Forward utilities and Mean-field games under relative performance concerns
aa r X i v : . [ q -f i n . P M ] M a y Forward utilities and Mean-field games underrelative performance concerns ∗ Gonc¸alo dos Reis † and Vadim Platonov Abstract
We introduce the concept of mean field games for agents using Forwardutilities to study a family of portfolio management problems under relative per-formance concerns. Under asset specialization of the fund managers, we solve theforward-utility finite player game and the forward-utility mean-field game. We studybest response and equilibrium strategies in the single common stock asset and theasset specialization with common noise. As an application, we draw on the corefeatures of the forward utility paradigm and discuss a problem of time-consistentmean-field dynamic model selection in sequential time-horizons.
Key words:
Forward utility, Mean-Field Games, social interactions, performanceconcerns.
Gonc¸alo dos ReisThe University of Edinburgh, School of Mathematics, Mayfield Road, The King’s Buildings, Ed-inburgh, EH9 3FD, United Kingdom, andCentro de Matem´atica e Aplicac¸ ˜oes (CMA), FCT, UNL, Quinta da Torre, 2829-516 Caparica, Por-tugal, e-mail:
Vadim PlatonovThe University of Edinburgh, School of Mathematics,Mayfield Road, The King’s Buildings, Edin-burgh, EH9 3FD, United Kingdom, e-mail: [email protected] ∗ The authors express sincerest gratitude to Thaleia Zariphopolou (University of Texas, US) andMichalis Anthropelos (University of Piraeus, GR) for the helpful discussions. † G. dos Reis acknowledges support from the
Fundac¸ ˜ ao para a Ci ˆ encia e a Tecnologia (Por-tuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centrode Matem´atica e Aplicac¸ ˜oes CMA/FCT/UNL) 1 Gonc¸alo dos Reis and Vadim Platonov This work focuses on bringing the concept of forward utilities to the mean-fieldgame setting in the limelight of competitive optimal portfolio management of agentsunder relative performance criteria and the analysis of the associated finite-playergame.There exists a very rich literature on portfolio management for agents with utilitypreferences and under performance concerns to which this short introduction cannotpossibly due justice. For a literature perspective of the financial setting including anin-depth discussion of agents with performance concerns and its impact in the utilitymaximization framework we refer to [12, 13, 4] and references therein. Additionally,we point the reader to the beautiful introductions of [19, 18] where those conceptsare brought to the framework of mean-field games. Further, those works also makefor an excellent review of mean-field games in the context of the Merton problemwhich is the framework underlying our work.In short, mean-field games (MFG), stochastic or not, gained renewed interest dueto their modelling power in crucially reducing the dimensionality of the underlyingproblem under the assumption of statistically equivalent populations [17, 5, 6]. Inother words, as long as the actions of a single agent do not affect the average inter-action of the agents in their whole, then, in principle, the MFG framework stands tobe more tractable than the n -agent games. See [19, 18].The novelty of our work is the conceptualization and analysis, simplified here,of the formulation of mean-field games within the so-called forward utilities frame-work. Further, we juxtapose our construction to the related finite-player game.The classical and ubiquitous approach of utility preferences , found throughoutthe literature ([12, 13, 4]), is that each agent, at an initial-time, specifies their risk-preferences to some future time T and proceeds to optimize their investment to thatinitial-time. This backward approach lacks flexibility to handle mid-time changesof risk-preferences by the agents, or, to allow an update of the underlying model:having in mind Covid-19, if the fund manager made investments in early 2019 tomature in the later part of 2020, how would one update the underlying model stockmodel to the change of parameters?These problems feature an inherently forward-in-time nature of investment. Aview that is particularly clear for (competitive) fund managers updating their in-vestment preferences frequently depending on market behavior. To cope with thelimitation of the backward-in-time view induced by the classical utility optimiza-tion formulation, and, to better address this forward view, the mathematical toolof forward utilities was developed. It was initially introduced for the analysis of theportfolio management problems in [20, 21, 22] and subsequently expanded [25, 1, 7]and [11, 9, 10]. The latter dealing with general forward utility Itˆo random fields andwith applications to longevity risk. Our approach builds from [14] where the firstforward-utility definition under competition appeared (for finite-player games); weadditionally refer the reader to the forthcoming [2] (who also builds from [14]).In essence, the concept of forward utility reflects that the utility map must beadaptive and adjusted to the information flow. The forward dynamic utility map is orward utilities for many player games and Mean-field games 3 built to be consistent with respect to the given investment universe and the approachwe discuss here is based on the martingale optimality principle (see Section 2.1).To the MFG context, the closest to our work we have found is the concept ofForward-Forward MFG concept of [16]. Organization of the paper.
In Section 2 we introduce the financial market. InSections 3 and 4 we study the finite-agent and mean-field game respectively. Westudy forward utilities of time-monotone type. In section 4.4 we discuss the mean-field investment problem with dynamic model selection in large time-horizons. Weconclude in Section 5 with a discussion of open questions and future research.
The market.
We consider a market environment with one riskless asset and n riskysecurities which serve as proxies for two distinct asset classes. We assume theirprices to be of log-normal type, each driven by two independent Brownian motions.More precisely the price ( S it ) t > of the stock i traded by the i -th agent solves dS it S it = µ i dt + ν i dW it + σ i dB t , (1)with constant parameters µ i > σ i > ν i > σ i + ν i >
0. The one-dimensional standard Brownian motions B , W , · · · , W n are independent. When σ i >
0, the process B induces a correlation between the stocks, and thus we call B the common noise and W i an idiosyncratic noise . The independent Brownian motions B , W , · · · , W n are defined on a probability space ( Ω , F , F , P ) endowed with thenatural filtration F = ( F t ) t > generated by them and satisfies the usual conditions.We recall the case of single common stock , where ∀ i = , . . . , n , ( µ i , σ i ) =( µ , σ ) , ν i = , for some µ , σ > i . The single common stockcase has been explored in great generality in [12, 13, 4] incorporating portfolio con-straints, general stock price dynamics and risk-sharing mechanisms. Agents’ wealth.
Each agent i = , . . . , n trades using a self-financing strategy, ( π it ) t > , which represent the (discounted by the bond) amount invested in the i -thstock. The i th agent’s wealth ( X it ) t > then solves dX it = π it (cid:16) µ i dt + ν i dW it + σ i dB t (cid:17) , with X i = x i ∈ R . (2)A portfolio strategy is said admissible if it belongs to the set A i , which consists of A i = n π i : F -progressively measurable R -valued processes ( π it ) t > , and self-financing such that E [ Z t | π s | ds ] < ∞ , ∀ t > o . Gonc¸alo dos Reis and Vadim Platonov
The Agents’ social interaction.
Each manager measures the performance of herstrategy taking into account the policy of the other. Each agent engages in a formof social interaction that affects that agent’s perception of wealth, all in an additivefashion modeled through the arithmetic average wealth of all agents (this modelis largely inspired in [12, 13, 4, 19]). Namely the relative performance metric ofmanager i ∈ { , . . . , n } , denoted e X i is defined to be e X i = X i − θ i X , where X : = n n ∑ k = X k and θ i ∈ [ , ] . (3)We easily obtain a dynamics for X and e X i , namely X t = (cid:16) n n ∑ k = x k (cid:17) + (cid:16) n n ∑ k = π kt µ k (cid:17) dt + (cid:16) n n ∑ k = π kt ν k dW kt (cid:17) + (cid:16) n n ∑ k = π kt σ k (cid:17) dB t = x + ( πµ ) t dt + (cid:16) n n ∑ k = π kt ν k dW kt (cid:17) + ( πσ ) t dB t , e X it = (cid:0) x i − θ i x (cid:1) + (cid:0) π it µ i − θ i ( πµ ) t (cid:1) dt + (cid:16) π it ν i dW it − θ i (cid:0) n n ∑ k = π kt ν k dW kt (cid:1)(cid:17) + (cid:0) π it σ i − θ i ( πσ ) t (cid:1) dB t , (4)where x , πµ and πσ are identified as averages (as seen from the 1st equation to the2nd). Similarly to [19, Remark 2.5], it is natural to replace the average wealth X in(3) by the average over all other agents. With that in mind we define for convenience X ( − i ) = n − ∑ k = i X k and Y ( − i ) = nn − X ( − i ) . This leads us to recast (3) as b X i = X i − θ i X ( − i ) , where X ( − i ) = n − ∑ k = i X k . (5)We easily obtain a dynamics for b X and X ( − i ) , namely X ( − i ) t = x ( − i ) + ( πµ ) ( − i ) t dt + (cid:16) n − ∑ k = i π kt ν k dW kt (cid:17) + ( πσ ) ( − i ) t dB t , b X it = (cid:0) x i − θ i x ( − i ) (cid:1) + (cid:0) π it µ i − θ i ( πµ ) ( − i ) t (cid:1) dt + (cid:16) π it ν i dW it − θ i (cid:0) n − n ∑ k = i π kt ν k dW kt (cid:1)(cid:17) + (cid:0) π it σ i − θ i ( πσ ) ( − i ) t (cid:1) dB t . (6)We also define the quantities c πσ ( − i ) : = n ∑ k = i π k σ k , ( πµ ) ( − i ) : = n ∑ k = i π k µ k and ( πν ) ( − i ) : = n ∑ k = i ( π k ν k ) , where we have the following relations between c πσ ( − i ) , πσ ( − i ) and πσ : orward utilities for many player games and Mean-field games 5 πσ ( − i ) = nn − πσ − n − π i σ i , πσ ( − i ) = nn − c πσ ( − i ) , (7)and c πσ ( − i ) = πσ − n π i σ i . We do not write it explicitly but we extend the samenotation and relations to c πµ ( − i ) , πµ ( − i ) and πµ . We recall, for reference, the classic forward utility formulation. We define a forwarddynamic utilities in the context of the probability space ( Ω , F , F , P ) . We denote by u : R → R the initial data. The forward utility is constructed based on the martingaleoptimality principle. Definition 1 (Forward dynamic utilities).
Let U : Ω × R × [ , ∞ ) → R be an F -progressively measurable random field. U is a forward dynamic utility if: • For all t > x U ( x , t ) is increasing and concave; • It satisfies U ( x , ) = u ( x ) ; • For all T > t and each self-financing strategy, represented by π , the associateddiscounted wealth process X π satisfies a supermartingale property E [ U ( X π T , T ) | F t ] U ( X π t , t ) ; • For all T > t there exists a self financing strategy, represented by π ∗ , for whichthe associated discounted wealth X ∗ satisfies a martingale property E [ U ( X ∗ T , T ) | F t ] = U ( X ∗ t , t ) . The above definition assumes the optimizer is attained. This is a somewhat strongassumption which is discussed in [25, 1]. There it is argued that such constraint isnot necessary for the forward utility construction in certain contexts.
Each manager measures the output of her relative performance metric using a for-ward relative one as modeled by an F t -progressively measurable random field U i : R × [ , ∞ ) → R for i ∈ { , . . . , n } . The below criteria follows those proposedin [14].The main idea here being a formulation inspired in the first step in the usualstrategy of solving a Nash game, namely the best response of an agent to the actions Gonc¸alo dos Reis and Vadim Platonov of all other agents. Take manager i and assume all other agents j = i have actedwith an investment policy π j then for any strategy π i ∈ A i , the process U i ( b X it , t ) isa (local) supermartingale, and there exists π i , ∗ ∈ A i such that U i ( b X i , ∗ t , t ) is a (local)martingale where b X i and b X i , ∗ solves (5) with strategies π i and π i , ∗ respectively.This version of a relative criterion is (implicitly and) exogenously parametrizedby the policies of all other managers j = i over which there is no assumption ontheir optimality. In Nash-game language, we solve the so-called best response. Definition 2 (Forward relative performance for the manager).
Each manager i ∈{ , · · · , n } satisfies the following. Let π j ∈ A j , ∀ j = i be arbitrary but fixed andadmissible policies for the other managers j = i .An F -progressively measurable random field U i ( x , t ) is a forward relative per-formance for manager i if, for all t >
0, the following conditions hold:i) The mapping x U i ( x , t ) , is strictly increasing and strictly concave;ii) For π i ∈ A i , U i ( b X it , t ) is a local supermartingale and b X i is the relative perfor-mance metric given in (5);iii) There exists π i , ∗ ∈ A i such that U i ( b X i , ∗ t , t ) is a (local) martingale where b X i , ∗ solves (5) with strategies π i , ∗ being used.In the above definition, we do not make explicit references to the initial conditions U k ( x , ) but we assume that admissible initial data exist such that the above defini-tion is viable. Contrary to the classical expected utility case, the forward volatilityprocess is an investor-specific input. Once it is chosen, the supermartingality andmartingality properties impose conditions on the drift of the process. Under enoughregularity, these conditions lead to the forward performance SPDE (see [24]).Since we are working in a log-normal market, it suffices to study smooth relativeperformance criteria of zero volatility (of the forward utility map). Such processesare extensively analyzed in [23] in the absence of relative performance concerns.There, a concise characterization of the forward criteria is given along (necessaryand sufficient) conditions for their existence and uniqueness. In that setting, thezero-volatility forward processes are always time-decreasing processes. We point tothe reader that this does not have to be case if relative performance concerns arepresent (see also [14]).We assume that the Itˆo decomposition of the forward utility map is dU i ( x , t ) = U it ( x , t ) dt , for i ∈ { , · · · , n } , (8)and the derivatives U it ( x , t ) , U ix ( x , t ) and U ixx ( x , t ) exists for t >
0. And we next derivea stochastic PDE and an optimal investment strategy for a smooth relative perfor-mance criteria of zero volatility of some agent i assuming that all other agents j = i have made their investment decisions. Proposition 1 (Best responses).
Fix i ∈ { , · · · , n } and the agent’s initial preferenceu i . Assume that each manager j = i follows π j ∈ A j . Consider the stochastic PDEfor ( x , t ) ∈ R × [ , ∞ ) orward utilities for many player games and Mean-field games 7 U it = (cid:16) θ i ( πµ ) ( − i ) t − µ i θ i σ i ( πσ ) ( − i ) t ν i + σ i (cid:17) U ix + µ i ( ν i + σ i ) ( U ix ) U ixx + U ixx h(cid:16) θ i ( πσ ) ( − i ) t (cid:17) (cid:16) σ i ν i + σ i − (cid:17) − θ i n − ( πν ) ( − i ) i , (9) and assume that for an admissible initial condition U ( · , ) = u i ( · ) , the SPDE has asmooth solution U i such that x U i ( x , t ) is strictly increasing (U x > ) and strictlyconcave (U xx < ) for each t > .Define the strategy π i , ∗ π i , ∗ t = ν i + σ i (cid:16) θ i σ i ( πσ ) ( − i ) t − µ i U ix ( b X i , ∗ t , t ) U ixx ( b X i , ∗ t , t ) (cid:17) , t > , where b X i , ∗ solves (6) with π i , ∗ being used.If π i , ∗ ∈ A i and b X i , ∗ are well-defined, then U i ( x , t ) is a forward utility perfor-mance process. Moreover, the policy π i , ∗ is optimal (in the sense of Definition 2). Using the language of [22, Section 5], define the local risk tolerance functionr i : Ω × R × [ , ∞ ) → R such that r i ( x , t ) : = − U ix ( x , t ) / U ixx ( x , t ) . Then, by directinspection of the expression for I π i , ∗ one sees that if the local risk tolerance func-tion r i ( x , t ) = r i = Const ∀ t > Corollary 1 (Constant strategies under CARA).
Assume that all agents j = i in-vest according to constant strategies α j ∈ R and that the local risk tolerance func-tion r i is constant. Then π i , ∗ is constant. We now prove the previous “best responses” proposition above.
Proof (Proof of Proposition 1).
From (5) we have the dynamics of d b X i (and hencethat of d ( X i − θ i X ( − i ) ) ). We now apply the Itˆo formula to U i ( b X it , t ) = U i ( X it − θ i X ( − i ) t , t ) , dU i ( b X it , t ) = U it ( b X it , t ) dt + U ix ( b X it , t ) d b X it + U ixx ( b X it , t ) d h b X it i = U it ( b X it , t ) dt + U ix ( b X it , t ) (cid:0) π it µ i − θ i ( πµ ) ( − i ) t (cid:1) dt + U ix ( b X it , t ) (cid:16) π it ν i dW it − θ i (cid:0) n − n ∑ k = i π kt ν k dW kt (cid:1)(cid:17) (10) + U ix ( b X it , t ) (cid:0) π it σ i − θ i ( πσ ) ( − i ) t (cid:1) dB t + U ixx ( b X it , t ) h ( π it ν i ) + θ i n − ( πν ) ( − i ) + (cid:0) π it σ i − θ i ( πσ ) ( − i ) t (cid:1) i dt , with U i ( b X i , ) = U i ( x i − θ i x ( − i ) , ) and we used that the B , W j are all i.i.d. Gonc¸alo dos Reis and Vadim Platonov
By Definition 2, the process U i ( b X it , t ) becomes a Martingale at the optimum π .Direct computations using first order conditions ( ∂ π i “drift” =
0) yield0 + U ix (cid:0) µ i − (cid:1) + U ixx h π i ν i + + (cid:0) π it σ i − θ i ( πσ ) ( − i ) t (cid:1) σ i i = ⇔ U ixx π i ( ν i + σ i ) = − U ix µ i + U ixx θ i σ i ( πσ ) ( − i ) t (11) ⇒ π it = ν i + σ i (cid:16) θ i σ i ( πσ ) ( − i ) t − µ i U ix ( b X it , t ) U ixx ( b X it , t ) (cid:17) . Injecting the expression of π it in the drift term of (10) and simplifying we arrive atthe consistency condition (9), we do not carry out this step explicitly, nonetheless,using that U i solves (9) equation (10) simplifies to (exact calculations are carriedout in the Section 6) dU i ( b X it , t )= U ix ( b X it , t ) (cid:16) π it ν i dW it − θ i (cid:0) n − n ∑ k = i π kt ν k dW kt (cid:1)(cid:17) + U ix ( b X it , t ) (cid:16) π it σ i − θ i ( πσ ) ( − i ) t (cid:17) dB t + U ixx ( b X it , t ) ν i + σ i (cid:12)(cid:12)(cid:12) π i ( ν i + σ i ) − (cid:16) θ i σ i ( πσ ) ( − i ) t − µ i U ix ( b X it , t ) U ixx ( b X it , t ) (cid:17)(cid:12)(cid:12)(cid:12) dt . (12)The concavity assumption of U i ( x , t ) implies that the drift term above is non-positiveand vanishes when (11) holds. We can conclude that, if π i , ∗ t = π it ∈ A i and the asso-ciated process b X i , ∗ is well-defined (solution to (6) with π i , ∗ ), the process U i ( b X i , ∗ t , t ) is a local-martingale, otherwise it is a local supermartingale. The result concludes. Example 1 (The classic CARA case - exponential case).
The exponential criteriontakes as initial condition the map U ( x , ) ( x ∈ R ) defined as U i ( x , ) = − e − x / δ , with δ > . (13)In this case, the local risk tolerance function r = − U ix / U ixx = δ .In our case accounting for social interaction between agents in the form of per-formance concerns, the i -th agent’s utility is a function U i : Ω × R × R × [ , ∞ ) → R of both her individual wealth x and the average wealth wealth of all agents, m . Theinitial/starting utility map is of the form U i ( x , m , ) = − exp n − δ i ( x − θ i m ) o , orward utilities for many player games and Mean-field games 9 where we refer to the constants δ i > θ i ∈ [ , ] as personal risk tolerance and competition weight parameters, respectively. Example 2 (The time-monotone forward utility with starting exponential).
For i ∈{ , · · · , n } , let the dynamics of U i be given by (8) and assume U i ( x , ) = − e − η x with η >
0. Then the solution to the SPDE (9) is given by U i ( x , t ) = − e − x δ i + f i ( t ) , with δ i > , (14)where ( f i ( t )) t > is the random map given below independent of x satisfying f i ( ) =
0, sufficiently integrable and t f i ( t ) is differentiable. Note that in this case, the local risk tolerance function satisfies r i = − U ix / U ixx = δ i .Injecting U i ( x , t ) above in (9) yields an ODE for f i (we omit the time variable), f ′ i = − θ i δ i (cid:16) ( πµ ) ( − i ) − µ i σ i ( πσ ) ( − i ) ν i + σ i (cid:17) + µ i ( ν i + σ i )+ θ i δ i h(cid:16) ( πσ ) ( − i ) (cid:17) (cid:16) σ i ν i + σ i − (cid:17) − n − ( πν ) ( − i ) i = − θ i δ i ( πµ ) ( − i ) + ( ν i + σ i ) (cid:16) µ i + θ i δ i σ i ( πσ ) ( − i ) (cid:17) − θ i δ i h(cid:16) ( πσ ) ( − i ) t (cid:17) + n − ( πν ) ( − i ) i = : λ i . Hence, f i ( t ) = R t λ i ( s ) ds . In particular, if all coefficients and strategies are con-stant, then (with a slight abuse of notation) f i ( t ) = t λ i for a constant λ i given by theRHS of the above ODE. Example 3 (No performance concerns: θ i = ). We continue to work under the time-monotone forward utility case of the previous example. Without performance con-cerns, i.e. θ i =
0, then λ i is just the Sharpe ratio λ i = µ i ( ν i + σ i ) and we recover knownresults. We have from Proposition 1 that π i , ∗· = µ i δ i ν i + σ i and U i ( x , t ) = − exp n − x δ i + t λ ( θ i = ) i o , with the constant λ ( θ i = ) i just being the Sharpe ratio, λ ( θ i = ) i = µ i ( ν i + σ i ) . In view of the best responses discussed in Proposition 1 we now investigate the simultaneous best responses as to establish the existence of a Nash equilibrium.
Definition 3 (Forward Nash equilibrium).
A forward Nash equilibrium consistsof n -pairs of F t -adapted maps ( U i , π i , ∗ ) such that for any t > • ∀ i ∈ { , · · · , n } , π i , ∗ ∈ A i ; • For each player i ∈ { , · · · , n } the following holds: given the strategies π j , ∗ ∈ A j (any j = i ) the processes U i ( b X it ( π ∗ , − i ) , t ) is a local supermartingale where b X i ( π ∗ , − i ) solves (6) with all managers j = i acting according to π j , ∗ ; • For each player i ∈ { , · · · , n } the following holds: the process U i ( b X i , ∗ t ( π ∗ , − i ) , t ) is a local martingale where b X i ( π ∗ , − i ) solves (6) with all managers j acting ac-cording to π j , ∗ .If all the optimal strategies are constant we say we have a constant forward Nashequilibrium .Under appropriate integrability conditions plus the martingale/supermartingale char-acterizations, we have for some agent i for any π i ∈ A i E [ U i ( b X i , ∗ t ( π ∗ , − i ) , t )] = E [ U i ( b X i , ∗ ( π ∗ , − i ) , )] = E [ U i ( x i − θ i x ( − i ) , )]= U i ( x i − θ i x ( − i ) , ) > E [ U i ( b X it ( π ∗ , − i ) , t )] . As expected, no manager can increase the expected utility of her relative perfor-mance metric by unilateral decision.The solvability of the general forward Nash equilibrium seems very difficult fora general forward criteria as one needs to solve the following system for the π i , ∗ (see Proposition 1, in particular (11)) and the corresponding SPDEs for the U i , i ∈{ , · · · , n } : π i , ∗ t ( ν i + σ i ) = θ i σ i (cid:16) n − n ∑ k = , k = i π k , ∗ t σ k (cid:17) − µ i U ix (cid:0) b X i , ∗ t ( π ∗ , − i ) , t (cid:1) U ixx (cid:0) b X i , ∗ t ( π ∗ , − i ) , t (cid:1) . (15) In order to obtain explicit results we focus on the time-monotone case presented inExample 2 for which U ix / U ixx = − δ i . More notably, at the level at which we haveformulated our problem we can easily recover the results of [19, Theorem 2.3] forwhich one has U ix / U ixx = − δ i ∀ t (note their Remark 2.5). Theorem 1.
Assume the conditions of Proposition 1 hold for all agents i ∈ { , · · · , n } .Assume furthermore that agents have time-monotone forward utility U i with initialcondition (13) .Define the quantities ϕ σ n and ψ σ n by orward utilities for many player games and Mean-field games 11 ϕ σ n : = n n ∑ i = δ i µ i σ i ν i + σ i (cid:0) + θ i n − (cid:1) and ψ σ n : = n − n ∑ i = θ i σ i ν i + σ i (cid:0) + θ i n − (cid:1) . (16) Then, if ψ σ n = then a constant forward Nash equilibrium exists, with the constantoptimal strategies π i , ∗ given by π i , ∗· = ν i + σ i (cid:0) + θ i n − (cid:1) (cid:16) θ i σ i (cid:16) + n − (cid:17) ϕ σ n − ψ σ n + µ i δ i (cid:17) . (17) The forward Nash equilibria is given by the n-pairs { ( U i , ∗ , π i , ∗ ) } i = , ··· , n where theU i , ∗ is the solution of (9) (see Example 2) under the optimal constant strategies π · , ∗ .The term λ i (see Example 2), at equilibrium, is given by λ i = − θ i δ i (cid:16)n nn − πµ − n − π i µ i o − µ i σ i ν i + σ i n nn − πσ − n − π i σ i o(cid:17) + µ i ( ν i + σ i ) + θ i δ i hn nn − πσ − n − π i σ i o (cid:16) σ i ν i + σ i − (cid:17) − n n ( n − ) ( πν ) − ( n − ) ( π i ν i ) oi , where the relevant expressions for πσ , πµ and ( πν ) are given below in (18) , (19) and (20) .Remark 1. We note that we solve not the same problem studied at [19], but an equiv-alent one. However imposing the scaling factor given by [19, Remark 2.5] we re-cover the same results as in [19, Theorem 2.3].
Proof.
Injecting the condition U x / U xx = − δ i in (15), the system to be solved inorder to ascertain the Nash equilibrium is, across i ∈ { , · · · , n } , π i , ∗ t ( ν i + σ i ) = θ i σ i (cid:16) n − n ∑ k = , k = i π k , ∗ t σ k (cid:17) + µ i δ i = θ i σ i (cid:16) nn − ( πσ ) t − n − π i , ∗ σ i (cid:17) + µ i δ i ⇔ π i , ∗ t = ν i + σ i (cid:0) + θ i n − (cid:1) (cid:16) θ i σ i nn − ( πσ ) t + µ i δ i (cid:17) . The final line yields the expression for π i , ∗ as a function of the unknown πσ . Todetermine the latter, multiply both sides by σ i and average over i ∈ { , · · · , n } , thisyields a solvability condition ( πσ ) t = ( πσ ) t ψ σ n + ϕ σ n ⇔ πσ = ϕ σ n − ψ σ n as long as ψ σ n = . (18) Plugging the expression ( πσ ) in that for π i , ∗ yields the result. That the optimalstrategies are constant is now obvious.It remains to derive the expression for the λ i ’s. Just like for πσ , we obtain anexpression for πµ by multiplying π i , ∗ by µ i and averaging on both sides, we have πµ = nn − · ϕ σ n − ψ σ n · ψ µ n + φ µ n and πµ ( − i ) = nn − πµ − n − π i µ i , (19)where we used (7) and the quantities ϕ µ n , ψ µ n are defined as ϕ µ n : = n n ∑ k = δ k µ k ν k + σ k ( + θ k n − ) and ψ µ n : = n n ∑ k = θ k µ k σ k ν k + σ k ( + θ k n − ) . Similarly, defining ( πν ) : = n − ∑ k = i ( π kt ν k ) we have ( πν ) = n n ∑ i = (cid:16) ν i θ i σ i · nn − · ϕ σ n − ψ σ n + ν i µ i δ i ν i + σ i (cid:0) + θ i n − (cid:1) (cid:17) . (20)Similarly to (7), we have ( πν ) ( − i ) = nn − ( πν ) − n − ( π i ν i ) . Replacing these ex-pressions in that for λ i in Example 2 the expression in the result’s statement follows.From the forward utility machinery one can easily recover the classical case ofutility optimization where one prescribes the utility map for the horizon time T thenproceeds to optimize. Example 4 (Recovering the classical utility problem from the forward one.).
If onewould start the forward utility with (for some 0 < T < ∞ ) u i ( x ) : = − e − x / δ i − T λ i , then computations like those presented yield the forward utility map U ( x , t ) as U i ( x , t ) = − e − x / δ i +( t − T ) λ i , t ∈ [ , T ] and in particular U ( x , T ) = − e − x / δ i . In other words, our forward utility recovers asa particular case the classical exponential utility maximization problem (discussedin [19]). Corollary 2 (Single stock).
Let µ i = µ > , σ i = σ > and ν i = , ∀ i = , . . . n.Defining constants as ϕ σ n : = n n ∑ i = δ i + θ i n − and ψ σ n : = n − n ∑ i = θ i + θ i n − . (21) Then, if ψ σ n = then a constant forward Nash equilibrium exists, with the constantoptimal strategies π i , ∗ given by orward utilities for many player games and Mean-field games 13 π i , ∗· = µσ (cid:0) + θ n − (cid:1) (cid:16) θ (cid:16) + n − (cid:17) ϕ σ n − ψ σ n + δ (cid:17) . By inspection of Theorem 1 one sees that the optimal strategy and forward utilitymap for some agent depend on that agent’s specific parameters (model parameters,initial wealth, risk tolerance and performance concern) and on certain averages ofthe parameters of all agents. This makes a case for a MFG approach to the game.In this section and inspired by the results in the previous one, we formalize theconcept of forward mean-field Nash game. We use the concept of type distributions introduced in [17] and [19]. We follow the construction presented in the latter.We focus on initial forward utilities at time t = U i ( x , m , ) = − exp n − δ i ( x − θ i m ) o , where we refer to the constants δ i > θ i ∈ [ , ] as personal risk tolerance and competition weight parameters, respectively.For the n -agent game, we define for each agent i = , . . . , n the type vector ζ i : = ( x i , δ i , θ i , µ i , ν i , σ i ) . These type vectors induce an empirical measure, called the type distribution , whichis the probability measure on the type space Z e : = R × ( , ∞ ) × [ , ] × ( , ∞ ) × [ , ∞ ) × [ , ∞ ) , (22)given by m n ( A ) = n n ∑ i = A ( ζ i ) , for Borel sets A ⊂ Z e . Assume now that as the number of agents becomes large, n → ∞ , the above empiricalmeasure m n has a weak limit m , in the sense that R Z e f dm n → R Z e f dm for everybounded continuous function f on Z e . For example, this holds almost surely if the ζ i ’s are i.i.d. samples from m . Let ζ = ( ξ , δ , θ , µ , ν , σ ) denote a random variablewith this limiting distribution m .The mean field game (MFG) defined next allows us to derive the limiting strategyas the outcome of a self-contained equilibrium problem, which intuitively representsa game with a continuum of agents with type distribution m . Rather than directlymodeling a continuum of agents, we follow the MFG paradigm of modeling a sin-gle representative agent, who we view as randomly selected from the population.The probability measure m represents the distribution of type parameters among thecontinuum of agents; equivalently, the representative agent’s type vector is a ran-dom variable with law m . Heuristically, each agent in the continuum trades in a single stock driven by two Brownian motions, one of which is unique to this agentand one of which is common to all agents. We extend the Forward Nash equilibriumof Definition 3 to the MFG setting below. To formulate the MFG, we now assume that the probability space ( Ω , F , P ) sup-ports yet another independent (one-dimensional) Brownian motion, W , as well as arandom variable ζ = ( ξ , δ , θ , µ , ν , σ ) , independent of W and B , and with values in the space Z e defined in (22). Thisrandom variable ζ is called the type vector , and its distribution is called the typedistribution .Let F MF = ( F MF t ) t ∈ [ , T ] denote the smallest filtration satisfying the usual as-sumptions for which ζ is F MF0 -measurable and both W and B are adapted. Let also F B = ( F Bt ) t ∈ [ , T ] denote the natural filtration generated by the Brownian motion B .The generic agent’s wealth process solves dX t = π t ( µ dt + ν dW t + σ dB t ) , X = ξ , (23)where the portfolio strategy must belong to the admissible set A MF of self-financing F MF -progressively measurable real-valued processes ( π t ) t > satisfying the square-integrability condition E [ R T | π t | dt ] < ∞ for any T ∈ [ , ∞ ) . The random variable ξ is the initial wealth of the representative agent, whereas ( µ , ν , σ ) are the marketparameters. In the sequel, the parameters δ and θ will affect the risk preferences ofthe representative agent. Note that each agent among the continuum may still havedifferent preference parameters, captured by the fact that δ and θ are random. The formulation of the forward Nash game of Section 3 drives the formulation of theMean-field game we discuss here. Recall that in the MFG-formulation the genericagent has no influence on the average wealth of the continuum of agents, as butone agent amid a continuum of agents. We next introduce the concept of mean-field(MF)-forward relative performance , π ∗ is the MF-equilibrium and, the main objectof interest the
MF-Forward relative performance equilibrium .We recall the framework. We assume that the Itˆo decomposition of the forwardutility map (without noise) is dU ( x , t ) = U t ( x , t ) dt and initial condition (13) , orward utilities for many player games and Mean-field games 15 where the derivatives U t ( x , t ) , U x ( x , t ) and U xx ( x , t ) exist for t >
0. Given the marketsetup we developed so far, we next define our concept of equilibrium.
Definition 4 (MF-Forward relative performance equilibrium (for the genericmanager)).
Let π ∈ A MF and X π solving (23) with π ; to ( π , X π ) we associate the F B -adapted square integrable stochastic process ( X t ) t > , representing the averagewealth of the continuum of agents, as X t : = E P ⊗ m [ X π t | F Bt ] for all t > F MF -progressively measurable random field ( U ( x , t )) t > is a MF-forwardrelative performance for the generic manager if, for all t >
0, the following condi-tions hold:i) The mapping x U ( x , t ) , is strictly increasing and strictly concave;ii) For each π ∈ A MF , U ( X π t − θ X t , t ) is a P -local supermartingale and X is the generic agent’s wealth process solving (23) for the strategy π ;iii) There exists π ∗ ∈ A MF such that U ( X ∗ t − θ X t , t ) is a P -local martingale where X ∗ solves (23) with π ∗ plugged in as the strategy;iv) We call π ∗ of point iii) a MF-equilibrium if X t = E P ⊗ m [ X ∗ t | F Bt ] for all t > X ∗ solves (23) with π ∗ plugged in as the strategy.We denote the triplet ( U , π ∗ , X ) satisfying i)-iv) the MF-Forward relative per-formance equilibrium . An MF-equilibrium is constant if there exists an F MF0 -measurable RV π ∗ such that π t = π ∗ , ∀ t > B each agent faces an independent noise W and an inde-pendent type vector ζ . As in MFG [19], conditionally on B , all agents faces i.i.d.copies of the same optimization problem. The law of large numbers suggests thatthe average terminal wealth of the whole population should be E P ⊗ m [ X ∗ t | F Bt ] .Our construction allows us to identify E P ⊗ m [ X ∗ t | F Bt ] with a certain dynamicsand, in turn, treat this component as an additional uncontrolled state process. Thisavoids altogether the conceptualization of the master equation for models with dif-ferent types of agents. The latter is left for future research. We now present the main result of this section which is the existence of a
MF-Forward relative performance equilibrium for the generic manager according toDefinition 4 within the context of time-monotone forward utilities.From the methodological point of view, the problem is solved as before. ApplyItˆo-Wentzell to U ( Z π t , t ) , determine the optimal strategy π ∗ and the consistency con-dition (the SPDE) for U such that the first three conditions of Definition 4 hold. Thelast condition, to show that π ∗ is indeed the MFG Forward equilibrium follows byconstruction as we will see. Theorem 2.
Assume that m-a.s. δ > , θ ∈ [ , ] , µ > , σ > , ν > such that σ + ν > .Assume the following constants are finite ψ σ : = E m h θ σ ν + σ i , ϕ σ : = E m h δ µσν + σ i , ψ µ : = E m h θ µσν + σ i , ϕ µ : = E m h δ µ ν + σ i . Assume that ψ σ = . Then there exists a unique constant MF-Forward relative per-formance equilibrium in the sense of Definition 4.The constant MF-equilibrium strategy is given by π ∗ = ν + σ (cid:16) θσ ϕ σ − ψ σ + µδ (cid:17) , (24) constrained to the identity E m [ σπ ∗ ] = ϕ σ − ψ σ < ∞ . The MF-forward CARA relative performance utility map is the solution ofU t = θ (cid:16) ϕ σ − ψ σ · ψ µ + ϕ µ − µ σν + σ · ϕ σ − ψ σ (cid:17) U x + µ ( ν + σ ) ( U x ) U xx + U xx · θ (cid:16) ϕ σ − ψ σ (cid:17) (cid:16) σ ν + σ − (cid:17) . (25) When the initial condition is U ( x , ) = u ( x ) = − e − x / δ , i.e. the exponential prefer-ences, U is given explicitly by U ( x , t ) = u ( x ) e t λ with λ given by λ = − θδ µα + ( ν + σ ) (cid:16) µ + θδσσα (cid:17) − θ δ (cid:16) σα (cid:17) , where σα and µα are given by (29) and (30) respectively.If ψ σ = , then there exists no constant MF-equilibrium. By comparing the statements of Theorem 1 and Theorem 2 (and same happensfor the respective Single (common) Stock Corollaries) one easily sees that as n → ∞ the strategies, weights ( φ · n and ψ · n ) and forward-utility map in Theorem 1 convergeto the respective quantities appearing in Theorem 2. Remark 2.
In contrast with Remark 1, here we recover the result from [19, Theorem2.10] as the scaling factors converge to 1 (as n → ∞ ). Hence, due to space constraintswe defer the reader to [19, Section 2.3] for the discussion of the equilibria. Proof.
We proceed in several steps in order to construct the constant MF-equilibrium.To that end we must solve ii)-iii) in Definition 4 for a given X process associated to orward utilities for many player games and Mean-field games 17 π ∈ A MF . Condition iv), for MF-equilibrium allows us to focus only on processesof the form X t = E P ⊗ m [ X α t | F Bt ] where X α solves (23) for a constant (i.e. F MF -measurable) strategy α satisfying E m [ α ] < ∞ . Step 0. The dynamics of the average wealth process.
To solve the above problemgiven ( X t ) t > it suffices to restrict ourselves to processes ( X t ) t > satisfying X t = E P ⊗ m [ X α t | F Bt ] P ⊗ m -a.s.. We then have P ⊗ m -a.s. X t = E P ⊗ m [ X α t | F Bt ] = E P ⊗ m h ξ + Z t µα ds + Z t να dW s + Z t σα dB s (cid:12)(cid:12)(cid:12) F Bt i = ¯ ξ + Z t µπ s ds + Z t σπ s dB s , (26)where, for consistency of notation wrt to the previous section, we denote¯ ξ : = E m [ ξ ] , µα : = E m [ µα ] and σα : = E m [ σα ] . Hence for π ∈ A MF and as in the previous section we can define the dynamics ofthe process Z π = X π − θ XdZ π t = (cid:0) µπ t − θµα (cid:1) dt + νπ t dW t + (cid:0) σπ t − θσα (cid:1) dB t , Z π = ξ − θξ , and solve the MFG Forward utility problem in Definition 4 with its help.Hence applying Itˆo-Wentzell to U ( Z π t , t ) yields dU ( Z π t , t ) = U t ( Z π t , t ) dt + U x ( Z π t , t ) dZ π t + U xx ( Z π t , t ) d h Z π t i = h U t ( Z π t , t ) + U x ( Z π t , t ) (cid:0) µπ t − θµα (cid:1) + U xx ( Z π t , t ) (cid:16) ( νπ t ) + (cid:0) σπ t − θσα (cid:1) (cid:17)i dt , (27) + U x ( Z π t , t ) νπ t dW t + U x ( Z π t , t ) (cid:0) σπ t − θσα (cid:1) dB t , with U ( Z π , ) = U ( ξ − θξ , ) = − exp {− ( ξ − θξ ) / δ } and we used that the B , W are all i.i.d. Exact calculations on deriving (27) are presented in the Section 6. Step 1. Finding the candidate optimal strategy π ∗ . As before, the process U ( Z π t , t ) becomes a Martingale at the optimum π . Direct computations using first order con-ditions ( ∂ π “drift” =
0) yield0 + U x · (cid:0) µ − (cid:1) + U xx h πν + (cid:0) σπ t − θσα (cid:1) σ i = ⇒ π ∗ t ( ν + σ ) = θσσα − µ U x ( Z π t , t ) U xx ( Z π t , t ) = θσσα + µδ , (28)where we injected the CARA constraint U x / U xx = − δ ∀ t . By inspection it is clearthat π ∗ is a F MF0 -measurable RV which is independent of time and is well-definedas long as σα is finite. Step 2. The optimality of the strategy.
The argument is similar to that in [19]. Theoriginal constant strategy α if a MF-equilibrium if and only if for all t > E P ⊗ m [ X α t | F Bt ] = E P ⊗ m [ X π ∗ t | F Bt ] a . s . ⇔ ¯ ξ + µα t + σα B t = ¯ ξ + µπ ∗ t + σπ ∗ B t a . s . Taking expectations on both sides implies that α is a MG-equilibrium if and only ifthe following two conditions holds µα = µπ ∗ and σα = σπ ∗ . Using (28) with U x / U xx = − δ and the expressions for ϕ σ , ψ σ one derives that σπ ∗ = θ σ ν + σ σα + δ µσν + σ ⇒ σπ ∗ = σαψ σ + ϕ σ , using that σα = σπ ∗ yields solvability if ψ σ = E m (cid:2) θ σ ν + σ (cid:3) =
1. The same proce-dure deals with the condition µα = µπ ∗ . We then have σπ ∗ = σα = ϕ σ − ψ σ = Const , (29) µπ ∗ = µα = ϕ σ − ψ σ · ψ µ + ϕ µ = Const. (30)Injecting these identities in the expression for π ∗ we find (24).For the non-solvability statement, if the equation (30) has ψ σ = ϕ σ = ψ σ = ϕ σ = µ > δ > σ = ψ σ = ψ σ = Step 3. Finding the consistency SPDE and the Utility map.
We do not carry outthis step explicitly, nonetheless, injecting the expression of π ∗ , σα and µα in thedrift term of (27) and simplifying, we find the necessary equation (25), i.e. the con-sistency condition the random field U must satisfy to that the required properties inDefinition 4 hold.Just like in Example 2, the time-monotone forward utility equation (25) can besolved and indeed one has a simplified version. We have U ( x , t ) = − e − x / δ + t λ , (31)where the F MF0 -measurable RV λ is given by (using (29) and (30)) orward utilities for many player games and Mean-field games 19 λ = − θδµα + ( ν + σ ) (cid:16) µ + θδσσα (cid:17) − θ δ (cid:16) σα (cid:17) (32) = − θδ (cid:16) ϕ σ − ψ σ · ψ µ + ϕ µ − µ σν + σ · ϕ σ − ψ σ (cid:17) + µ ( ν + σ ) + θ δ (cid:16) ϕ σ − ψ σ (cid:17) (cid:16) σ ν + σ − (cid:17) . Step 4. The MFG forward utility dynamics.
Injecting the consistency SPDE (25)in the expression for dU ( Z π t , t ) given in (27) yields, dU ( Z π t , t ) = U xx ( Z π t , t )( ν + σ ) (cid:12)(cid:12)(cid:12) π t ( ν + σ ) − (cid:16) θσ · ϕ σ − ψ σ + µδ (cid:17)(cid:12)(cid:12)(cid:12) dt + U x ( Z π t , t ) νπ t dW t + U x ( Z π t , t ) (cid:16) σπ t − θ · ϕ σ − ψ σ (cid:17) dB t . We close with a corollary regarding the since common stock case.
Corollary 3 (Single stock).
Let µ , σ , ν be deterministic with ν = , µ , σ > . Defin-ing constants as ϕ : = E m [ δ ] and ψ : = E [ θ ] . (33) Then, if ψ = then a constant MF-equilibrium exists, with the constant optimalstrategy π ∗ given by π ∗· = µσ (cid:16) θ ϕ − ψ + δ (cid:17) . Over the time interval [ , ∞ ) our generic agent selects a sequence of horizon time ( T j ) j ∈ N (such that T = T j + − T j > j T j = ∞ ) on which the agent as-sesses and updates the market model by adjusting the model’s coefficients. Compar-ing with (23) the agent models the stock as dS jt S jt = µ j dt + ν j dW t + σ j dB t , S T j = s j , t ∈ [ T j , T j + ] , (34)where the index j represents the model specification at time T j . The associatedwealth process of the generic agent is dX jt = π t ( µ j dt + ν j dW t + σ j dB t ) , X T j = ξ j , t ∈ [ T j , T j + ] . Following the earlier constructions of this section, assume that at time T = u ( x ) = − e − x / δ . Then using the results of Theorem2, the agent’s forward utility map is given by U ( x , t ) = − e x / δ e tB = u ( x ) e t λ , t ∈ [ T , T ] = [ , T ] , where λ is the version of (32) for the type of the agent over the time interval [ T , T ] and all the coefficients correspond to a type ζ , i.e. λ ( ζ ) = λ . λ = λ ( ζ ) : = − θδµα + ( ν + σ ) (cid:16) µ + θδσσα (cid:17) − θ δ (cid:16) σα (cid:17) . (35)At time T , the generic agent assesses the previous model specification and choosesnew coefficients (leading to a change in type, say from ζ to ζ ). The agent thencarries out the optimization program over t ∈ [ T , T ] but starting from initial utility U ( x , T ) . Under the assumption of constant coefficients Theorem 2, yields, U ( x , t ) = (cid:16) u ( x ) e T λ (cid:17) e ( t − T ) λ , t ∈ [ T , T ] , where λ = λ ( ζ ) (given by (35)) depends only on information at time T . Quickcalculations generalize to any time horizon T j . Assume we work on the time inter-val [ T j , T j + ] . Stemming from previous calculations, it is easy to see that the initialcondition for the forward utility problem is U ( x , T j ) = u ( x ) j ∏ k = e ( T k − T k − ) λ k − (with the convention that if j < ∏ jk = · · · =
0) and the MFG forward utility is ∀ t ∈ [ T j , T j + ] , j > λ j = λ ( ζ j ) . U ( x , t ) = U ( x , T j ) e ( t − T j ) λ j = u ( x ) j ∏ k = e ( T k − T k − ) λ k − · e ( t − T j ) λ j , = u ( x ) exp n T ( λ − λ ) + T ( λ − λ ) + · · · + T j ( λ j − − λ j ) o e t λ j . There are two points to highlight. Firstly, the agent needs to carry information ofwhat happened in the past in order to have time-consistency at present time. Sec-ondly, this construction also allows the agents to change not just the model speci-fication ( µ , ν , σ ) but also their type including risk parameter δ and performance-concern level θ . The initial wealth is fixed from the previous time interval. In this work we considered two optimal portfolio management problems under for-ward utility performance concerns. We presented a simplified setting allowing forexplicit calculations of the optimal control value function, strategies and an intuitivevalidation that the finite-play game reaches the mean-field game in the limit. orward utilities for many player games and Mean-field games 21
This work provides a proof-of-concept for the forward mean-field utility con-struction leaving open many questions. Generalizing the dynamics of the forwardutility (8) to a fully Itˆo-dynamics and stochastic strategies is also open. A crucialtool for such would be a general Itˆo-Wenzell-Lions chain rule as developed in [8].Such an approach would require [25], [11].Here we addressed only the exponential-utilities (CARA) and left the power-case(CRRA) open. Even within (8), one can build towards the CRRA case in [19] orinclude the consumption problem [18]; for the general forward utility case see [9].Also open is the so-called mean-field aggregation problem where different agentsuse utility maps from different families, e.g. CRRA and CARA: [10] would be astarting point for the finite-player case while the mean-field case would requiresthe multi-class approach of [3, Section 8] with the parameterization technique offrom our Section 4. Many other questions can be posed in this context of mean-field forward utilities, ranging from possible non-solvability [13], to risk-sharing[4], ergodic problems [7] and associated numerics [15].
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Proof (of Proposition 1).
We recall the optimal strategy is given by (11), where we define b σ : = ( πσ ) ( − i ) t , B ν t : = θ i ( n − ) ∑ k = i ( π kt ν k ) , M µ t : = θ i ( πµ ) ( − i ) t = θ i n − ∑ k = i π kt µ k . The drift of (10) becomes (we omit the argument in U t , U x , U xx and use b σ : = ( πσ ) ( − i ) t ) U it + U ix (cid:0) π it µ i − M µ t (cid:1) + U ixx h ( π it ν i ) + B ν t + (cid:0) π it σ i − θ i ( πσ ) ( − i ) t (cid:1) i = (cid:16) U it − M µ t U ix U ixx B ν t (cid:17) + U ixx h(cid:0) θ i b σ (cid:1) − ( π it ) ( ν i + σ i ) i = U it + U ix h θ i σ i b σµ i ν i + σ i − M µ t i − µ i ν i + σ i ( U ix ) U ixx + U ixx n B ν t + (cid:0) θ i b σ (cid:1) − ν i + σ i (cid:0) θ i σ i b σ (cid:1) o = U it + U ix h µ i θ i σ i b σν i + σ i − θ i ( πµ ) ( − i ) t i − µ i ( ν i + σ i ) ( U ix ) U ixx + U ixx n θ i ( n − ) ∑ k = i ( π kt ν k ) + (cid:0) θ i b σ (cid:1) h − σ i ν i + σ i io . orward utilities for many player games and Mean-field games 23Equation (9) now follows as U it needs to be chosen such that the equation is zero. We inject inthe drift of (10) the expression (9) and obtain a simplified version − n U ix h θ i σ i b σµ i ν i + σ i − M µ t i − µ i ν i + σ i ( U ix ) U ixx + U ixx n B ν t + (cid:0) θ i b σ (cid:1) − ν i + σ i (cid:0) θ i σ i b σ (cid:1) oo + U ix (cid:0) π it µ i − M µ t (cid:1) + U ixx h ( π it ν i ) + B ν t + (cid:0) π it σ i (cid:1) − π it σ i θ i b σ + (cid:0) θ i b σ (cid:1) i = U ixx ν i + σ i (cid:16)(cid:0) π it ) ( ν i + σ i ) − (cid:0) π it ( ν i + σ i ) (cid:1)(cid:16) σ i θ i b σ − µ i U ix U ixx (cid:17)(cid:17) + − ν i + σ i U ixx U ixx n U ix h θ i σ i b σµ i i − µ i ( U ix ) U ixx + U ixx n − (cid:0) θ i σ i b σ (cid:1) oo = U ixx ν i + σ i (cid:12)(cid:12)(cid:12) π it ( ν i + σ i ) − (cid:16) σ i θ i b σ − µ i U ix U ixx (cid:17)(cid:12)(cid:12)(cid:12) , which results in (12). Proof (of Equation (27) ). We take up the drift of (27) and we have just by re-organizing the terms0 = U t ( Z π t , t ) + U x ( Z π t , t ) (cid:0) µπ t − θµπ t (cid:1) + U xx ( Z π t , t ) (cid:16) ( νπ t ) + (cid:0) σπ t − θσπ t (cid:1) (cid:17) = (cid:16) U t − U x θµπ t + U xx θ ( σπ t ) (cid:17) + U xx ( ν + σ ) (cid:16) π t ( ν + σ ) − π t ( ν + σ ) n θσσπ t − µ U x U xx o(cid:17) We recall the optimal strategy given by (28), where we complete the square inside the U xx term inthe SPDE above we have0 = ( U t + U x · (cid:16) µ θσσπ t ( ν + σ ) − θµπ t (cid:17) + U xx · θ ( σπ t ) (cid:16) − σ ν + σ (cid:17) − µ ( ν + σ ) ( U x ) U xx ) + U xx ( ν + σ ) (cid:12)(cid:12)(cid:12) π t ( ν + σ ) − (cid:16) θσσπ t − µ U x U xx (cid:17)(cid:12)(cid:12)(cid:12) Under the CARA condition U x / U xx = − δ and the choice of the optimal strategy, the remainingdrift must zero-out. We then have U t = − U x ( ν + σ ) · (cid:16) µθσσπ t + δµ (cid:17) + U xx ( θσσπ t ) ( ν + σ ) − U xx · ( θσπ t ) + U x (cid:16) θµπ t (cid:17)(cid:17)