Ergodic mean-field games of singular control with regime-switching

Dianetti, Jodi; Ferrari, Giorgio; Tzouanas, Ioannis Working Paper Ergodic mean-field games of singular control with regimeswitching Center for Mathematical Economics Working Papers, No. 681 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Dianetti, Jodi; Ferrari, Giorgio; Tzouanas, Ioannis (2023) : Ergodic mean-field games of singular control with regime-switching, Center for Mathematical Economics Working Papers, No. 681, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29797309 This Version is available at: https://hdl.handle.net/10419/278595 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 681 July 2023 Ergodic Mean-Field Games of Singular Control with Regime-Switching (extended version) Jodi Dianetti, Giorgio Ferrari, and Ioannis Tzouanas Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en ERGODIC MEAN-FIELD GAMES OF SINGULAR CONTROL WITH REGIME-SWITCHING (EXTENDED VERSION) JODI DIANETTI, GIORGIO FERRARI, AND IOANNIS TZOUANAS ABSTRACT. This paper studies a class of stationary mean-field games of singular stochastic control with regime-switching. The representative agent adjusts the dynamics of a Markov-modulated Itˆ odiffusion via a two-sided singular stochastic control and faces a long-time-average expected profit criterion. The mean-field interaction is of scalar type and it is given through the stationary distribution of the population. Via a constructive approach, we prove the existence and uniqueness of the stationary mean-field equilibrium. Furthermore, we show that this realizes a symmetric εN-Nash equilibrium for a suitable ergodic N-player game with singular controls. The proof hinges on the characterization of the optimal solution to the representative player’s ergodic singular stochastic control problem with regime switching, which is of independent interest and appears here for the first time. Keywords: stationary mean-field games; singular control; regime-switching; ergodic criterion; εNash equilibrium. MSC subject classification: 49L20, 91A15, 91A16, 60G40, 35R35, 93C30. 1. INTRODUCTION Mean-field games (MFGs in short) have been introduced independently in 2006 by Larsy-Lions [38] and Caines et al. [13] as limit models for symmetric N-player differential games, where the interaction is through the empirical distribution of the states (and possibly of the actions) of the players. In MFGs, a representative agent determines her best reply to a given flow of probability measures – e.g., representing the distribution of the states of the indistinguishable rivals – so that the counterpart to the Nash equilibrium concept arising in N-player games takes now the form of a consistency condition: The law of the optimally controlled state of the representative agent must agree with the given flow of probability measures. Since their introduction, because of their tractability, their relation to the theory of propagation of chaos and of forward-backward systems, and their ability to reproduce εN-Nash equilibria for suitably related symmetric N-player games, MFGs have attracted large attention in the mathematical and applied literature. We refer to the two-volume book by Carmona and Delarue [19] for a comprehensive presentation of results, approaches, and techniques, as well as to the paper by Carmona [18] for a review of applications of MFGs in Economics, Finance, and Engineering. In stationary MFGs, the representative player interacts with the long-run distribution of the population. Such a concept has a long tradition in economic theory: Stationary equilibria appeared already in the 1980s in the context of games with a continuum of players (see [31] and [33]), and also play an important role in the analysis of competitive market models with heterogeneous agents (see, e.g., [1] and [41], amongst many others). Closely connected is also the concept of stationary oblivious equilibria, introduced by Adlakha et al. in [2]. Within the mathematical literature, stationary MFGs have been approached both via analytic and probabilistic methods. Among those papers adopting a partial differential equations (PDE) approach, we refer to the works of Bardi and Feleqi [7] for the study of the forward-backward system arising in stationary MFGs with regular controls, Gomes et al. [29] for extended stationary MFGs, Cardaliaguet and Porretta [17] for the study of the long-term behavior of the master equation arising in MFG theory, and to Bertucci [10] for the study of stationary mean-field Date: July 27, 2023. 1 2 DIANETTI, FERRARI, AND TZOUANAS optimal stopping games. On the other hand, a probabilistic approach is followed in a series of recent contributions dealing with stationary MFGs with singular and impulsive controls, see A¨ ıd et al. [3], Cao and Guo [16], Cao et al. [15], and Christensen et al. [20]. 1.1. Our results. In this paper, we study a class of stationary MFGs where the underlying state process is a general singularly controlled one-dimensional diffusion whose coefficients are modulated by a continuous-time Markov chain with d≥2states. More precisely, the representative agent optimally controls a Markov-modulated real-valued Itˆ o-diffusion through a two-sided singular control in order to maximize an ergodic reward functional. This is given by the long-time-average of the timeintegral of a running profit function, net of the proportional costs of actions. The mean-field interaction is of scalar type and comes through a real-valued parameter denoted by θ, which, at equilibrium, has to identify with a suitable generalized moment of the stationary distribution of the optimally controlled state process. From the economic point of view, θcan be thought of as a stationary price index arising from the aggregate productivity through an isoelastic demand function ` a la Spence-Dixit-Stiglitz (see pp. 7-8 in [1]), or of as a stationary demand due to aggregate advertising (see pp. 595-596 in [39]). Under suitable assumptions on the problem’s data, by employing mainly probabilistic means, we prove the existence and uniqueness of a stationary MFG equilibrium for the considered game. Furthermore, we show that this realizes an ε-Nash equilibrium for a related symmetric ergodic N-player game with singular controls. Our first contribution consists in studying the representative player’s optimal control problem and thus in providing, for the first time in the literature, the complete solution to a two-sided ergodic singular stochastic control problem with regime-switching (cf. Proposition 3.5 below). This is accomplished through the study of an auxiliary optimal stopping game, which is then tackled through probabilistic arguments similarly to Ferrari and Rodosthenous in [25] (where, however, the considered Dynkin game was related to a discounted singular stochastic control problem). Through a novel verification argument (see Propositions 3.4 and 3.5 below), we show that, for a given and fixed meanfield parameter θ, the optimal control is of barrier-type. That is, the optimal control uniquely solves a Skorokhod reflection problem (see e.g. Burdzy et al. [12]) at endogenously determined barriers (free boundaries), which depend on the underlying Markov chain and on the given and fixed mean-field parameter θ. As a byproduct, we also show that the optimal upwards and downwards reflection policies satisfy a couple of functional equations resembling those in Theorem 2 of [35]. The next step deals with the construction of the MFG equilibrium and with the proof of its uniqueness. To that end, we first show that the joint process constituted by the optimally controlled Markovmodulated diffusion process and the Markov chain admits a stationary distribution (cf. Proposition 4.2 below). As a matter of fact, we prove that its cumulative distribution function is the unique classical solution to a weakly-coupled system of ordinary differential equations (cf. (4.10) below). This is in line with the result in the Brownian setting obtained by D’Auria and Kella in Theorem 1 of [23]. Clearly, the stationary distribution and its cumulative distribution depend on the fixed mean-field parameter θ, since the optimally controlled state does. In order to proceed with the equilibrium analysis, we thus study the stability of the stationary distribution with respect to θand actually prove its continuity with respect to such a parameter (cf. Theorem 4.1). Further exploiting the connection to the aforementioned Dynkin game of optimal stopping, we are then able to determine an invariant compact set where any equilibrium value of θ(if one exists) should lie. Combining those continuity and compactness results, an application of the Schauder-Tychonof fixed point theorem allows us to prove that there exists a unique stationary equilibrium (cf. Theorem 4.1). The analysis of the considered stationary MFG is finally justified by the fact that its unique stationary mean-field equilibrium is able to realize an εN-Nash equilibrium for an N-player symmetric game with singular controls in which each player faces an ergodic net profit functional. It is worth noticing that in the N-player game the interaction comes through a suitable time-dependent average ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 3 of the players’ states (see Eqs. (5.1) and (5.2) below), and it is therefore given in terms of the empirical distribution of players’ states at the current time. 1.2. Related literature. Ergodic singular stochastic control problems for one-dimensional diffusions have been treated in general settings, including state-dependent costs of actions, and with different applications, in [5], [40], [42] and [32], [37], among others. However, in all those papers, no regime switching is included. Our paper is placed within the recent bunch of literature dealing with MFGs with singular controls by following a probabilistic approach; see A¨ ıd et al. [3], Cao and Guo [16], Cao et al. [15], Campi et al. [14], Dianetti et al. [22], Fu [26], Fu and Horst [27], and Guo and Xu [30]. Amongst those, the works that most relate to ours are those by A¨ ıd et al. [3] and by Cao et al. [15]. Cao et al. consider in [15] ergodic MFGs involving a one-dimensional singularly controlled Itˆ o-diffusion. However, differently to us, the control in [15] can be exerted only upwards and no regime-switching process is considered therein. In our work, similarly to [3], we consider a stationary MFG involving a singularly controlled one-dimensional diffusion whose coefficients are modulated by a continuous-time Markov chain. However, differently to [3], here the control is two-sided, rather than only increasing, the performance criterion is of ergodic type, rather than of discounted type, the dynamics of the underlying state process are general, rather than geometric, and the Markov chain has d≥2states, rather than only two regimes. We also clearly relate to those works dealing with MFGs involving regime-switching regular control models. Wang and Zhang [50] consider social optima of mean-field linear-quadratic-Gaussian control models with Markov jump parameters, while distributed games for large-population multiagent systems with random time-varying parameters are investigated in [49]. Bensoussan et al. [9] focus on MFGs of risk-sensitive type with jump-diffusions and regime-switching. Furthermore, due to the application in networks with switching mechanism, mean-field control problems with regimeswitching became recently of particular interest: see, among others, Bayraktar et al. [8], Zhang et al. [52] and Nguyen et al. [43]. 1.3. Organization of the paper. The rest of the paper is organized as follows. In Section 2, we introduce the probabilistic setting and the MFG under study. Next, in Section 3, for a given and fixed mean-field parameter, we solve the ergodic stochastic control problem faced by the representative player. In Section 4we then prove the existence and uniqueness of the mean-field equilibrium, while in Section 5we provide the approximation result for a related N-player symmetric game. Finally, technical proofs are collected in the Appendices Aand B. 2. PROBLEM FORMULATION 2.1. Probabilistic Setting. Let (Ω,F,P)be a probability space which satisfies the usual conditions, on which it is defined a one-dimensional Brownian motion {Wt}t≥0and an independent irreducible continuous-time Markov chain {Yt}t≥0. Denote by F:= {FW,Y t}t≥0the filtration which is generated by Wand Y, as usual augmented by P-null sets of F. The Markov chain Yhas state space Y:= {1, ..., d}and transition matrix Q:= {qij}1≤i,j≤d. The transition rates are such that κi:= −qii >0 and the condition Pj∈Yqij = 0 holds for every i∈Y. Accordingly, the transition probabilities of Y are defined as P(Yt+∆t=j|Yt=i) := (qij∆t+o(∆t), j 6=i, 1 + qii∆t+o(∆t), j =i. For future frequent use, we denote by p(i) := κi Pd j=1 κj the i-th component of the stationary distribution of Y. Let (2.1) A:= { {ξt}t≥0,F-adapted, with bounded-variation, left-continuous, ξ0= 0,a.s. } 4 DIANETTI, FERRARI, AND TZOUANAS and notice that any ξ∈ A admits the Jordan decomposition ξ=ξ+−ξ−, for ξ±nondecreasing. Also, let {|ξ|t}t≥0:= ξ++ξ−denote the variation of ξ∈ A. Then, for given ξ∈ A and Borel-measurable functions b:R×Y→R,σ:R×Y→(0,∞), we introduce the process Xξwith state space I:= (x, x)⊆Rand dynamics (2.2) dXξ t=b(Xξ t, Yt)dt +σ(Xξ t, Yt)dWt+dξ+ t−dξ− t,(Xξ 0, Y0) = (x, i)∈ I × Y. The following assumption in particular ensures that there exists a unique strong solution to (2.2), for every ξ∈ A and (x, i)∈ I × Y(see Theorem 7 Chapter V in [46]). In the following, we shall denote such a strong solution by (Xx,ξ, Y i), when needed. Assumption 2.1. The following hold: (1) The functions b(·, i)and σ(·, i)are twice continuously differentiable, for every i∈Y. (2) There exists C > 0,such that |b(x, i)|+|σ(x, i)| ≤ C(1 + |x|),for any (x, i)∈ I × Y. (3) There exists c > 0, such that bx(x, i)≤ −c, for any (x, i)∈ I × Y. (4) For any (x, i)∈ I × Y, σ(x, i)>0. Denoting by (X0, Y )the unique strong solution to (2.2) with ξ≡0, Conditions (1) and (2) in Assumption 2.1 imply that (X0, Y )is regular, meaning that there exists a sequence of stopping times {βn}n≥0, with βn:= inf{t≥0 : |X0 t|=n}, such that β∞:= limn→∞ βn=∞,P-a.s.; for further details see Section 2.3 in [51]. For our subsequent analysis, we introduce the F-adapted process {b Xt}t≥0, which evolves as (2.3) db Xt= (b(b Xt, Yt) + σσx(b Xt, Yt))dt +σ(b Xt, Yt)dc Wt,(b X0, Y0) = (x, i)∈ I × Y, for an F-adapted Brownian motion c W. Notice that Equation (2.3) also admits a unique strong solution (b Xx, Y i)which is regular, due to Assumption 2.1. For f:R×Y→Rsuch that f(·, i)∈C2(R),for any i∈Y, the infinitesimal generator of the uncontrolled process (X0, Y )is denoted by L(X,Y )and it is such that (2.4) L(X,Y )f(x, i) = 1 2σ2(x, i)fxx(x, i) + b(x, i)fx(x, i) + X j6=i qij(f(x, j)−f(x, i)), while the infinitesimal generator L(b X,Y )for the process (b X, Y )is such that (2.5) L(b X,Y )f(x, i) = 1 2σ2(x, i)fxx(x, i) + (b(x, i) + σσx(x, i))fx(x, i) + X j6=i qij(f(x, j)−f(x, i)). In the rest of the paper, we adopt the following notation: P(x,i)[·] := P[· |Xξ 0=x, Y0=i] and E(x,i)[·] := EP[· |Xξ 0=x, Y0=i]for the hybrid-diffusion process (Xξ, Y ), and b P(x,i)[·] := b P[· | b X0=x, Y0=i]and b E(x,i)[·] := Eb P[· | b X0=x, Y0=i]for (b X, Y ). We also set Pi[·] := P[· |Y0=i]and denote by Eithe corresponding expectation. 2.2. The Ergodic Mean-Field Game. Within the previous probabilistic setting, we now introduce the ergodic mean-field game (ergodic MFG for short) which will be the main object of our study. For (x, i)∈ I × Y=: O,ξ∈ A and θ∈R+, we introduce the ergodic profit functional (2.6) J(x, i;ξ, θ) := lim sup T↑∞ 1 TE(x,i)ZT 0 π(Xξ t, θ)dt −k1ξ+ T+k2ξ− T, ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 5 where 0< k2< k1. In (2.6), πis the instantaneous profit function satisfying Assumption 2.2 below, and θis (for the moment) a fixed nonnegative number. We will see later, that θdrives the mean-field interaction (cf. Definition 2.1 below). The instantaneous profit function πfulfills the following conditions. Assumption 2.2. The function π:R×R+7→ R+is such that: (1) π(·, θ)∈C2(I), for any θ∈R+; (2) π(·, θ)is non-decreasing and concave, for any θ∈R+; (3) πxθ is continuous and it is such that πxθ(x, θ)<0, for any (x, θ)∈ I × R+; (4) for every (x, i, θ)∈ O × R+, lim x↓xb E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt=∞, and, lim x↑xb E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt= 0; (5) for every (x, i, θ)∈ O × R+, b E(x,i)Z∞ 0 e−ct|πx(b Xt, θ)|dt<∞, and, for some ǫ0:= ǫ0(x)∈(0,1), b EZ∞ 0 e−ct sup x′∈(x,x+ǫ0)|πxx(b Xx′ t, θ)|∂xb Xx sx=x′dt+ +b EZ∞ 0 e−ct|πx(b Xx t, θ)|sup x′∈(x,x+ǫ0)Zt 0 |bxx(b Xx′ s, Y i s)|∂xb Xx sx=x′dsdt<∞, where ∂xb Xx t:= exp Rt 0bx+∂x(σσx)−1 2σ2 x(b Xx s, Ys)ds +Rt 0σx(b Xx s, Ys)dc Wsand c > 0is the same constant as in Assumption 2.1-(3), (6) for every (x, i)∈ O, and for some ǫ0:= ǫ0(x)∈(0,1), it holds that (2.7) b EZ∞ 0 e−ct sup x′∈(x,x+ǫ0)|bxx(b Xx′ t, Y i t)∂xb Xx tx=x′dt<∞. We are now ready to introduce the notion of stationary mean-field equilibrium. To that end, we provide the following assumption for the functions Fand fthat will appear in Definition 2.1 below. Furthermore, we restrict to those ξ∈ A belonging to (2.8) Ae:= ξ∈ A :E|ξ|T<∞for any T < ∞,lim sup T↑∞ 1 TE|Xξ T|= 0. Assumption 2.3. F:R+→R+,f:I → R+are such that: (1) Fand fare strictly increasing continuously differentiable functions; (2) for β∈(0,1), there exists C > 0such that: (a) |f(x)| ≤ C(1 + |x|β),|F(x)| ≤ C(1 + |x|1 β), (b) F(x)−F(y)≤C(1 + |x|+|y|)1 β−1|x−y|; (3) limx↑∞ F(x) = limx↑xf(x) = ∞. Definition 2.1 (Ergodic MFG Equilibrium).For (x, i)∈ O, a couple (ξ∗(θ∗), θ∗)∈ Ae×R+is said to be an equilibrium of the ergodic MFG for the initial condition (x, i)if (1) J(x, i;ξ∗(θ∗), θ∗)≥J(x, i;ξ, θ∗),for any ξ∈ Ae. 6 DIANETTI, FERRARI, AND TZOUANAS (2) The optimally controlled state process (Xξ∗(θ∗), Y )admits a limiting stationary distribution µθ∗and θ∗=FPd i=1 RIf(x)µθ∗(dx, i). In the sequel, our solution plan will be as following: (1) For a fixed mean-field parameter θ∈R+, we solve the ergodic control problem aiming at maximizing (2.6) over Ae. (2) We then impose the consistency condition (2) in Definition 2.1 and we prove the existence and uniqueness of the mean-field parameter θ∗via a fixed-point argument. (3) We finally show that the ergodic MFG equilibrium realizes an ǫN-Nash equilibrium for a suitable ergodic N-player game of singular control with regime-switching. 3. THE ERGODIC OPTIMAL CONTROL PROBLEM Recalling (2.6), in this section we fix θ∈R+and solve the ergodic control problem. In particular, we want to find (3.1) ¯ λ(θ) := sup ξ∈Ae J(x, i;ξ, θ). Notice that the value ¯ λis independent of (x, i)∈ O, since an initial jump does not alter the value of the limit in (2.6). In order to solve (3.1), we let V:O × R+→Rand λ:R+×Y→Rto be determined such that V(·, i;θ)∈C2(I),for any (i, θ)∈Y×R+and the pair (V, λ)solves the variational inequality (3.2) max L(X,Y )V(x, i;θ) + π(x, θ)−λ(θ, i), Vx(x, i;θ)−k1, k2−Vx(x, i;θ)= 0. It will be shown in Proposition 3.4 below that a solution (V, λ)to (3.2) allows to obtain ¯ λin the sense that (3.3) ¯ λ(θ) := d X i=1 p(i)λ(θ, i). The following additional assumption on πholds throughout the rest of this paper. Assumption 3.1. For x−(θ) := (x−(1, θ), ..., x−(d, θ)), x+(θ) := (x+(1, θ), ..., x+(d, θ)), with x−(i, θ)< x+(i, θ), i ∈Y, it holds: πx(x, θ) + k1bx(x, i)     >0, x ∈(x, x−(i, θ)), = 0, x =x−(i, θ), <0, x ∈(x−(i, θ),x), πx(x, θ) + k2bx(x, i)     >0, x ∈(x, x+(i, θ)), = 0, x =x+(i, θ), <0, x ∈(x+(i, θ), x). Assumption 3.1 together with Assumption 2.2-(4) guarantee that a solution to (3.2) will be of threshold type, meaning that there shall exist α(i, θ)< β(i, θ), i ∈Y, such that Vx(x, i;θ)< k1 on {(x, i)∈ O :x < α(i, θ)}and Vx(x, i;θ)> k2on {(x, i)∈ O :x > β(i, θ)}. 3.1. An auxiliary optimal stopping game. To deal with (3.2), we introduce the auxiliary optimal stopping game (Dynkin game) with stopping functional b J(x, i;τ, σ, θ) := b E(x,i)Zτ∧σ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt +k1eRτ 0bx(b Xt,Yt)dt1{τ<σ} (3.4) +k2eRσ 0bx(b Xt,Yt)dt1{σ<τ}, ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 7 where (b Xt, Yt)t≥0is the unique strong solution to (2.3) and τ, σ ∈ T , where T:= {ρ: Ω →[0,∞] : ρis an F-stopping time}. In (3.4), we use the convention eRρ 0bx(b Xx s,Y i s)ds = 0 on {ρ=∞}. The Dynkin game is such that Player 1 aims at minimizing (3.4) over τ∈ T , while Player 2 at maximizing (3.4) over σ∈ T . Theorem 2.1 in [44] allows to show that such a game indeed admits a value. Theorem 3.1. Let (x, i, θ)∈ O × R+. Then, (3.5) inf τ∈T sup σ∈T b J(x, i;τ, σ, θ) = sup σ∈T inf τ∈T b J(x, i;τ, σ, θ), and we define the value function (3.6) v(x, i;θ) := inf τ∈T sup σ∈T b J(x, i;τ, σ, θ) = sup σ∈T inf τ∈T b J(x, i;τ, σ, θ). Moreover, letting the continuation region be Cθ:= {(x, i)∈ O :k2< v(x, i;θ)< k1}, and the stopping regions be Sθ inf := {(x, i)∈ O :v(x, i;θ)≥k1},Sθ sup := {(x, i)∈ O :v(x, i;θ)≤k2}, the stopping times τ∗(x, i;θ) := inf{t≥0 : ( b Xt, Yt)∈ Sθ inf}, σ∗(x, i;θ) := inf{t≥0 : ( b Xt, Yt)∈ Sθ sup},b P(x,i)-a.s., realize a saddle-point. In particular, v(x, i;θ) = b J(x, i;τ∗(x, i;θ), σ∗(x, i;θ), θ). Proof. See Appendix A. The following proposition is easily proved thanks to Conditions (2) and (3) in Assumption 2.2. Proposition 3.1. It holds: (1) The mapping x7→ v(x, i;θ)is decreasing for every (i, θ)∈Y×R+. (2) The mapping θ7→ v(x, i;θ)is decreasing for every (x, i)∈ O. The next result excludes the possibility of empty stopping regions. Proposition 3.2. The following hold: Sθ inf 6=∅and Sθ sup 6=∅. Proof. See Appendix A. We define the free-boundaries (3.7) αi(θ) := sup{x∈ I :v(x, i;θ)≥k1}, βi(θ) := inf{x∈ I :v(x, i;θ)≤k2}, for any (i, θ)∈Y×R+, with the conventions sup ∅=xand inf ∅=x. Then, thanks to the monotonicity of v(·, i;θ),(i, θ)∈Y×R+, we have (3.8) Cθ={(x, i)∈ O :x∈(αi(θ), βi(θ))}, (3.9) Sθ inf ={(x, i)∈ O :x∈(x, αi(θ)]}and Sθ sup ={(x, i)∈ O :x∈[βi(θ), x)}. Notice that, due to Assumption 3.1,x−(i, θ)< α(i, θ)and β(i, θ)< x+(i, θ), for any (i, θ)∈Y×R+. The next proposition provides regularity results about the value of the Dynkin game (3.6). The proof is in the same spirit as that of Theorem 4.3 in [25]. Proposition 3.3. For any (i, θ)∈Y×R+we have v(·, i;θ)∈C1(I)∩C2(I \ {αi(θ), βi(θ)}). Proof. See Appendix A. 14 DIANETTI, FERRARI, AND TZOUANAS Proof. Take arbitrary i∈Yand let θ1, θ2∈R+such that θ1≤θ2. Then, by Proposition 3.1, we have that the map θ7→ v(x, i;θ)is decreasing, for any (x, i)∈ O. Hence, {x∈ I :v(x, i;θ2)≥k1} ⊆ {x∈ I :v(x, i;θ1)≥k1}, which, by (3.7), implies that αi(θ2)≤αi(θ1). The monotonicity of θ7→ βi(θ), i ∈Y, can be proved similarly.  Lemma 4.2. The following hold. (1) For any compact X, the map θ7→ v(x, i;θ)is locally Lipschtiz continuous, uniformly for (x, i)∈ X × Y. (2) The function vx(x, i;θ)is nonpositive for any (x, i, θ)∈ O × R+and for fixed θ∈R+is strictly negative for any (x, i)∈ Cθ, where Cθis the continuation region as in Theorem 3.1. Proof. Take a compact X × Θ⊂ I × R+and generic (x, i, θ)∈ X × Y×Θ. Following the proof of Proposition 3.3 (cf. Appendix A), for ǫ > 0small enough, by mean value theorem, we have that v(x, i;θ+ǫ)−v(x, i;θ)≤b E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ +ǫ)−πx(b Xt, θ)dt =ǫb E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπxθ(b Xt,˜ θ)dt ≤ǫb E(x,i)Z∞ 0 e−ctπxθ(b Xt,˜ θ)dt≤ǫ C(X,Θ∗), for ˜ θ∈(θ, θ +ǫ)and a compact Θ∗such that θ+ǫ∈Θ∗for any θ∈Θ. Thus, since C(X,Θ∗)<∞by Assumption 4.1-(2), we deduce that the map θ7→ v(x, i;θ)is locally Lipschtiz continuous, proving Claim (1). We next prove Claim (2). Using the representation of vxin Lemma A.1, Assumption 2.2-(2) and Assumption 4.1-(2), one can easily check (by observing that ∂xb Xx t>0,b P⊗dt-a.s., since ∂xb Xxis a solution to a linear equation) that the function vxis nonpositive for any (x, i, θ)∈ O × R+. For arbitrary fixed θ∈R+and for any (x, i)∈ Cθwe know that τ∗(x, i;θ)∧σ∗(x, i;θ)>0,b P-a.s., hence by representation of vx(cf. Lemma A.1) we conclude that vx(x, i, ;θ)<0. Proposition 4.1. The map θ7→ (αi(θ), βi(θ))i∈Yis continuous. Proof. For fixed i∈Ywe, first prove that the map θ7→ αi(θ)is left-continuous. Fix θ∈R+, a sequence {θn}n∈N, and a compact Θ⊂R+such that θnրθ, and θ, θn∈Θfor any n∈N. Since the map θ7→ αi(θ)is nonincreasing (cf. Lemma 4.1), arguing by contradiction, we assume that there exists δ > 0such that αi(θ) + δ≤αi(θn)for any n∈N. Observe that, since the function αiis finite and monotone (cf. Lemma 4.1), one has (4.1) αi(θn)∈hinf y∈Θαi(y),sup y∈Θ αi(y)i=: K,for any n, with Kbeing compact. Thanks to the fourth display equation in (3.10) we have that v(αi(θ), i;θ) = v(αi(θn), i;θn) = k1for any n∈N. Hence, using Proposition 3.3 and Lemma 4.2 we can write 0 = v(αi(θn), i;θn)−v(αi(θ), i;θ) =v(αi(θn), i;θn)−v(αi(θn), i;θ)+v(αi(θn), i;θ)−v(αi(θ), i;θ) =−Zθ θn vθ(αi(θn), i;y)dy +Zαi(θn) αi(θ) vx(y, i;θ)dy ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 15 or, equivalently, (4.2) Zαi(θn) αi(θ)−vx(y, i;θ)dy =Zθ θn−vθ(αi(θn), i;y)dy. By Lemma 4.2 we have vx(x, i;θ)<0for fixed θ∈Θand for any (x, i)∈ Cθ, then for ǫ∈(0, δ) small enough we have that Zαi(θn) αi(θ)−vx(y, i;θ)dy ≥Zαi(θ)+δ αi(θ)+ǫ−vx(y, i;θ)dy(4.3) ≥min y∈[αi(θ)+ǫ,αi(θ)+δ]|vx(y, i;θ)|(δ−ǫ)>0, and (4.4) Zθ θn−vθ(αi(θn), i;y)dy ≤max y∈Θ|vθ(αi(θn), i;y)|(θ−θn). Combining (4.3) and (4.4) with (4.2) we arrive at the inequality, 0<min y∈[αi(θ)+ǫ,αi(θ)+δ]|vx(y, i;θ)|(δ−ǫ)≤max (x,y)∈K×Θ|vθ(x, i;y)|(θ−θn)≤C(θ−θn), for a constant Cnot depending on n(thanks to Lemma 4.2). Thus, letting n→ ∞ leads to a contradiction. We next show that the map θ7→ αi(θ)is right-continuous. Fix a sequence {θn}n∈N⊆Θsuch that θnցθ. Again by the monotonicity of the map θ7→ αi(θ), arguing by contradiction we assume that there exists δ > 0such that αi(θn)≤αi(θ)−δfor any n∈N. Similarly to the first part of the proof, we find Zθn θ−vθ(αi(θn), i;y)dy =Zαi(θ) αi(θn)−vx(y, i;θn)dy. which leads to the following inequality 0<min y∈[αi(θ)−δ,αi(θ)−ǫ]|vx(y, i;θ)|(δ−ǫ)≤max (x,y)∈K×Θ|vθ(x, i;y)|(θn−θ)≤C(θ−θn), for Kas in (4.1) and a constant Cnot depending on n. Thus, letting n→ ∞, we obtain a contradiction. By repeating the same argument, one can show that the map θ7→ βi(θ)is continuous, thus completing the proof.  4.2. Analysis of the stationary distribution. First of all, we show that a stationary distribution for the process (Xξ∗(θ) t, Yt)t≥0exists. Proposition 4.2. For any θ∈R+, there exists a unique stationary distribution νθ∈ P(O)for the hybrid-reflected process (Xξ∗(θ) t, Yt)t≥0. Proof. In light of Theorem 2.1 in [47], in order to establish the existence and uniqueness of a stationary distribution, it is sufficient to show that (Xξ∗(θ) t, Yt)t≥0is a regenerative process (cf. Definition (D1) in [47]) with finite length regenerative epochs. In particular, for (x, i)∈ O, setting η:= inf{t > 0 : Xξ∗(θ) t=αYt(θ)},P(x,i)-a.s., by strong Markov property we have that the processes (Xξ∗(θ) t+η, Yt+η)t≥0and the collection of random variables ((Xξ∗(θ) t, Yt)t<η, η)are independent. Moreover, when starting from points in the set {(αi(θ), i)|i∈Y}, the process (Xξ∗(θ) t, Yt)t≥0has the same distribution as the process (Xξ∗(θ) t+η, Yt+η)t≥0. Thus, (Xξ∗(θ) t, Yt)t≥0is a regenerative process, with regenerative epoch η. In order to conclude the proof, it remains to show that E(x,i)[η]<∞, for any (x, i)∈ O. 16 DIANETTI, FERRARI, AND TZOUANAS For (x, i)∈ O and y∈[minj∈Yαj(θ),maxj∈Yβj(θ)], we define the F-hitting time η(y) := inf{t > 0 : Xξ∗(θ) t=y}. If Y0=iby Proposition B.2 we have that Xx,ξ∗(θ) t≤Xβi(θ),ξ∗(θ) t,Pi-a.s.. Thus, we have (4.5) E(x,i)[η]≤E(βi(θ),i)[η]≤E(βi(θ),i)[η(αi(θ))]. We now estimate E(βi(θ),i)[η(αi(θ))]. Let τbe the first jump of the process Yand denoting by Z the solution to the SDE (4.6) dZt=b(Zt, i)dt +σ(Zt, i)dWt, Z0=βi(θ), for ˜η:= inf{t > 0 : Zt=αi(θ)}, by a comparison principle (a slightly different version of Proposition B.2) we also have Xβi(θ),ξ∗(θ) t≤Zt,for any t∈[0, τ ∧˜η),Pi-a.s., which in turn implies P(βi(θ),i)η(αi(θ)) < τ≥P(βi(θ),i)˜η < τ. Therefore, we find (4.7) P(βi(θ),i)η(αi(θ)< τ≥P(βi(θ),i)˜η < τ=Z∞ 0 E[1{˜η<t}]Fτ(dt) := ρ > 0, where Fτdenotes the distribution function of τand the last inequality follows from the fact that the SDE (4.6) induces a regular diffusion (cf. Assumption 2.1). Using (4.7), we have E(x,i)η(αi(θ))=E(x,i)η(αi(θ))1{η(αi(θ))<τ}+E(x,i)η(αi(θ))1{η(αi(θ))≥τ} (4.8) ≤Eiτ+E(x,i)E(Xξ∗(θ) τ,Yτ)η(αi(θ))(1 −1{η(αi(θ))<τ}) =Eiτ+E(x,i)(1 −1{η(αi(θ))<τ})E(x,i)E(Xξ∗(θ) τ,Yτ)η(αi(θ)) =Eiτ+1−P(x,i)η(αi(θ)) < τE(x,i)η(αi(θ)), ≤Eiτ+ (1 −ρ)E(x,i)η(αi(θ)), where in the second equality we use the strong Markov property of (Xξ∗(θ) t, Yt)t≥0. Then, since (4.8) holds for any (x, i)∈ O, we obtain sup (x,i)∈O E(x,i)η(αi(θ))≤Eiτ+ (1 −ρ) sup (x,i)∈O E(x,i)η(αi(θ)). Equivalently, we write (4.9) sup (x,i)∈O E(x,i)η(αi(θ))≤1 ρEiτ<∞, where, in the last inequality, we have used that the process (Yt)t≥0is irreducible and positive recurrent. Finally, plugging (4.9) into (4.5), we conclude that E(x,i)[η]<∞, which in turn implies that (Xξ∗(θ) t, Yt)t≥0is a regenerative process with finite mean regenerative epochs. Hence, by Theorem 2.1 in [47], there exists a unique stationary distribution, concluding the proof.  Next, we characterize the stationary distribution and we study its stability with respect to changes of the boundary. Theorem 4.1. Let νθ∈ P(O)be the stationary distribution of (Xξ∗(θ) t, Yt)t≥0. The following hold: (1) For any i∈Y, the cumulative distribution function µθ(x, i) := νθ((x, x], i), x ∈ I has regularity µθ(·, i)∈C1([αi(θ), βi(θ)]) ∩C2((αi(θ), βi(θ)) \Sj∈Y{αj(θ), βj(θ)})and it is the unique nondecreasing solution of the equation (4.10) 1 2σ2(x, i)µθ xx(x, i)−(b(x, i)−σσx(x, i))µθ x(x, i)+X j∈Y qjiµθ(x, j) = 0, x ∈(αi(θ), βi(θ)), ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 17 satisfying the boundary conditions µθ(x, i) = 0, x ≤αi(θ)and µθ(x, i) = p(i), x ≥βi(θ). (2) The map ν:R+→ P(O), θ 7→ νθis continuous. Proof. The proof of the first item follows closely the proof of Theorem 1 in [23], and it is proved in Appenidx C. Therefore, we only provide a proof of Claim 2. For any (α, β) = (αi, βi)i∈Y, we denote by µ(·,·;α, β)the solution of (4.10) with (α(θ), β(θ)) = (α, β). Then for Θ⊂R+compact such that θ∈Θand for {θn}n∈N⊂Θsuch that θn→θ, as n↑ ∞, we know by Proposition 4.1 that αi(θn)→αi(θ) = αiand βi(θn)→βi(θ) = βifor any i∈Yas n↑ ∞. For simplicity we set (αn i, βn i)i∈Y:= (αi(θn), βi(θn))i∈Yfor any n∈N. Unless repeating the subsequent argument with minor modifications, we assume that there exists n0∈Nsuch that, for any n≥n0and for any i∈Y, we have (αn i, βn i)⊆(αi, βi). For any n∈N, let µn(x, i) := νn((x, x], i),(x, i)∈ O be the cumulative distribution function of the stationary distribution of the solution to SP(αn, βn, x, i)(cf. Proposition 4.2). According to Claim 1,µnsatisfies (4.10) with boundary condition on (αn, βn). Next, for any n∈Nand for any (x, i)∈Sd i=1(αn i, βn i)× {i}, define vn(x, i) := µ(x, i)−µn(x, i). Since µand µnare solutions to the linear system (4.10), for each n∈N,vnis a solution to the boundary value problem (4.11) (L∗ (b X,i)vn(x, i) + Pj∈Yqjivn(x, j) = 0, x ∈(αn i, βn i), i ∈Y, vn(αn i, i) = µ(αn i, i), vn(βn i, i) = µ(βn i, i)−p(i), where L∗ (b X,i)vn(x, i) := 1 2σ2(x, i)vn xx(x, i)−(b(x, i)−σσx(x, i))vn x(x, i). Now, thanks to Assumption 2.1 and Assumption 4.1-(3), we can apply Theorem 1 in [48] to (4.11) which gives us sup x∈(αn i,βn i) |vn(x, i)| ≤ Cmax |µ(αn i)|,|µ(βn i, i)−p(i)|. Therefore, for any x∈(αi, βi), we deduce that µn(x, i)→µ(x, i)as n→ ∞. Hence, by an application of Theorem 5.25 in [34] we obtain that νn⇀ νθas n→ ∞ thus completing the proof of Claim 2. 4.3. Existence and uniqueness of the MFG equilibrium. Next, we turn our attention to the main result of this section. To this end, we introduce the operator T:R+→R+, as (4.12) Tθ:= Fd X i=1 ZI f(x)νθ(dx, i)=Fhf, νθi, θ ∈R+ where hf, νθi:= Pi∈YRIf(x)νθ(dx, i). Thanks to the previous results, we can now prove the existence and uniqueness of a stationary mean-field equilibrium. Theorem 4.2. There exists a unique MFG equilibrium θ∗; i.e., a unique θ∗∈R+such that Tθ∗=θ∗. Proof. We divide the proof in three steps. Step 1: Set of relevant θ.We introduce the following auxiliary Dynkin game v(x, i) := inf τ∈T sup σ∈T b E(x,i)Zτ∧σ 0 eRt 0bx(b Xs,Ys)dsκ(b Xs)ds(4.13) +k1eRτ 0bx(b Xs,Ys)ds1{τ<σ}+k2eRσ 0bx(b Xs,Ys)ds1{σ<τ}. Arguing as in Section 3.1, the Dynkin game (4.13) admits a value and a saddle point. In particular, we can define αi:= sup{x∈ I :v(x, i)≥k1}and βi:= inf{x∈ I :v(x, i)≤k2}, as well as the stopping regions Sinf := {(x, i)∈ O :v(x, i)≥k1}and Ssup := {(x, i)∈ O :v(x, i)≤k2}. Thus, we have that the stopping times τ∗(x, i) := inf{t≥0 : ( b Xt, Yt)∈ Sinf}and σ∗(x, i) := inf{t≥ 0 : ( b Xt, Yt)∈ Ssup}realize a saddle-point for (4.13). Also by Condition (3) in Assumption 2.2 and 18 DIANETTI, FERRARI, AND TZOUANAS Condition (1) in Assumption 4.1 we conclude that πx(x, θ)≥κ(x), for any θ∈R+, so that (cf. (3.4) and (3.6)) (4.14) v(x, i)≤v(x, i;θ),for any (x, i;θ)∈ O × R+. Then, for any (i, θ)∈Y×R+, we have (4.15) αi(θ) = sup{x∈ I :v(x, i;θ)≥k1} ≥ sup{x∈ I :v(x, i)≥k1}=αi. Analogously, we find (4.16) βi(θ)≥βi,for any (i, θ)∈Y×R+. We then define (4.17) θ:= Fd X i=1 ZI f(x)ν(dx, i) where νis the unique stationary distribution of the process (Xt, Yt)t≥0and Xis the solution to the SP((αi, βi)i∈Y;X0, Y0). Then, by Proposition B.2, since fis increasing, we obtain (4.18) Ef(Xt)≤Ef(Xξ∗(θ) t),for any t≥0. Hence, by Proposition 4.2, we can employ the ergodic theorem (see pg.274 in [47]), in order to deduce that hf, νi= lim t→∞ 1 tZt 0 Ef(Xt)dt ≤lim t→∞ 1 tZt 0 Ef(Xξ∗(θ) t)dt =hf, νθi, which in turn, by monotonicity of F, leads to (4.19) θ≤ T θ. To find an upper bound for Tθ, we proceed as follows. Since, Xξ∗(θ) t∈[αYt(θ), βYt(θ)],for any t≥ 0,P-a.s. for any θ∈R+, thanks to the monotonicity of Fand f(see Condition (1) Assumption 2.3), we have Tθ=Fd X i=1 ZI f(x)νθ(dx, i)≤Fd X i=1 p(i)f(βi(θ))≤Fd X i=1 p(i)f(βi(θ)), for any θ≥θ. In the last inequality above we have used the fact that the map θ7→ (βi(θ))i∈Yis nonincreasing (cf. Lemma 4.1). Hence, we define θ:= FPd i=1 p(i)f(βi(θ))and for any θ≤θwe find that (4.20) Tθ≤θ. Thus, combining (4.19) and (4.20), we conclude that any potential fixed point of Tmust lie in the convex, compact set (4.21) K:= [θ, θ]⊂R+. Step 2: Continuity of T.We begin by observing that the barriers related to θ∈K, belong to a compact b K, defined in terms of the compact set Kin (4.21). Indeed, by Step 1 we know that αi≤αi(θ)for any (i, θ)∈Y×K, and, by monotonicity of the map θ7→ βi(θ)for i∈Ywe have βi(θ)≤βi(θ)for i∈Y. Thus, for each θ∈Kwe have, (4.22) αi(θ), βi(θ)∈hmin j∈Yαj,max j∈Yβj(θ)i=: b K, for any i∈Y. Define now the map T1:K→ P(b K×Y)by T1(θ) := νθ, θ ∈K. ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 19 By (4.22), such a map is well defined. Moreover, thanks to Claim 2in Theorem 4.1, the map T1is continuous. Next, we denote by T2:P(b K×Y)→Kthe map T2(ν) := Fd X i=1 ZI f(x)ν(dx, i). Since the functions fand Fare continuous and the probability measures have compact support, the map T2is clearly continuous. Concluding, the map T:= T2◦ T1:K→Kis continuous in the convex compact set Kand, by Schauder-Tychonof fixed point theorem ( Corollary 17.56 in [4]), there exist θ∗∈K, such that Tθ∗=θ∗. Step 3: Uniqueness. Let θ∗∈Kbe the fixed-point of Tand let θ′∈Kanother fixed-point of T such that θ∗6=θ′. Without loss of generality we can assume that θ∗< θ′(the opposite inequality can be treated similarly). Following the same arguments as in Step 1 we conclude that αi(θ∗)≥αi(θ′), βi(θ∗)≥βi(θ′),∀i∈Y, so that θ′=Tθ′≤ T θ∗=θ∗, which leads to a contradiction.  5. N-PLAYER GAME AND APPROXIMATE NASH EQUILIBRIA In this section we establish the classical connection between MFG and N-player game, by constructing approximate Nash equilibria starting from the MFG equilibrium. 5.1. N-player game. Let Wand Ybe as in Section 2and assume the filtered probability space (Ω,F,F={Ft}t≥0,P)to be large enough to accommodate a sequence of independent and identically distributed F-adapted processes {(Wn, Y n)}n∈Nas well as independent and identically distributed I × Y-random variables {(xn 0, in 0)}n∈N,(x0, i0). Each (Wn, Y n)has the same distribution as (W, Y )and the random variables (Wn, Y n),(W, Y ),{(xn 0, in)}n∈Nand (x0, i0)are assumed to be independent. When player ndoes not intervene, its state process Xnevolves accordingly to the SDE dXn t=b(Xn t, Y n t)dt +σ(Xn t, Y n t)dWn t,(Xn 0, Y n 0) = (xn 0, in 0). Given a boundary vector (αn, βn) = (αn j, βn j)j∈Y∈R2d, the (αn, βn)-barrier-type strategy for player nis the reflection process for the state of player nin the regime-switching domain (αn Yt, βn Yt)t≥0; that is, the ξn-component of the solution (Xn, ξn)to the Skorokhod problem SP(αn, βn, x, i)for the noise (Wn, Y n), accordingly to Definition 3.1. Without carrying the dependence on the index of the player, we denote by Abthe set of barrier-type strategies. Thus, when player nchoose a strategy ξn∈ Ab, its state process Xn,ξnevolves following the regime-switching (reflected) SDE dXn,ξn t=b(Xn,ξn t, Y n t)dt +σ(Xn,ξn t, Y n t)dWn t+dξn,+ t−dξn,− t,(Xn,ξn 0−, Y n 0) = (xn 0, in 0). The (generic) vector ξ:= (ξ1, ..., ξN)∈ AN brepresents a tuple of control policies of the Nplayers. We denote by ξ−n= (ξℓ)ℓ6=na vector of strategies of the opponents of player n, and we use the notation ξ= (ξn,ξ−n). The profit functional of player nis defined as (5.1) Jn(ξn,ξ−n) := lim sup T↑∞ 1 TEZT 0 π(Xn,ξn t, θN ξ−n(t))dt −k1ξn,+ T+k2ξn,− T, where θN ξ−ndenotes the mean-field interaction term between the players and has the form (5.2) θN ξ−n(t) := F1 N−1X ℓ6=n f(Xℓ,ξℓ t), t ≥0. 20 DIANETTI, FERRARI, AND TZOUANAS Our aim is to use the solution of the ergodic MFG as an approximating solution of the N-player game. In particular, for b Kbeing the compact set as in (4.22), defining the set of restricted barrier strategies (5.3) b Ab:= {(αn, βn)-barrier-type strategy with (αn j, βn j)j∈Y∈b K2d}, we give the following definition of ǫ-Nash equilibrium. Definition 5.1 (ǫ-Nash equilibrium).For ǫ > 0,¯ ξ= (¯ ξ1, ..., ¯ ξN)∈b AN bis called ǫ-Nash equilibrium for the N-player game if, for any n= 1, ..., N, one has (5.4) Jn(¯ ξn,¯ ξ−n)≥Jn(ξn,¯ ξ−n)−ǫ, ∀ξn∈b Ab. Let θ∗be the unique MFG equilibrium as in Theorem 4.2 and (α(θ∗), β(θ∗)) = (αi(θ∗), βi(θ∗))i∈Y be the related boundary (see (3.7)). We now use the MFG equilibrium θ∗in order to construct profile strategies for the N-player games. For any N≥1and n= 1, ..., N, define the strategy ¯ ξnas the (α(θ∗), β(θ∗))-barrier strategy for the noise (Wn, Y n). Remark 5.1 (On the initial distribution).We point out that all the results in the previous sections hold true also if the deterministic initial condition (X0−, Y0) = (x, i)∈ I × Yis replaced by the random initial condition (X0−, Y0) = (x0, i0). Indeed, by considering (with slight abuse of notation) the payoff J(x0, i0;ξ, θ) := lim sup T↑∞ 1 TEE(x0,i0)ZT 0 π(Xξ t, θ)dt −k1ξ+ T+k2ξ− T, by the Markov property of the solution to the reflected Skorokhod problem the MFG equilibrium (ξ(θ∗), θ∗)is still given by Theorem 4.2, with ξ(θ∗)characterized as in in Proposition 3.5. In light of the definition of b Kin (4.22), we have ¯ ξn∈b Ab. Let ¯ ξbe the related profile strategy; i.e., set ¯ ξ:= (¯ ξ1, ..., ¯ ξN). We first show the following preliminary result. Proposition 5.1. For any N≥1and {ξn}n≤N∈b AN b, the ergodic limit lim T↑∞ 1 TZT 0 G(X1,ξ1 t, Y 1 t, ..., XN,ξN t, Y N t)dt =ZON G(x1, i1, ..., xN, iN)⊗N ℓ=1 νℓ(dxℓ, iℓ),P-a.s. holds for any bounded function G:ON→R. Proof. Since {(Wn t, Y n t)t≥0}n∈Nis i.i.d., the family {(Xn,ξn t, Y n t)t≥0}n≤Nis i.i.d. For any fixed 1≤n≤N, by the proof of Proposition 4.2 we know that the process (Xn,ξn t, Y n t)t≥0has a unique stationary distribution νn. Thus, by independence of the processes (Xn,ξn t, Y n t)t≥0, the stationary distribution for (X1,ξ1, Y 1, ..., XN,ξN, Y N), denoted by ¯νN∈ P(ON), exists and it is given by ¯νN(dx1, i1, ..., dxN, iN) = ⊗N ℓ=1νℓ(dxℓ, iℓ). Clearly, this distribution is unique. Thus, by Theorems 3.2.6 and 3.3.1 in [21], the ergodic limit holds.  5.2. Approximation result. We enforce the following condition. Assumption 5.1. There exists C > 0such that, |π(x, θ1)−π(x, θ2)| ≤ C(1 + |x|β)|θ1−θ2|, for any θ1, θ2∈R+and x∈ I. In the spirit of [15] and [16], we can now state and prove the main result of this section. ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 21 Theorem 5.1. The profile strategy ¯ ξ:= (ξ1, ..., ξN)∈b AN bis an ǫN-Nash equilibrium for the ergodic N-player game, with ǫN→0as N→ ∞. Proof. We first introduce some notation. Take barriers (α, β) = (αi, βi)i∈Y∈b K2d, with b Kas in (4.22). Denote by (Xn,ξn, ξn)the solution to SP(α, β, x, i)for the noise (Wn, Y n)of player n, and denote by (˜ X, ˜ ξ)the solution to SP(α, β, x, i)for the noise (W, Y )of the MFG. By uniqueness in law of the solution to the Skorokhod problem, the processes (Xn,ξn, ξn)and (˜ Xn,ξn,˜ ξn)have the same law, hence Jn(ξn, θ∗) := lim sup T↑∞ 1 TEZT 0 π(Xn,ξn t, θ∗)dt −k1ξn,+ T+k2ξn,− T (5.5) = lim sup T↑∞ 1 TEZT 0 π(˜ Xn,ξn t, θ∗)dt −k1˜ ξn,+ T+k2˜ ξn,− T=: J(˜ ξ, θ∗). Next, introduce the error estimate between the functionals of the N-player game and the MFG. For N∈Nand 1≤n≤N, set RN(ξn) := Jn(ξn,¯ ξ−n)−Jn(ξn, θ∗), ξn∈b Ab. Since the control ξ(θ∗)is optimal for θ∗(cf. Proposition 3.5), using (5.5), we have that Jn(¯ ξn,¯ ξ−n)−Jn(ξn,¯ ξ−n) = RN(¯ ξn)−RN(ξn) + Jn(¯ ξn, θ∗)−Jn(ξn, θ∗) =RN(¯ ξn)−RN(ξn) + J(ξ(θ∗), θ∗)−J(˜ ξn, θ∗) ≥RN(¯ ξn)−RN(ξn),for any ξn∈b Ab. Therefore, in order to complete the proof it is sufficient to show that (5.6) lim N↑∞ sup ξn∈b Ab RN(ξn) = 0. By using that limT↑∞αT+βT≥limT↑∞(αT) + limT↑∞(βT), write RN(ξn) = lim sup T↑∞ 1 TEZT 0 π(Xn,ξn t, θN ¯ ξ−n(t))dt −k1ξn,+ T+k2ξn,− T (5.7) −lim sup T↑∞ 1 TEZT 0 π(Xn,ξn t, θ∗)dt −k1ξn,+ T+k2ξn,− T ≤lim sup T↑∞ 1 TEZT 0π(Xn,ξn t, θN ¯ ξ−n(t)) −π(Xn,ξn t, θ∗)dt. Next, by Proposition 4.2, there exists a stationary distribution νξnfor the process (Xn,ξn t, Yt)t≥0. Moreover, Proposition 5.1 ensures that the process (Xn,ξn, Y n),{(Xℓ,¯ ξℓ, Y ℓ)t≥0}ℓ6=nadmits an ergodic distribution ⊗ℓ6=nνℓ⊗νξnwith support in the compact set (b K×Y)N. Thus, by continuity of the functions f, F and π, Proposition 5.1 also allows to rewrite (5.7) as RN(ξn)≤ZOZON−1πxn, FX ℓ6=n f(xℓ) N−1−π(xn, θ∗)⊗ℓ6=nνℓ(dxℓ, iℓ)⊗νξn(dxn, in). 22 DIANETTI, FERRARI, AND TZOUANAS We proceed by further estimating supξn∈b AbRN(ξn). From the latter inequality, since ⊗ℓ6=nνℓ⊗νξn has compact support, we can use Assumption 5.1 to obtain RN(ξn)≤CZON (1 + |xn|β)FX ℓ6=n f(xℓ) N−1−Fhf, νθ∗i⊗ℓ6=nνℓ(dxℓ, iℓ)⊗νξn(dxn, in) ≤CZO (1 + |xn|β)νξn(dxn, in) ×ZON−1FX ℓ6=n f(xℓ) N−1−Fhf, νθ∗i⊗ℓ6=nνℓ(dxℓ, iℓ) ≤Cb KZON−1FX ℓ6=n f(xℓ) N−1−Fhf, νθ∗i⊗ℓ6=nνℓ(dxℓ, iℓ), where the constant Cb Kdepends only on the compact b K, since νξnha support in the compact b K (ξn∈b Abas in (5.3)). Next, using the local Lipschitz property of F(cf. Assumption (2b)-(2.3)), and again the fact that νℓare supported in the compact b K×Y, we obtain sup ξn∈b Ab RN(ξ) ≤Cb KZON−11 + X ℓ6=n |f(xℓ)| N−1+hf, νθ∗i1 β−1X ℓ6=n f(xℓ) N−1− hf, νθ∗i⊗ℓ6=nνℓ(dxℓ, iℓ) ≤Cb KZON−1 1 N−1X ℓ6=n f(xℓ)− hf, νθ∗i⊗ℓ6=nνℓ(dxℓ, iℓ). Finally, by using the strong law of large numbers (cf. Theorem 4.23 in [34]) and the dominated convergence theorem, a limit as N→ ∞ in the latter inequality leads to (5.6). This completes the proof of the theorem.  APPENDIX A. RESULTS ON THE OPTIMAL STOPPING GAME Proof of Theorem 3.1.Let (x, i, θ)∈ O × R+and τ, σ ∈ T given and fixed. We aim at applying Theorem 2.1 in [44], and for this we first notice that b J(x, i;τ, σ, θ) = b E(x,i)Zτ∧σ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt +k1eRτ 0bx(b Xs,Ys)ds1{τ≤σ} +k2eRσ 0bx(b Xs,Ys)ds1{σ<τ}=b E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt −Z∞ τ∧σ eRt 0bx(b Xs,Ys)dtπx(b Xt, θ)dt +k1eRτ 0bx(b Xs,Ys)ds1{τ≤σ}+k2eRσ 0bx(b Xs,Ys)ds1{σ<τ} =b E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt−b E(x,i)b E(x,i)Z∞ τ∧σ eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dtFτ∧σ +b E(x,i)k1eRτ 0bx(b Xs,Ys)ds1{τ≤σ}+k2eRσ 0bx(b Xs,Ys)ds1{σ<τ}. Then, defining G0(x, i;θ) := b E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt, ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 23 by the strong Markov property of (b Xt, Yt)t≥0we obtain (A.1) b J(x, i;τ, σ, θ) = G0(x, i;θ)−b E(x,i)eRτ∧σ 0bx(b Xs,Ys)dsG0(b Xτ∧σ, Yτ∧σ;θ) +b E(x,i)k1eRτ 0bx(b Xs,Ys)ds1{τ≤σ}+k2eRσ 0bx(b Xs,Ys)ds1{σ<τ}. Therefore, (A.2) b J(x, i;τ, σ, θ) = G0(x, i;θ) + b E(x,i)eRτ 0bx(b Xs,Ys)dsk1−G0(b Xτ, Yτ;θ)1{τ≤σ} +b E(x,i)eRσ 0bx(b Xs,Ys)dsk2−G0(b Xσ, Yσ;θ)1{σ<τ}; that is, b J(x, i;τ, σ, θ) = G0(x, i;θ) + b E(x,i,1)G1(¯ Xt;θ)1{τ≤σ}+G2(¯ Xt;θ)1{σ<τ}, where we have set G1(x, i, z;θ) := z(k1−G0(x, i;θ)),G2(x, i, z;θ) := z(k2−G0(x, i;θ)), we have introduced the 3-dimensional right-continuous strong Markov process (A.3) ¯ Xt:= ( b Xt, Yt, Zt), with Zt:= z·eRt 0bx(b Xs,Ys)ds, and b E(x,i,1) is the expectation with the respect to b P(x,i,1)[·] := b P[· | b X0= x, Y0=i, Z0= 1]. We observe that by Condition (3) in Assumption 2.1 and Condition (5) in Assumption 2.2, b E(x,i,1)sup t≥0Gi(¯ Xt;θ)<∞, i = 0,1,2, where as well as, G2(¯ Xt;θ)≤G1(¯ Xt;θ)for any t≥0. Also, lim t↑∞ G1(¯ Xt;θ) = 0 and lim t↑∞ G2(¯ Xt;θ) = 0,b P(x,i,1)-a.s. Hence, we can apply Theorem 2.1 from [44] and complete the proof.  Proof of Proposition 3.2. We argue by contradiction and we suppose that Sθ sup =∅. This implies that σ∗=∞,b P(x,i)-a.s. and for any (x, i)∈ O, and we thus obtain that k2< v(x, i;θ) = inf τ∈T b E(x,i)Zτ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt +k2eRτ 0bx(b Xs,Ys)ds ≤b E(x,i)Z∞ 0 eRt 0bx(b Xs,Ys)dsπx(b Xt, θ)dt. By Condition (4) and Condition (5) in Assumption 2.2 we reach a contradiction with k2>0by taking x↑x. The proof of Sθ inf 6=∅follows by similar arguments.  The next result, which will be used in the proof of Proposition 3.3. below, provides the so-called semiharmonic characterization of v(see [44]). Its proof is direct consequence of Theorem 2.1 in [44]. Proposition A.1. For any (x, i)∈ O, we have under b P(x,i)that (1) Rt∧τ∗(θ) 0πx(b Xt, θ)dt+eRt∧τ∗(θ) 0bx(b Xt,Yt)dtv(b Xt∧τ∗(θ), Yt∧τ∗(θ))t≥0 is an F-submartingale; (2) Rt∧σ∗(θ) 0πx(b Xt, θ)dt+eRt∧σ∗(θ) 0bx(b Xt,Yt)dtv(b Xt∧σ∗(θ), Yt∧σ∗(θ))t≥0 is an F-supermartingale; (3) Rt∧τ∗(θ)∧σ∗(θ) 0πx(b Xt, θ)dt+eRt∧τ∗(θ)∧σ∗(θ) 0bx(b Xt,Yt)dtv(b Xt∧τ∗(θ)∧σ∗(θ), Yt∧τ∗(θ)∧σ∗(θ))t≥0 is an F-martingale. 30 DIANETTI, FERRARI, AND TZOUANAS APPENDIX C. PROOF OF (1)IN THEOREM 4.1: THE FOKKER-PLANCK COUPLED SYSTEM We follow closely the proof of Theorem 1 in [23]. Let µθbe the cumulative distribution function of the stationary distribution νθ(cf. Proposition 4.2). To simplify the notation, we drop the dependence on θ; that is, we set µ:= µθas well as αi:= αi(θ),βi:= βi(θ)for any i∈Y. We divide the rest of the proof in three steps. Step 1. We first derive a variational equation for the function µ. To this end, introduce the generator Lα,β (X,Y )of the process (Xξ, Y ), where Xξis reflected at the boundaries (α, β) := {(αi, βi)}i∈Y. Namely, for any φ:= (φ(·, i))i∈Y∈C2(I;Rd)define Lα,β (X,Y )φ(x, i) := 1 2σ2(x, i)φxx(x, i) + b(x, i)φx(x, i) + X j∈Y qij b φ(x, j),(x, i)∈ O, where b φ(x, i) := φ(α− i∨x∧βi, i). From Chapter IV in [24], we have that µsolves the equation (C.1) X i∈YZβi α− i Lα,β (X,Y )φ(x, i)dµ(x, i) = 0,for any φ:= (φ(·, i))i∈Y∈C2(I;Rd). Hence, by Theorem 5.3.4 in [6] we obtain that µis absolutely continuous with respect to the Lebesgue measure with density function µx. Next, split the interval [mini∈Yαi,maxi∈Yβi]in the disjoint sub-intervals Ik= (lk−1, lk), k = 1, .., K, with l0= mini∈Y{αi}, lK= maxi∈Y{βi}and lk+1 = min i∈Y: αi>lk {αi} ∧ min i∈Y: βi>lk {βi}. Introduce the set of test functions (C.2) V:= φ∈C2(I;Rd) : φx(x, i) = φxx(x, i) = 0, x ∈ I \ (αi(θ), βi(θ)),for any i∈Y. Since µ(α− i, i) = 0, for φ∈ V we observe that Zβi α− ib φ(x, j)dµ(x, i) = µ(βi, i)b φ(βj, j)−Zβj αj µ(α− i∨x∧βi, i)φx(x, j)dx, so that, exchanging the indexes in th sum, we find X i∈YX j∈Y qij Zβi α− ib φ(x, j)dµ(x, i) = X i∈YX j∈Y qijµ(βi, i)b φ(βj, j)(C.3) −X i∈YX j∈Y qji Zβi αi µ(α− j∨x∧βj, j)φx(x, i)dx. By substituting (C.3) in (C.1) and rearranging the terms, for any φ∈ V, we obtain (C.4) X i∈YIi w(µ, φ) + Σi(µ, φ)= 0, ERGODIC MFGS OF SINGULAR CONTROL WITH REGIME-SWITCHING 31 where Ii w(µ, φ) := Zβi αi 1 2σ2(x, i)µx(x, i)φxx(x, i)dx +Zβi αib(x, i)µx(x, i)−X j∈Y qjiµ(α− j∨x∧βj, j)φx(x, i)dx, Σi(µ, φ) := X j∈Y qjiµ(βj, j)b φ(βi, i). Step 2. The aim of this step is to gradually use the variational equation (C.4) to obtain the system (4.10) and the proper regularity of µ. We argue as follows: (1) First, for any i∈Yand ksuch that Ik⊂(αi, βi), choose φ(·, j) = 0 for j6=iand φ(·, i) such that supp φ(·, i)⊂Ik. For such a choice of φ, we have Σi(µ, φ) = 0, so that from (C.4) we find Ii w(µ, φ) = 0. Thus, setting ψ:= φx(·, i), we deduce that the function µ(·, i)is a solution to the variational equation ZIk1 2σ2(x, i)µx(x, i)ψx(x, i)dx +b(x, i)µx(x, i)−X j∈Y qjiµ(α− j∨x∧βj, j)ψ(x, i)dx = 0, for any ψ∈C2 c(Ik). Therefore, by the interior regularity for elliptic equations (see Theorem 8.10 in [28]), we obtain that µ(·, i)∈C2(Ik;R). Since the interval Ikis arbitrary, we obtain (C.5) µ(·, i)∈C2(αi, βi)\[ j∈Y {αj, βj}, for any i∈Y. (2) Next, for i∈Ywe use the enumeration lk=αi< lk+1 < ... < lm−1< lm=βi and define ∆f(lj, i) = f(l+ j, i)−f(l− j, i)for k < j < m,∆f(lk, i) = −f(l− k, i)and ∆f(lm, i) = f(l+ m, i). Since µ(·, i)∈C2(Iℓ)for ℓ=k+ 1, ..., m, dividing the domain of integration into the Iℓ’s and applying integration by parts, for φ∈ V we obtain Ii w(µ, φ) = Ii s(µ, φ) + Ψi σ(µ, φ),(C.6) where Ii s(µ, φ) := −Zβi αi1 2σ2(x, i)µxx(x, i)−(b(x, i)−σσx(x, i))µx(x, i) +X j∈Y qjiµ(α− j∨x∧βj, j)φx(x, i)dx, Ψi σ(µ, φ) := − m X j=k 1 2σ2(lj, i)∆µx(lj, i)φx(lj, i). In particular, for any i∈Yand ℓsuch that Iℓ⊂(αi, βi), by choosing φ(·, j) = 0 for j6=i and φ(·, i)such that supp φ(·, i)⊂Iℓ,µsolves the equation Ii s(µ, φ) = 0. Hence, since the i, Iℓare arbitrary, we conclude that 1 2σ2(x, i)µxx(x, i)−(b(x, i)−σσx(x, i))µx(x, i) + X j∈Y qjiµ(α− j∨x∧βj, j) = 0, in (αi, βi), for any i∈Y; that is, µsolves the equation (4.10) pointwise. 32 DIANETTI, FERRARI, AND TZOUANAS (3) Furthermore, using the latter equation, we deduce that Ii s(µ, φ) = 0 for any φ∈ V. Therefore, by choosing φ∈ V such that φx(lj, i) = φxx(lj, i) = 0 for any j, i ∈Y, from (C.4) and (C.6) we obtain Σi(µ, φ) = 0. Since the values of φ(βi, i)are arbitrary, this in turn implies that X j∈Y qjiµ(βj, j) = 0, i ∈Y. Such a system of equations corresponds to the eigenvector problem with zero eigenvalue of the matrix Qand its solution is given by the stationary distribution of Y. Hence, we have µ(βi, i) = p(i)for any i∈Y, proving that the boundary conditions of (4.10) are met. (4) Finally, using Ii s(µ, φ) = Σi(µ, φ) = 0 in (C.4), we have X i∈Y m X j=k: lk=αi, lm=βi 1 2σ2(lj, i)∆µx(lj, i)φx(lj, i) = 0,for any φ∈ V, from which we deduce (thanks also to (C.5)) that µ∈C1(I;Rd). Concluding, µ(·, i)∈C1([αi, βi])∩C2((αi, βi)\Sj∈Y{αj, βj})is a classical solution to the boundary value problem in (4.10). Step 3. Now we prove that (4.10) admits a unique solution. Assume that (4.10) has two solutions, denoted by µ1= (µ1(·, i))i∈Yand µ2= (µ2(·, i))i∈Y, then the difference ˜µ:= µ2−µ1= (µ2(·, i)− µ1(·, i))i∈Ysolves (4.10) with ˜µ(x, i) = 0 for x∈ {αi, βi}. 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A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type. SIAM J. Control Optim., 56(4):2563–2592. J. DIANETTI: CENTER FOR MATHEMATICAL ECONOMICS (IMW), BIELEFELD UNIVERSITY, UNIVERSIT ¨ ATSSTRASSE 25, 33615, BIELEFELD, GERMANY Email address:[email protected] G. FERRARI: CENTER FOR MATHEMATICAL ECONOMICS (IMW), BIELEFELD UNIVERSITY, UNIVERSIT ¨ ATSSTRASSE 25, 33615, BIELEFELD, GERMANY Email address:[email protected] 34 DIANETTI, FERRARI, AND TZOUANAS I. TZOUANAS: CENTER FOR MATHEMATICAL ECONOMICS (IMW), BIELEFELD UNIVERSITY, UNIVERSIT ¨ ATSSTRASSE 25, 33615, BIELEFELD, GERMANY Email address:[email protected]