱti=σYis;0≤s≤t,i=1,2 and ℱti≡ℱ0i≠ϕ, ∀t∈−δ,0. The finite time duration is defined by T>0, and the constant time delays are defined by 0<δ,δ1,δ2<T, respectively. The expectation on Ω,ℱ,P is denoted by E, and the conditional expectation under ℱt is denoted by Eℱt≔E.ℱt. In ℝ and ℝn×d, ⋅,⋅ is the usual inner product and ⋅ is the Euclidean norm. The symbol “⊤” that appears in the superscript represents the transpose of the matrix. In this article, all of the equalities and inequalities are in the sense of dt×dℙ almost surely on 0,T×Ω.We introduce the following notations:(1)L2ℱT;ℝ=ξ:ξis anℝ−valued,ℱT−measurable random variable satisfyingEξ2<∞,Lℱ2s,r;ℝ=vt,s≤t≤r:vtis anℝ−valued,ℱt−adapted process satisfyingE∫srvt2dt<∞.Let the nonempty setUi∈ℝi=1,2 be convex and the admissible control set be defined as the following:(2)Ui0,T=vi:0,T×Ω→Uiviisℱti−adapted,E∫0Tvit4dt<∞,i=1,2.Every element inUi is known as an admissible control to Player ii=1,2. And U1×U2 is known as the admissible control set to the players.This work pays attention to a kind of partially observed games of BSDDE, which stems from some attractive financial scenarios. Now let us elaborate on the problem. Take into account the following BSDDE:(3)−dyv1,v2t=fΘv1,v2tdt−zv1,v2tdWt,t∈0,T,yv1,v2T=ξ,yv1,v2t=ψ0t,t∈−δ,0,v1t=ψ1t,t∈−δ1,0,v2t=ψ2t,t∈−δ2,0,where(4)Θv1,v2t=t,yv1,v2t,yv1,v2t−δ,zv1,v2t,v1t,v1t−δ1,v2t,v2t−δ2,and f:0,T×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ⟶ℝ, ξ∈L2ℱT;ℝ, and ψ0⋅∈Lℱ2−δ,0;ℝ, ψ1⋅∈Lℱ2−δ1,0;ℝ, and ψ2⋅∈Lℱ2−δ2,0;ℝ are the initial paths of y, v1, andv2, respectively. We suppose that ψ1t∈ℱ01 and ψ2t∈ℱ02 are measurable continuous functions such that E∫0Tψ1t4dt<∞ and E∫0Tψ2t4dt<∞. The control processes for Player 1 and Player 2 are v1⋅ and v2⋅, and v⋅=v1⋅,v2⋅. Subscript 1 presents the variables to Player 1, and subscript 2 presents the variables to Player 2, respectively. BSDDE game system (3) means that the two players complete a common target ξ in the end time T.Suppose that the two participants cannot directly observe the state processesyv1,v2⋅, but they can be aware of related noisy processes Y1⋅ and Y2⋅ of yv1,v2⋅, which are described as follows:(5)dYit=hit,yv1,v2t,yv1,v2t−δ,zv1,v2t,v1t,v2tdt+dWit,Yi0=0,i=1,2,where W1⋅andW2⋅ are ℝ-valued stochastic processes depending on v1⋅ and v2⋅ and hi:0,T×ℝ×ℝ×ℝ×ℝ×ℝ⟶ℝ, i=1,2, are continuous functions.We assume(H1) (i) f and hi,i=1,2, are continuously differentiable with respect to y,yδ,z,v1,v1δ,v2,v2δ(ii)fy,fyδ,fz,fv1,fv1δ,fv2,fv2δ,hiy,hiyδ,hiz,hiv1,hiv2,i=1,2, are bounded by c>0Now, if (H1) is true and both v1⋅ and v2⋅ are admissible controls, then BSDDE (3) has a unique solution yv1,v2⋅,zv1,v2⋅∈Lℱ2−δ,T;ℝ×Lℱ2−δ,T;ℝ (see [14]).Definedℙv1,v2≐Zv1,v2tdℙ, where(6)Zv1,v2t=exp∑j=12∫0thjs,yv1,v2s,yv1,v2s−δ,zv1,v2s,v1s,v2sdYjs−12∑j=12∫0thjs,yv1,v2s,yv1,v2s−δ,zv1,v2s,v1s,v2s2ds.Obviously,Zv1,v2t satisfies the subsequent SDE:(7)dZv1,v2t=∑j=12hjt,yv1,v2t,yv1,v2t−δ,zv1,v2t,v1t,v2tZv1,v2tdYjt,Zv1,v20=1.Hence, by (H1) and Girsanov’s theorem, we obtain a three-dimensional Brownian motion W⋅,W1⋅,W2⋅ built on the probability space Ω,ℱ,ℙv1,v2, in which ℙv1,v2 is a probability measure.Making sure to accomplish the targetξ, each player owns his individual interest, which is the cost functional as follows:(8)Jiv1⋅,v2⋅=Ev1,v2∫0TliΘv1,v2tdt+Φiyv1,v20,i=1,2,where Ev1,v2 is the expectation on Ω,ℱ,ℙv1,v2 and(9)li:f:0,T×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ⟶ℝ,Φi:ℝ⟶ℝ,i=1,2.We also assume fori=1,2,(H2) (i) li are continuously differentiable with respect to y,yδ,z,v1,v1δ,v2,v2δ, and their partial derivatives are continuous in y,yδ,z,v1,v1δ,v2,v2δ and bounded by c1+y+yδ+z+v1+v1δ+v2+v2δ(ii)Φi are continuously differentiable, and Φiy are bounded by c1+yAssume that every player wants to minimize the cost functionalJiv1⋅,v2⋅ by picking the appropriate admissible control vi⋅i=1,2. Then, our partially observed nonzero-sum stochastic differential game problem is to find out a pair of admissible controls u1⋅,u2⋅∈U1×U2 such that(10)J1u1⋅,u2⋅=minv1⋅∈U1J1v1⋅,u2⋅,J2u1⋅,u2⋅=minv2⋅∈U2J2u1⋅,v2⋅.Obviously, cost functional (8) can be converted to(11)Jiv1⋅,v2⋅=E∫0TZv1,v2tliΘv1,v2tdt+Φiyv1,v20,i=1,2.So, the original problem (10) is the same thing as minimizing (11) over v1⋅,v2⋅∈U1×U2 subject to (3) and (7). For the sake of convenience, we refer to the above game problem as Problem (POBNZ). If an admissible control u⋅=u1⋅,u2⋅ which satisfied (10) can be found, then it is called as an equilibrium point of Problem (POBNZ), and the corresponding state trajectory is denoted by y⋅,z⋅=yu⋅,zu⋅.


## 3. A Partially Observed Necessary Maximum Principle

In the case of a convex admissible control set, the convex perturbation method is the classical method to obtain the necessary optimality condition. Let the equilibrium point of Problem (POBNZ) beu⋅=u1⋅,u2⋅, and the corresponding optimal trajectory is y⋅,z⋅. Let v˜1⋅,v˜2⋅ be such that u1⋅+v˜1⋅,u2⋅+v˜2⋅∈U1×U2. Since U1 and U2 are convex, for any 0≤ρ≤1, u1ρ⋅,u2ρ⋅=u1⋅+ρv˜1⋅,u2⋅+ρv˜2⋅ is also in U1×U2. For the controls u1ρ⋅,u2⋅ and u1⋅,u2ρ⋅, the corresponding state trajectories of game system (3) are denoted by yu1ρ⋅,zu1ρ⋅ and yu2ρ⋅,zu2ρ⋅.We introduce the subsequent symbols:(12)φt=φt,yt,yt−δ,zt,u1t,u1t−δ1,u2t,u2t−δ2,φv1,v2t=φt,yt,yt−δ,zt,v1t,v1t−δ1,v2t,v2t−δ2,φu1ρ,u2t=φt,yt,yt−δ,zt,u1ρt,u1ρt−δ1,u2t,u2t−δ2,φu1,u2ρt=φt,yt,yt−δ,zt,u1t,u1t−δ1,u2ρt,u2ρt−δ2,where φ denotes one of f,li,i=1,2, and(13)hit=hit,yt,yt−δ,zt,u1t,u2t,hiv1,v2t=hit,yt,yt−δ,zt,v1t,v2t,hiu1ρ,u2t=hit,yt,yt−δ,zt,u1ρt,u2t,hiu1,u2ρt=hit,yt,yt−δ,zt,u1t,u2ρt,i=1,2.The variational equations are as follows:(14)−dyi1t=fytyi1t+fyδtyi1t−δ+fztzi1t+fvitv˜it+fviδtv˜it−δidt−zi1tdWt,t∈0,T,yi1T=0,yi1t=0,t∈−δ,0,v˜1t=0,t∈−δ1,0,v˜2t=0,t∈−δ2,0,i=1,2,(15)dZi1t=∑j=12Zi1thjt+Zthjytyi1t+hjyδtyi1t−δ+hjztzi1t+hjvitv˜itdYjt,Zi10=0,i=1,2.From (H1), it is easy to see that (14) and (15) admit unique solutions yv1,v2⋅,zv1,v2⋅∈Lℱ2−δ,T;ℝ×Lℱ2−δ,T;ℝ and Z1t∈Lℱ20,T;ℝ, respectively.Fort∈0,T and ρ>0, we set(16)y˜iρt=yuiρt−ytρ−yi1t,z˜iρt=zuiρt−ztρ−zi1t,Z˜iρt=Zuiρt−Ztρ−Zi1t,i=1,2.Similar to the arguments in Lemmas 3.1 and 3.2 in [34], it is easy to obtain subsequent Lemmas 1 and 2. Thus, we omit the details for simplicity.Lemma 1.
Assume (H1) and (H2) are true. Then,(17)limρ→0sup0≤t≤TEy˜iρt2=0,limρ→0E∫0Tz˜iρt2dt=0,limρ→0sup0≤t≤TEZ˜iρt2=0,i=1,2.Sinceu1⋅,u2⋅ is a Nash equilibrium point, then(18)ρ−1J1u1ρ⋅,u2⋅−J1u1⋅,u2⋅≥0,ρ−1J2u1⋅,u2ρ⋅−J2u1⋅,u2⋅≥0.LetΓit=Z−1tZi1t,i=1,2. From Itô’s formula, we deduce(19)dΓit=∑j=12hjytyi1t+hjztzi1t+hjvtv˜itdWjt,Γi0=0,i=1,2.From this and Lemma1, we have the following.Lemma 2.
Assume (H1) and (H2) are true. Then, we get the following variational inequality:(20)Eu1,u2∫0TlitΓit+liytyi1t+liyδtyi1t−δ+liztzi1t+livitv˜it+liviδtv˜it−δidt+Eu1,u2Φiyy0yi10≥0,i=1,2.Our Hamiltonian functionHi:0,T×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ→ℝ,i=1,2, is defined as follows:(21)Hit,y,yδ,z,v1,v1δ,v2,v2δ,pi,Q1i,Q2i=−pit,ft,y,yδ,z,v1,v1δ,v2,v2δ+∑j=12Qjit,hjt,y,yδ,z,v1,v2+lit,y,yδ,z,v1,v1δ,v2,v2δ,i=1,2.DenoteHit≡Hit,y,yδ,z,v1,v1δ,v2,v2δ,pi,Q1i,Q2i and its derivatives.We note that the adjoint equation to (19) is a BSDE, whose solution is Pi⋅,Q1i⋅,Q2i⋅:(22)−dPit=litdt−∑j=12QjitdWjt,PiT=0,i=1,2,and the adjoint equation to (14) is an SDE, whose solution is pi⋅:(23)dpit=−Hiyt+EℱtHiyδt+δdt−HiztdWt,pi0=−Φiyy0,i=1,2.Remark 1.
It is easy to see that equation (23) is a linear anticipated SDE. Under (H1) and (H2), the unique solvability of equation (23) is assured by Theorem 2.2 in [14].
Based on variational inequality (20), we set out the main result of this section.Theorem 1 (partially observed necessary maximum principle).
Assume (H1) and (H2) are true, an equilibrium point of Problem (POBNZ) is u1⋅,u2⋅, the optimal trajectory is y⋅,z⋅, and the solution of (7) is Z⋅. Let Pi⋅,Q1i⋅,Q2i⋅,i=1,2, be the solution of (22) and pi⋅ be the solution of adjoint equation (23). Then, the following maximum principle(24)Eu1,u2H1v1t+Eu1,u2H1v1t+δ1ℱt1v1−u1tℱt1≥0,(25)Eu1,u2H2v2t+Eu1,u2H2v2t+δ2ℱt2v2−u2tℱt2≥0,for any v1,v2∈U1×U2, a.e., t∈0,T, in which the Hamiltonian function H is defined as (21).Proof.
Fori=1, using Itô’s formula to y11t,p1t+Γt,P1t, from variational equations (14) and (15), variational inequality (20), and adjoint equations (22) and (23), we obtain(26)Eu1,u2∫0Tl1tΓ1t+l1yty11t+l1yδty11t−δ+l1ztz11t+l1v1tv˜1t+l1v1δtv˜1t−δ1dt+Eu1,u2Φ1yy0y110=Eu1,u2∫0Tfv1⊤tp1t+∑j=12hjv1⊤tQj1t+l1v1t+Eu1,u2fv1⊤t+δp1t+δ+∑j=12hjv1⊤t+δQj1t+δ+l1v1t+δℱt1,v˜1tdt=Eu1,u2∫0TH1v1t+Eu1,u2H1v1t+δ1ℱt1,v˜1tdt≥0.
Becausev˜1t satisfies u1t+v˜1t∈U1, we have(27)Eu1,u2∫0TH1v1t+Eu1,u2H1v1t+δ1ℱt1,v1−u1tdt≥0,∀v1∈U1.
This implies that(28)Eu1,u2H1v1t+Eu1,u2H1v1t+δ1ℱt1,v1−u1t≥0,∀v1∈U1.
Now, assume thatF is an arbitrary element of σ-algebra ℱt1 and v1t∈U1 is a deterministic element. Let(29)w1t=v1t1F+u1t1Ω−F.
Obviously,w1 is an admissible control.
Using the above inequality tow1, we obtain(30)Eu1,u21FH1v1t+Eu1,u2H1v1t+δ1ℱt1,v1−u1t≥0,∀F∈ℱt1,which implies that(31)Eu1,u2H1v1t+Eu1,u2H1v1t+δ1ℱt1,v1−u1tℱt1≥0,∀v1∈U1,a.e.t∈0,T,a.s.
Similar to the aforementioned method, we can get the other inequality for anyv2∈U2. The proof of Theorem 1 is completed.


## 4. A Partially Observed Sufficient Maximum Principle

In this section, we explore a sufficient maximum principle to Problem (POBNZ). Letyt,zt,u1t,u2t be a quintuple that satisfies (3), and assume that there is a solution pit corresponding to adjoint SDE (23). We assume the following:(H3) For i=1,2, for all t∈0,T, Hit,⋅,⋅,⋅,⋅,⋅,⋅,⋅,pi,Q1i,Q2i is convex in y,yδ,z,v1,v1δ,v2,v2δ, and Φiy is convex in yFori=1,2, let(32)Hit=Hit,yt,yt−δ,zt,u1t,u1t−δ,u2t,u2t−δ,pit,Q1it,Q2it,Hiv1t=Hit,yt,yt−δ,zt,v1t,v1t−δ1,u2t,u2t−δ2,pit,Q1it,Q2it,Hiv2t=Hit,yt,yt−δ,zt,u1t,u1t−δ1,v2t,v2t−δ2,pit,Q1it,Q2it,hit=hit,yt,yt−δ,zt,u1t,u2t,hiv1t=hit,yt,yt−δ,zt,v1t,u2t,hiv2t=hit,yt,yt−δ,zt,u1t,v2t,φt=φt,yt,yδt,zt,u1t,u1t−δ1,u2t,u2t−δ2,φv1t,⋅=φt,yt,yδt,zt,v1t,v1t−δ1,u2t,u2t−δ2,φv2t,⋅=φt,yt,yδt,zt,u1t,u1t−δ1,v2t,v2t−δ2,where φ=f,li,i=1,2.Theorem 2 (partially observed sufficient maximum principle).
Assume (H1)–(H3) are true. Moreover, the maximum conditions (24) and (25) of the partial observation are true; then, u1⋅,u2⋅ is the equilibrium point to Problem (POBNZ).Proof.
For anyv1⋅∈U1, we consider(33)J1v1⋅,u2⋅−J1u1⋅,u2⋅=E∫0Tl1tZv1t−Ztdt+EΦ1yv10−Φ1y0+Ev1,u2∫0Tl1v1t−l1tdt=I1+I2+I3.
Using Itô’s formula toP1t,Zv1,u2t−Zt on 0,T, we deduce(34)I1=Ev1,u2∫0T∑j=12Qj1thjv1t−hjtdt.
Using Itô’s formula top1t,yv1t−yt on 0,T, from the convexity of Φ1, we have(35)I2≥Ev1,u2Φ1yy0,yv10−y0=−Ev1,u2∫0Tyv1t−yt,H1yt+EℱtH1yδt+δdt−Ev1,u2∫0Tzv1t−zt,H1ztdt+Ev1,u2∫0Tp1t,fv1t−ftdt.
Then, we have(36)J1v1⋅,u2⋅−J1u1⋅,u2⋅≥E∫0TH1v1t−H1tdt−E∫0Tyv1t−yt,H1yt+EℱtH1yδt+δdt−E∫0Tzv1t−zt,H1ztdt.
By the virtue of convexity ofH1 to y,yδ,z,v1,v1δ,v2,v2δ, we deduce(37)H1v1t−H1t≥yv1t−yt,H1yt+yδv1t−yδt,H1yδt+zv1t−zt,H1zt+v1t−u1t,H1v1t+v1δt−u1δt,H1v1δt.
Notice the truth that(38)E∫0Tyδv1t−yδt,H1yδtdt−E∫0Tyv1t−yt,EℱtH1yδt+δdt=E∫0Tyδv1t−yδt,H1yδtdt−E∫δT+δyδv1t−yδt,H1yδtdt=E∫0δyδv1t−yδt,H1yδtdt−E∫TT+δyδv1t−yδt,H1yδtdt=0.
Then, we get(39)J1v1⋅,u2⋅−J1u1⋅,u2⋅≥E∫0TH1v1t+EℱtH1v1δt+δ,v1t−u1tdt.
Finally, by necessary optimality conditions (24), we obtain(40)J1v1⋅,u2⋅−J1u1⋅,u2⋅≥0.
Then, it implies(41)J1u1⋅,u2⋅=minv1⋅∈U1J1v1⋅,u2⋅.
In the same way,(42)J2u1⋅,u2⋅=minv2⋅∈U2J2u1⋅,v2⋅.
So, we come to the expected conclusion. The proof is completed.


## 5. Application

In this section, we construct a partially observed LQ differential game with regard to backward stochastic systems with time delays. Using the classical filtering theory and the aforementioned theoretical results, we attempt to give a specific expression of the Nash equilibrium point. Let us think about the subsequent linear BSDDE:(43)−dyv1,v2t=Atyv1,v2t+A¯tyv1,v2t−δ+Btzv1,v2t+C1tv1t+C2tv2tdt−zv1,v2tdWt,t∈0,T,yv1,v2T=ξ,yv1,v2t=ψt,t∈−δ,0,and the observation(44)dYit=Ditdt+dWit,Yiv1,v20=0,i=1,2.We introduce the following cost functional:(45)Jiv1⋅,v2⋅=12Ev1,v2∫0TMitvi2tdt+Niyv02,i=1,2,where constant Ni≥0, functions A⋅,A¯⋅,B⋅,Ci⋅,Di⋅,Mi⋅,i=1,2, are deterministic and bounded, and Mi−1⋅ is bounded. Our partially observed nonzero-sum LQ differential game is to find out a pair of admissible controls u1⋅,u2⋅∈U1×U2 satisfying(46)J1u1⋅,u2⋅=minv1⋅∈U1J1v1⋅,u2⋅,J2u1⋅,u2⋅=minv2⋅∈U2J2u1⋅,v2⋅.Similarly to [34], with the help of the necessary maximum principle (Theorem 1), we have the explicit expression to a Nash equilibrium point with regard to the above LQ game problem.Theorem 3.
For the above LQ game, we find out a Nash equilibrium point as(47)u1t,u2t=M1−1tC1⊤tEv1,v2p1tℱt1,M2−1tC2⊤tEv1,v2p2tℱt2,t∈0,T,in which yt,zt,p1t,p2t satisfy the general FBSDE:(48)−dyv1,v2t=Atyv1,v2t+A¯tyv1,v2t−δ+Btzv1,v2t+C1tM1−1tC1⊤tEv1,v2p1tℱt1+C2tM2−1tC2⊤tEv1,v2p2tℱt2dt−zv1,v2tdWt,t∈0,T,dpit=A⊤tpit+EℱtA¯⊤tpiδ+tdt+B⊤tpitdWt,t∈0,T,i=1,2,yv1,v2T=ξ,yv1,v2t=ψt,t∈−δ,0,pi0=Niy0,pit=0,t∈T,T+δ,i=1,2.


## 6. Conclusion

In this research, we have explored a class of partially observed game problem of the backward stochastic system with delay. More specially, based on the convex variational method, we establish the necessary and sufficient conditions with regard to Nash equilibrium in our game issue. The theoretical results of this paper are applied to an LQ game, for which the unique equilibrium point is expressed explicitly. On account of that the LQ model is usually used to depict many financial and economic phenomena, we expect that our LQ game result of BSDDEs can be widely used in these fields. As far as we know, the partially observed nonzero-sum backward game problem with the time-delay generator is firstly investigated in our paper.Notwithstanding that we are committed to the above game problem, we are likewise able to progress some consequences of optimal control for BSDDEs, for example, [14, 34].



---

*Source: 3518961-2020-06-19.xml*