Relative Arbitrage Opportunities in N Investors and Mean-Field Regimes
aa r X i v : . [ q -f i n . M F ] J un Relative Arbitrage Opportunities in N Investors and Mean-FieldRegimes
Tomoyuki Ichiba ∗ Tianjiao Yang † June 30, 2020
Abstract
This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrageboth in the N investors and mean field regimes. An investor competes with a benchmark of market and peerinvestors, expecting to outperform the benchmark and minimizing the initial capital.With market price of risk processes depending on the stock market and investors respectively, the minimalinitial capital required is the optimal cost in the N -player games and mean field games. It can be characterizedas the smallest nonnegative continuous solution of a Cauchy problem. The measure flow of wealth appearsin the cost, while the joint flow of wealth and strategy is in state processes. We modify the extended meanfield game with common noise and its notion of the uniqueness of Nash equilibrium. There is a uniqueequilibrium in N -player games and mean field games with mild conditions on the equity market. This paper discusses a differential game system of relative arbitrage problems where investors aim to outperformnot only the market index but also peer investors.The relative arbitrage problem is defined in stochastic portfolio theory, see Fernholz [9], in the sense thata strategy outperforms a benchmark portfolio at the end of a certain time span. It shows in [11] that relativearbitrage can exist in equity markets that resemble actual markets, and that the relative arbitrage resultsfrom market diversity, a condition that prevents the concentration of all the market capital into a single stock.Specific examples of market including the stabilized volatility model, in which relative arbitrage exists, areintroduced in [10]. Our model arises from the pioneering work of Fernholz and Karatzas [7], which characterizesthe best possible relative arbitrage with respect to the market portfolio, and derives nonanticipative investmentstrategies of the best arbitrage in a Markovian setting. The best arbitrage opportunity is further analyzed in [8]in a market with Knightian uncertainty. The smallest proportion of the initial market capitalization is describedas the min-max value of a zero-sum stochastic game between the investor and the market. Further investigationof exploiting relative arbitrage opportunities has been done in [1, 12, 26, 27]. The papers [24] and [29] connectrelative arbitrage with information theory and optimal transport problems.However, most of the literature on relative arbitrage uses an absolute benchmark such as market portfo-lio. To the best of our knowledge, this is the first paper that discusses relative arbitrage with respect to arelative benchmark – matching the performances of a group of investors in a stochastic differential game. Ourpaper modifies the original relative arbitrage problem to provide a general structure of the market and optimalportfolios that allows the interaction among investors.This paper first considers N investors in an equity market M over a time horizon r , T s . We consider N is bigso that the equity trading of this group as a whole contributes to the evolution of the market; whereas individualsamong the group are too “small” to bring changes to the market. These investors interact with the marketthrough a joint distribution of their wealth and strategies, particularly for example, through the total investmentsof this group to the assets. There are n stocks with prices-per-share driven by n independent Brownian motions ∗ Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 93106, USA (E-mail:[email protected]) Part of research is supported by National Science Foundation grants DMS-1615229 and DMS-2008427. † Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 93106, USA (E-mail:[email protected]) “ p W , . . . , W n q . The n -dimensional price process X N “ p X N , . . . , X Nn q follows a nonlinear stochasticdifferential equation d X N p t q “ X N p t q β p t, X N p t q , ν Nt q dt ` X N p t q σ p t, X N p t q , ν Nt q dW t (1)in which its drift β and diffusion σ coefficients depend on the joint empirical measure ν Nt : “ N N ÿ ℓ “ δ p V ℓ p t q ,π ℓ p t qq (2)of portfolio strategy and wealth of N investors. To emphasize the dependence of wealth on the initial capital-ization and portfolio, we write V ℓ “ V v ℓ ,π ℓ for ℓ “ , . . . , N . We show the market model is well-posed througha finite dynamical system.To specify what we mean by relative arbitrage opportunities in this problem set-up, we first define a bench-mark process V N by the weighted average performance of the market and the investors V N p t q “ δ ¨ X N p t q ` p ´ δ q ¨ N N ÿ ℓ “ V ℓ p t q{ v ℓ , ď t ď T , with a fixed weight δ P r , s . An investor achieves the relative arbitrage if his/her terminal wealth can out-perform this benchmark by c ℓ , a constant personal index for the investor ℓ , given at time 0. Furthermore, A N denotes all admissible, self-financing long-only portfolios for N investors.The first question raised in this paper is: What is the best strategy one can take, so that the arbitrage relativeto the above benchmark can be attained?
Specifically, every investor we study aims to outperform the marketand their competitors, starting with as little proportion of the benchmark capital as possible. Mathematically,given the other p N ´ q investors’ portfolios π ´ ℓ P A N ´ , the objective of investor ℓ , ℓ “ , . . . , N , is formulatedas u ℓ p T q “ inf ! ω ℓ P p , ˇˇˇ D π ℓ p¨q P A such that v ℓ “ ω ℓ V N p q , V v ℓ ,π ℓ p T q ą e c ℓ V N p T qu Since the interactions of a large group of investors are through stocks, portfolios and wealth, the next questionarises is:
Is it possible for every investor to take the optimal strategy in the market M ? We characterize theoptimal wealth one can achieve by the unique Nash equilibrium of the finite population game. Here we denote theobjective as ˜ u ℓT ´ t to emphasize the time homogeneity in the coefficients of the market model, see Assumption 6.Using open loop or closed loop controls arrives at the same expression of ˜ u ℓT ´ t , which is the smallest nonnegativesolution of a Cauchy problem B ˜ u ℓ p τ, x N , y qB t “ A ˜ u ℓ p τ, x N , y q , p τ, x N , y q P p , T s ˆ p , n ˆ p , n , ˜ u ℓ p , x N , y q “ e c ℓ , p x N , y q P p , n ˆ p , n , wherein A is an operator defined in (27), Y p t q is the empirical mean of ν Nt , see (7). The resulting strategy weconsider is π ℓ ‹ i “ m i p t q ` X Ni p t q D i ˜ v p t q ` τ i p t q σ ´ p t q D p i ˜ v p t q , where ˜ v p t q “ log ˜ u ℓT ´ t ` ´ δδX Nt ¨ N N ÿ ℓ “ V ℓt log ˜ u ℓT ´ t . It turns out that π ℓ ‹ i and ˜ u ℓT ´ t are proportional to c ℓ . We show the existence of relative arbitrage throughmodified portfolio generating functions and the Fichera drift.After the discussion of N -player games of relative arbitrage, this paper applies the philosophy of mean fieldgames from [4] and [15] to search for approximate Nash equilibrium when N Ñ 8 . It is expected to be moretractable than the N -player games and might give us more information of the finite investors situation. Thisapproach of comparing N -player game and the corresponding mean field game is also discussed in [20], wherethe Merton problems with constant equilibrium strategies are studied. General results on limits of N -playergame are first developed in [16] and [21]. The large population system in these papers are reformulated by [3]2nto the stochastic version to accommodate with the common noise. With the notion of weak MFG, [18] and[5] study mean field game with common noise in open loop equilibrium.The relative arbitrage problem provide a new application and some modifications in mean field games.Because of the special problem set-up, there are two mean field measures evolve in different directions, while theuniqueness of Nash equilibrium depend on one of the measures. In particular, the objective relies on a weightedaverage of stock prices and the distribution of wealth J ν p π q : “ inf ! ω ą ˇˇ v “ ω V p q , V v,π p T q ě e c ¨ V p T q ) , where the mean field benchmark is given by V p T q “ δ ¨ X p T q ` p ´ δ q ¨ m T , m : “ E r V | F B s . On the other hand, the state processes depend on the conditional law of wealth and strategies ν : “ Law p V, π | F B q with respect to the Brownian motion B . This yields the McKean-Vlasov SDEs of stock prices and a represen-tative player’s wealth d X t “ β p X t , ν t , m t q dt ` s p X t , ν t , m t q dB t , X p q “ x ; dV t “ π p t q β p X t , ν t , m t q dt ` π p t q σ p X t , ν t , m t q dB t , V p q “ ˜ u p T q V p q . A modified extended mean field game model is introduced. Both open and closed loop equilibrium are con-sidered here regarding the well-posedness of mean field system and the approximation of games. We summarizethese results in the following diagram.Market dynamics Relative arbitrage of N investors §§§đ §§§đ N -particle dynamics T heorem . ÝÝÝÝÝÝÝÝÝÑ N -player Nash Equilibrium P roposition D. §§§đ P roposition . §§§đ P roposition . ݧ§§ -particle dynamics T heorem . ÝÝÝÝÝÝÝÝÑ
Mean Field Equilibrium
Main Contributions
From the perspective of portfolio theory, we construct a general framework for multi-player portfolio opti-mization problem with no assumption on the equivalent martingale measure. We propose an interactive marketmodel and introduce a relative arbitrage benchmark including peers and the market. The model is characterizedas N -player games and mean field games in both open and closed loop, Markovian and non-Markovian case.Additionally, the portfolio generated functionals in SPT are defined accordingly in the multi-player settings. Toour knowledge, this is the first paper to study Stochastic Portfolio Theory with market-investors interactions.From the perspective of stochastic games, this paper contributes to mean field vs N -player game approachand its applications. Firstly, we establish a modified extended mean field game and accommodate a scheme toseek the mean field equilibrium: The infinite-player system involves two different fixed point conditions aboutthe cost functional and the state processes, whereas only one of them is required to be unique. Secondly, we usea stochastic cost function instead of deterministic functions of states and controls, and demonstrate a Cauchyproblem path to solve N -player and mean field games instead of the typical HJB or FBSDEs approaches. Organization of this Paper
The organization of this paper is as follows. Section 2 introduces the market with N investors as a well-posed interacting particles system. Section 3 discusses the relative arbitrage problem and market price of riskprocesses. In Section 4, the existence and optimization of relative arbitrage is derived in N -player games.The functional generated portfolios in N -player set-up are constructed. Section 5 presents the problem underextended mean field games. We show in Section 6 that mean field game limit is indeed a nice approximation tothe N -players game. Finally, we include theoretical supports of the model in Appendix.3 The Market Model
We consider an equity market and focus on the market behavior and a group of investors in this market. Thenumber of investors is large enough to affect the market. Nevertheless, there are other investors apart from thisvery group we consider.
For a given time horizon r , T s , an admissible market model M we use in this paper is consisted of a given n dimensional Brownian motion W p¨q “ p W p¨q , . . . , W n p¨qq on the probability space p Ω , F , P q . Filtration F represents the “flow of information” in the market, where F “ t F p t qu ď t ă8 “ t σ p ω p s qq ; 0 ă s ă t u with F p q “tH , Ω u , mod P . W p¨q is adapted to the P -augmentation of F . All the local martingales and supermartingalesare with respect to the filtration F if not written out specifically.Thus, there are n risky assets (stocks) with prices-per-share X N p¨q “ p X N p¨q , . . . , X Nn p¨qq driven by n independent Brownian motions as follows: for t P r , T s , ω P Ω, dX Ni p t q “ X Ni p t qp β i p t, ω q dt ` n ÿ k “ σ ik p t, ω q dW k p t qq , i “ , . . . , n, (3)or X Ni p t q “ x Ni exp " ż t p β i p s, ω q ´ n ÿ k “ p σ ik p s, ω qqq dt ` n ÿ k “ ż t σ ik p s, ω q dW k p s q * , with initial condition X Ni p q “ x Ni . We assume in this paper that dim p W p t qq “ dim p X N p t qq “ n , that is, wehave exactly as many sources of randomness as there are stocks in the market M . The market M is hence acomplete market. The dimension n is chosen to be large enough to avoid unnecessary dependencies among thestocks we define.Here, β p¨q “ p β p¨q , . . . , β n p¨qq : r , T s ˆ Ω Ñ R n as the mean rates of return for n stocks and σ p¨q “p σ ik p¨qq n ˆ n : r , T s ˆ Ω Ñ GL p n q as volatilities are assumed to be invertible, F -progressively measurable inwhich GL p n q is the space of n ˆ n invertible real matrices. Then W p¨q is adapted to the P -augmentation of thefiltration F . To satisfy the integrability condition, we assume n ÿ i “ ż T ˆ | β i p t, ω q| ` α ii p t, ω q ˙ dt ă 8 , (4)where α p¨q : “ σ p¨q σ p¨q , α ii is the covariance process of X Ni . In the above market model, there are N small investors, “small” is in the sense that each individual of these N investors has very little influence on the overall system. An investor ℓ uses the proportion π ℓi p t q of currentwealth V ℓ p t q to invest in the stock i at each time t for ℓ “ , . . . , N . The wealth process V ℓ of an individualinvestor ℓ is dV ℓ p t q V ℓ p t q “ n ÿ i “ π ℓi p t q dX Ni p t q X Ni p t q , V ℓ p q “ v ℓ . (5)Since equity prices move according to the supply and demand for stock shares, we consider the average capitalinvested as a factor in the price processes. Definition 2.1 (Investment strategy, long only portfolio and average capital invested) . (1) An F -progressivelymeasurable and adapted process π ℓ : r ,
8q ˆ Ω Ñ R n is called an investment strategy if ż T p| π ℓ p t, ω q β p t, ω q| ` π ℓ p t, ω q α p t, ω q π ℓ p t, ω qq dt ă 8 , T P p , , ω P Ω , a.s . (6) The strategy here is self-financing, since the wealth at any point of time is obtained by trading the initialwealth according to the strategy π p¨q . π ℓ p¨q “ p π ℓ p¨q , . . . , π ℓn p¨qq is a long-only portfolio if it is a portfolio that takes values in the set ∆ n : “ t π “ p π , ..., π n q P R n | π ě , . . . , π n ě π ` . . . ` π n “ u . An investment strategy that takes value in ∆ n is called an admissible strategy, and we denote the admissibleset as A . If π ℓ P A , for all ℓ “ , . . . , N , then p π , . . . , π N q P A N . In the rest of the paper, we only considerstrategies in the admissible set A .(3) Each investor ℓ uses the proportion π ℓi p t q of current wealth V ℓ p t q to invest in the i th stock at each time t .The average amount Y i p t q invested by N players on stock i is assumed to satisfy Y i p t q “ N N ÿ ℓ “ V ℓ p t q π ℓi p t q “ ż t γ i p r, ω q dr ` ż t n ÿ k “ τ ik p r, ω q dW k p r q , t P p , N N ÿ ℓ “ V ℓ p q π ℓi p q “ y ,i , (7) where γ p¨q and τ p¨q follow n ÿ i “ ż T ˆ | γ i p t, ω q| ` ψ ii p t, ω q ˙ dt ă 8 (8) for every T P r , , ψ p¨q : “ τ p¨q τ p¨q . In fact, the average capitalization Y p t q is depending entirely upon X N p t q and π p t q . The process in Defini-tion 2.1(3) is defined for simplicity. The interaction between the players is of the mean field type, in that whenever an individual player has to makea decision, he or she sees the average of functions of the private states of the other players. Here we use meanfield interaction particle models from statistical physics to describe the market - We model the N investors as N particles, for fixed N .For any metric space p X , d q , P p X q denotes the space of probability measures on X endowed with the topologyof weak convergence. P p p X q is the subspace of P p X q of the probability measures of order p . Then µ P P p p X q holds ş X d p x, x q p µ p dx q ă 8 , where x P X is an arbitrary reference point. For p ě µ , ν P P p p X q , The p -Wasserstein metric on P p p X q is defined by W p p ν , ν q p “ inf π P Π p ν ,ν q ż X ˆ X d p x, y q p κ p dx, dy q , where d is the underlying metric on the space. Π p ν , ν q is the set of Borel probability measures π on X ˆ X with first marginal ν and second marginal ν . Precisely, κ p A ˆ X q “ ν p A q and κ p X ˆ A q “ ν p A q for everyBorel set A Ă X .Also, denote by C pr , T s ; R d q the space of continuous functions from r , T s to R d . In this paper, we oftentake X “ R d when considering a real-valued random variable or take X as the path space X “ C pr , T s ; R d q for a process, where a fixed number d will be specified later. P p p R d q equipped with the Wasserstein distance W p is a complete separable metric space, since R d is complete and separable. Definition 2.2 (Empirical measure in the finite N -particle system) . Consider p V ℓ , π ℓ q P C pr , T s ; R ` q ˆ C pr , T s ; A q that are F -measurable random variables, for every investor ℓ “ , . . . , N . We define empiricalmeasures ν N P P p C pr , T s , R ` q ˆ C pr , T s , A qq – P p C pr , T s , R ` ˆ A qq of the random vectors p V ℓ p t q , π ℓ p t qq as ν Nt : “ N N ÿ ℓ “ δ p V ℓ p t q ,π ℓ p t qq , @ ℓ “ , . . . , N, where δ x is the Dirac delta mass at x P R ` ˆ A . Thus for any Borel set A Ă R ` ˆ A , ν Nt p A q “ N N ÿ ℓ “ δ A p V ℓ p t q ,π ℓ p t qq “ N ¨ t ℓ ď N : p V ℓ p t q , π ℓ p t qq P A u . π p t q might have different structures given the accessible information at time t . Definition 2.3.
A control π p t q P A is an open loop control if it is a function of time t and initial states v .It is called a closed loop feedback control if π p t q P A is a function of time t and states of every controller V p t q . It is specified by feedback functions φ ℓ : r , T s ˆ Ω ˆ R n ` Ñ A , for ℓ “ , . . . , N , to be evaluated along thepath of the state process. Denote X Nt “ p X p t q , . . . , X n p t qq , V t “ p V p t q , . . . , V N p t qq . For a fixed N , with ν Nt in definition 2.2 thatgeneralizes Y p t q , we can generalize the N -particle system as d X Nt “ X Nt β p t, X Nt , ν Nt q dt ` X Nt σ p t, X Nt , ν Nt q dW t ; X N “ x N (9)and dV ℓt “ V ℓt π ℓt β p t, X Nt , ν Nt q dt ` V ℓt π ℓt σ p t, X Nt , ν Nt q dW t ; V ℓ “ v ℓ . (10)A strong solution of the conditional Mckean-Vlasov system (9)-(10) is a triplet p X N , V , ν N q , with X N takingvalues in C pr , T s , R n ` q , V in C pr , T s , R N ` q , ν N P P p C pr , T s , R ` q ˆ C pr , T s , A qq – P p C pr , T s , R ` ˆ A qq .The following assumptions on the triplet ensure that the system is well-posed. Assumption 1.
The initial wealth and strategies of the N players are i.i.d samples from ν N the distributionof p v , π q . The stock prices at time , x , has a finite second moment, E | x | ă 8 , and is independent ofBrownian motion t W t u . Assumption 2.
The following Lipschitz conditions are satisfied with Borel measurable mappings β , σ from r , T s ˆ C pr , T s , R ` q ˆ P p C pr , T s , R ` ˆ A qq to R n , i.e., there exists a constant L P p , , such that | xβ p t, x, ν q ´ x β p t, x , ν q| ` | xσ p t, x, ν q ´ x σ p t, x , ν q| ď L r| x ´ x | ` W p ν, ν qs| vβ p t, x, ν q ´ v β p t, x , ν q| ` | vσ p t, x, ν q ´ v σ p t, x , ν q| ď L r|p x, v q ´ p x , v q| ` W p ν, ν qs and the growth conditions for a constant C G P p , , | xβ p t, x, ν q| ` | xσ p t, x, ν q| ď C G p ` | x | ` M p ν qq , | vβ p t, x, ν q| ` | vσ p t, x, ν q| ď C G p ` | x | ` | v | ` M p ν qq , for every t P r , T s , x P R n ` , ν P P p C pr , T s , R ` ˆ A q , where M p ν q “ ˆ ż C pr ,T s , R ` ˆ A q | x | dν p x q ˙ { ; ν P P p C pr , T s , R ` ˆ A qq . Assumption 3.
For a closed loop feedback control, we assume π ℓ is Lipschitz on v , i.e., there exists a mapping φ ℓ : R n ` ˆ R N ` ˆ P p C pr , T s , R ` ˆ A qq Ñ A such that π ℓt “ φ ℓ p V t q . | φ ℓ p x, v, ν q ´ φ ℓ p x , v , ν q| ď L r|p x, v q ´ p x , v q| ` W p ν, ν qs for every x, x P R n ` , v, v P R n ` , ν, ν P P p C pr , T s , R ` ˆ A qq Proposition 2.1.
Under Assumption 2 and 3, the N -particle SDE system (9) - (10) admits a unique strongsolution, for each N .Proof. For simplicity, we discuss the time homogeneous case, whereas the inhomogeneous case can be proved inthe same fashion. Rewrite the system as a n ` N -dimension SDE system: d ˆ X Nt V t ˙ “ ¨˚˚˚˚˚˚˝ X N p t q β p X Nt , ν Nt q dt ` X p t q ř nk “ σ k p X Nt , ν Nt q dW k p t q . . .X Nn p t q β n p X Nt , ν Nt q dt ` X n p t q ř nk “ σ nk p X Nt , ν Nt q dW k p t q V t π t β p X Nt , ν Nt q dt ` V t π t σ p X Nt , ν Nt q dW t . . .V Nt π N t β p X Nt , ν Nt q dt ` V Nt π N t σ p X Nt , ν Nt q dW t ˛‹‹‹‹‹‹‚ : “ f p X Nt , V t , ν Nt q dt ` g p X Nt , V t , ν Nt q dW t , f p X Nt , V t , ν t q “ p f p¨q , . . . , f n ` N p¨qq , f i p¨q “ X Ni p t q β i p¨q for i “ , . . . , n , f j p¨q “ π j ´ nt β p¨q for j “ n ` , . . . , n ` N . Similiarly, g p X Nt , V t , ν t q “ p g p¨q , . . . , g n ` N p¨qq , g i p¨q “ X Ni p t q σ i p¨q for i “ , . . . , n , g j “ V j ´ nt π j ´ nt σ p X Nt , ν t q for j “ n ` , . . . , n ` N .Let us consider a closed loop strategy π ℓt “ φ ℓ p V t q . Open loop strategies case can be show in the same way.Define a mapping L N : R N ` Ñ P p C pr , T s , R ` ˆ A qq , L N p V t q “ N N ÿ ℓ “ δ p V ℓt ,φ ℓ p V t qq “ ν Nt . Define F : R N ` n ` Ñ R N ` n , G : R N ` n ` Ñ R N ` n ˆ R n , with F p X t , V t q “ f p X t , V t , L N p V t qq ; G p X t , V t q “ g p X t , V t , L N p V t qq . Let p x, v q “ p x , . . . , x n , v , . . . , v N q and p y, u q “ p y , . . . , y n , u , . . . , u N q be two pairs of values of p X p¨q , V p¨qq .By the inequality p a ` b q ď p a ` b q , uniformly boundedness and Lipschitz condition of φ ℓ , | F p x, v q ´ F p y, u q| ď n ÿ i “ | x i β i p x, L N p v qq ´ y i β i p y, L N p u qq| ` N ÿ ℓ “ | v ℓ φ ℓ p v q β p x, L N p v qq ´ u ℓ φ ℓ p u q β p y, L N p u qq| ď L r| x ´ y | ` | v ´ u | ` N W p L N p v q , L N p u qs . Denote the empirical measure induced by the joint distribution of random variable u and v by˜ π “ N N ÿ ℓ “ δ p u ℓ ,v ℓ q . It is a coupling of the function L N p v q and L N p u q . By the definition of Wasserstein distance, W p L N p v q , L N p u qq ď ż R N ˆ R N | v ´ u | ˜ π p dv, du q ď N | v ´ u | We treat G p¨q in the same fashion, and consequently, | F p x, v q ´ F p y, u q| ď L r| x ´ y | ` | v ´ u | s , | G p x, v q ´ G p y, u q| ď L r| x ´ y | ` | v ´ u | s . Then according to the existence and uniqueness conditions of multi-dimensional SDEs, the system (9)-(10)admits a unique strong solution.
We first recall the definition of relative arbitrage in Stochastic Portfolio Theory.
Definition 3.1 (Relative Arbitrage) . Given two investment strategies π p¨q and ρ p¨q , with the same initial capital V π p q “ V ρ p q “ , we shall say that π p¨q represents an arbitrage opportunity relative to ρ p¨q over the timehorizon r , T s , with a given T ą , if P ` V π p T q ě V ρ p T q ˘ “ and P ` V π p T q ą V ρ p T q ˘ ą , The market portfolio m is used to describe the behavior of the market: By investing in proportion to themarket weight of each stock, π m i p t q : “ X Ni p t q X N p t q , i “ , . . . , n, t ě , (11)it amounts to the ownership of the entire market - the total capitalization X N p t q “ X N p t q ` . . . ` X Nn p t q , t ě
0, since dV m p t q V m p t q “ n ÿ i “ π m i p t q ¨ dX Ni p t q X Ni p t q “ dX N p t q X N p t q , t ě V m p q “ X N p q . (12)The performance of a portfolio is measured withh respect to the market portfolio and other factors. Forexample, asset managers improves not only absolute performance comparing to the market index, but alsorelative performance with respect to all collegial managers - they try to exploit strategies that achieve anarbitrage relative to market and peer investors. We next define the benchmark of the overall performance. Definition 3.2 (Benchmark) . Relative arbitrage benchmark V N p T q , which is the weighted average of perfor-mances of the market portfolio and the average portfolio of N investors, is defined as, V N p T q “ δ ¨ X N p T q ` p ´ δ q ¨ N N ÿ ℓ “ V ℓ p T q v ℓ , T P p , , (13) with a given constant weight ă δ ă . We assume each investor measures the logarithmic ratio of their own wealth at time T to the benchmarkin (13), and searches for a strategy with which the logarithmic ratio is above a personal level of preferencealmost surely. For ℓ “ , . . . , N , we denote the investment preference of investor ℓ by c ℓ , a real number given at t “
0. Note that c ℓ is investor-specific constant, and so it might be different among individuals ℓ “ , . . . , N .An arbitrary investor ℓ tries to achievelog V ℓ p T q V N p T q ą c ℓ , a.s. or equivalently, V ℓ p T q ě e c ℓ V N p T q , a.s. . (14)Thus V N p T q is the benchmark and an investor ℓ aims to match e c ℓ V N p T q based on their preferences. Assumption 4.
Assume that the preferences of investors c ℓ are statistically identical and independent samplesfrom a common distribution Law p c q . Assumption 5.
We assume the existence of a market price of risk processes θ, λ : r , Ω ˆ P p C pr , T s , R ` ˆ A qq Ñ R n , an F -progressively measurable process such that for any p t, ω, ν q P r ,
8q ˆ Ω ˆ P p C pr , T s , R ` ˆ A qq , σ p t, ω, ν q θ p t, ω, ν q “ β p t, ω, ν q , τ p t, ω, ν q λ p t, ω, ν q “ γ p t, ω, ν q ; (15) P ˆ ż T || θ p t, ω, ν q|| ` || λ p t, ω, ν q|| dt ă 8 , @ T P p , ˙ “ . The new price of risk process λ p t q is from the fact that the market is simultaneously defined by the stocksand the investors. In future sections, We shall see λ p t q is a key entity for more tractable and practical resultsin game formations. Next we define the deflator based on the market price of risk processes. Definition 3.3.
We define a local martingale L p t q , dL p t q “ Θ p t q L p t q dW t , where Θ p t q “ a || θ p t q|| ` || λ p t q|| . Equivalently, L p t q : “ exp ! ´ ż t p θ p s q ` λ p s qq dW p s q ´ ż t p|| θ p s q|| ` || λ p s q|| q ds ) , ď t ă 8 . Thus under Assumption 5, the market is endowed with the existence of a local martingale L with E r L p T qs ď p V ℓ p¨q : “ V ℓ p¨q L p¨q , and p X p¨q : “ X p¨q L p¨q . p V ℓ p¨q admits d p V ℓ p t q “ p V ℓ p t q π ℓ p t q ` β p t q ´ σ p t q Θ p t q ˘ dt ` p V ℓ p t q ` π ℓ p t q σ p t q ´ Θ p t q ˘ dW p t q ; p V ℓ p q “ p v ℓ . (16) Proposition 3.1.
We have the following properties of c ℓ and δ .1. In [7] a special case is considered, when c ℓ “ c for every ℓ “ , . . . , N , and δ “ ; . If every investor achieves relative arbitrage opportunity in the sense of (14) , then p ´ δ q N ÿ ℓ “ e c ℓ v ℓ ă
1; (17)
3. Relative arbitrage is guaranteed, if p c , . . . , c N q satisfies that c ℓ ď log ˆ V ℓ p T q min t X N p T q , V p T q , . . . , V N p T qu ˙ a.s. ; (18)
4. When c ℓ ě log v ℓ ´ log p δv ` ´ δ q , if L p T q is a martingale, then no arbitrage relative to the market andinvestors is possible. We already know from [7] and [8] that any c ℓ ď c ℓ can be a small positive number. Investors pursuing relative arbitrage should follow the condition (17)for c ℓ .Now, we shall answer the questions posed in the introduction - Given the portfolios π ´ ℓ p¨q : “ p π p¨q , . . . , π ℓ ´ p¨q , π ℓ ` p¨q , . . . , π N p¨qq , of all but investor ℓ , what is the best strategy to achieve relative arbitrage for investor ℓ “ , . . . , N , and if thereexists such optimal strategy, is it possible for all N investors to follow it? We first utilize an idea in the samevein of optimal relative arbitrage in [7], i.e., using the optimal strategy π ℓ ‹ , the investor ℓ will start with theleast amount of the initial capital (or initial cost) relative to V N p q , in order to match or exceed the benchmark e c ℓ V N p T q at the terminal time T , that is, given π ´ ℓ p¨q , each investor ℓ optimizes u ℓ p T q “ inf " ω ℓ P p , ˇˇˇ D π ℓ p¨q P A such that v ℓ “ ω ℓ V N p q , V v ℓ ,π ℓ p T q ą e c ℓ ¨ V N p T q * . (19)To use martingale representation results in a complete market, we assume F “ F X N ,Y “ F W , where F X N ,Y is t σ p X N p s q , Y p s qq ; 0 ă s ă t u . The following proposition is essential to allow a PDE characterization of theobjective u ℓ p T q . This result follows from the supermartingale property of p V ℓ p¨q and martingale representationtheorem, see Appendix B for the details of the proof. Proposition 3.2. u ℓ p T q in (19) can be derived as e c ℓ V N p T q ’s discounted expected values over P . u ℓ p T q “ E “ e c ℓ V N p T q L p T q ‰ { V N p q , (20) Starting from this section, we consider X N p t q and Y p t q are time homogeneous processes. Assumption 6. β p t q , σ p t q , γ p t q and τ p t q are time-homogeneous, i.e., X Ni p t q β i p t q “ b i p X N , Y q , X Ni p t q σ ik p t q “ s ik p X N , Y q , n ÿ k “ s ik p t q s jk p t q “ a ij p X N , Y q ,γ i p t q “ γ i p X N , Y q , τ i p t q “ τ i p X N , Y q . where b i , s ik , a ij , γ i , τ i : p , n ˆ p , n Ñ R are H¨older continuous and Y : “ p Y pr , T sq , . . . , Y n pr , T sqq is thetotal trading volume defined in (7) . Market price of risk is Θ p x N , y q : “ σ ´ p x N , y q b p x N , y q , for each T P p , . We define ˜ u ℓ : p ,
8q ˆ p , n ˆ p , n Ñ p , from the processes p X N p¨q , Y p¨qq starting at p x , y q Pp , n ˆ p , n , and write the terminal values˜ u ℓ p T q “ ˜ u ℓ p T, x N , y q ; ℓ “ , . . . N. (21)9 .2.1 Open loop and closed loop control problem We use the notation D i and D ij for the partial and second partial derivative with respect to the i th or the i thand j th variables in X N p t q , respectively; D p and D pq for the first and second partial derivative in Y p t q . Assumption 7.
There exist two functions
H, I : R n ` Ñ R of class C , such that b p x N , y q “ a p x N , y q DH p x q , γ p x N , y q “ ψ p x N , y q DI p y q , i.e., b i p¨q “ ř nj “ a ij p¨q D j H p¨q , γ i p¨q “ ř nj “ ψ ij p¨q D j I p¨q in component wise for i “ , . . . , n . Hence the infinitesimal operator can be written as L f “ n ÿ i “ n ÿ j “ a ij p x N , y q “ D ij f ` D i f D j H p x N , y q ‰ ` n ÿ p “ n ÿ q “ ψ pq p x N , y q “ D pq f ` D p f D q I p x N , y q ‰ , and by the definition of θ p¨q and λ p¨q in (15), θ p x N , y q ` λ p x N , y q “ s p x N , y q DH p x q ` τ p x N , y q DI p y q . (22)Then it follows from Ito’s lemma applying on H p¨q and I p¨q that L p t q “ exp " ´ ż t θ p s q ` λ p s q dW p s q ´ ż t || θ p s q|| ` || λ p s q|| ds * “ exp " ´ H p X N p t qq ´ I p Y p t qq ` H p x q ` I p y q ´ ż t k p X N p s qq ` ˜ k p Y p s qq ds * , where k p x q : “ ´ n ÿ i “ n ÿ j “ a ij r D ij H p x q ` D i H p x q D j H p x qs , ˜ k p y q : “ ´ n ÿ i “ n ÿ j “ ψ pq r D pq I p y q ` D p I p y q D q I p y qs . Denote g ℓ p x N , y , π q : “ e c ℓ V N p q e ´ H p x q´ I p y q , G ℓ p T, x N , y q : “ E P “ g ℓ p X N p T q , Y p T qq e ´ ş T k p X N p t qq` ˜ k p Y p t qq dt ‰ . (23)Based on [14] Section 6.4, we have the following assumptions of make sure the solvability of the Cauchyproblem. Assumption 8.
Assume E P ˇˇ g ℓ p X N p t q , Y p t qq e ´ ş T k p X N p t qq` ˜ k p Y p t qq dt ˇˇ ă 8 . The functions b i p¨q , σ ik p¨q are of class C pp , n ˆ p , n q and satisfy the linear growth condition || b p x N , y q|| ` || s p x N , y q|| ď C p ` || x || ` || y ||q , p x N , y q P R n ` ˆ R n ` .a ij p¨q satisfy the nondegeneracy condition, i.e., if there exists a number ǫ ą such that a ij p x N , y q ě ǫ p|| x || ` || y || q , p x N , y q P R n ` ˆ R n ` .g ℓ p¨q is H¨older continuous, uniformly on compact subsets of R n ` ˆ R n ` , ℓ “ , . . . , N . k p¨q and ˜ k p¨q are continuousand lower bounded, G ℓ p¨q is continuous on p ,
8q ˆ p , n ˆ p , n , of class C pp ,
8q ˆ p , n ˆ p , n q . Under Assumption 8, (20) becomes ˜ u ℓ p T, x N , y q “ G ℓ p T, x N , y q g ℓ p x N , y q , (24)where ˜ u ℓ p τ, x N , y q P C pp ,
8q ˆ p , n ˆ p , n q is bounded on K ˆ p , n ˆ p , n for each compact K Ă p , . By Feynman-Kac formula, the function G ℓ p¨q solves B G ℓ B τ p τ, x N , y q “ L G ℓ p τ, x N , y q ´ ` k p x q ` ˜ k p y q ˘ G ℓ p τ, x N , y q , t P p , , p x N , y q P p , n ˆ p , n , ℓ p , x N , y q “ g p x N , y q , p x N , y q P p , n ˆ p , n . This yields a Cauchy problem B ˜ u ℓ p τ, x N , y qB τ “ A ˜ u ℓ p τ, x N , y q , τ P p , , p x N , y q P p , n ˆ p , n , (25)˜ u ℓ p , x N , y q “ e c ℓ , p x N , y q P p , n ˆ p , n , (26)where A ˜ u ℓ p τ, x N , y q “ n ÿ i “ n ÿ j “ a ij p x N , y q ´ D ij ˜ u ℓ p τ, x N , y q ` δD i ˜ u ℓ p τ, x N , y q ¨ r V N p qs ´ ¯ ` n ÿ p “ n ÿ q “ ψ pq p x N , y q D pq ˜ u ℓ p τ, x N , y q . (27)We emphasize that (25) is determined entirely from the volatility structure of X N p¨q and Y p¨q . Moreover, c ℓ enters into the initial condition (26). Assumption 6 ensures that the Cauchy problem is solvable. Remark 1.
If the market price of risk process depended solely on θ p¨q in Assumption 5, then the Cauchy problem B ˜ u ℓ p τ, x N , y qB t “ A ˜ u ℓ p τ, x N , y q involves a terminal term ˜ u ℓ p T q which would largely increase the intractability. Theorem 3.1.
Under Assumption 6, the function ˜ u ℓ : r ,
8q ˆ p , n ˆ p , n Ñ p , s is the smallestnonnegative continuous function, of class C on p ,
8q ˆ p , n , that satisfies ˜ u ℓ p , ¨q ” e c ℓ and B ˜ u ℓ p τ, x N , y qB t ě A ˜ u ℓ p τ, x N , y q , (28) where A p¨q follows (27) . The Cauchy problem (25)-(26) admits a trivial solution ˜ u ℓ p τ, x , y q ” e c ℓ . Meanwhile, We use portfolio generatingfunctionals, as shown in Section 4, to construct relative arbitrage portfolios for a certain time span. This resultindicates that ˜ u p τ, x , y q could take values less than 1, that is, the uniqueness of Cauchy problem fails.Through the F¨ollmer exit measure [13] we can relate the solution of Cauchy problem u ℓ p¨q to the maximalprobability of a supermartingale process staying in the interior of the positive orthant through r , T s . Followingthe route suggested by [7] and [26], there exists a probability measure Q on p Ω , F q , such that P is locallyabsolutely continuous with respect to Q : P ăă Q , Λ ℓ p T q is a Q -martingale, and d P “ Λ ℓ p T q d Q holds on each F T , T P p , . We can characterize ˜ u ℓ p t q by an auxiliary diffusion which takes values in the nonnegativeorthant r , n {t u . Definition 3.4 (Auxiliary process and the Fichera drift) . We define the following1. The auxiliary process ζ ℓ “ p ζ ℓ , . . . , ζ ℓ n q is defined as dζ ℓi p¨q “ ˆ b i p ζ p¨qq dt ` ˆ σ ik p ζ p¨qq dW k , ζ ℓi p q “ ζ ℓi , i “ , . . . , n, where ˆ b i p x N , y q “ δ V N p q ř nj “ a ij p x N , y q if i = 1, . . . , n, if i = n+1, . . . , 2n, ˆ a ij p x N , y q “ $’&’% a ij p x N , y q if i,j = 1, . . . , n, ψ ij p x N , y q if i,j = n+1, . . . , 2n, otherwise.2. The Fichera drift is defined as f i p¨q : “ ˆ b i p x N , y q ´ n ÿ j “ D j ˆ a ij p x N , y q ,i “ , . . . , n , p x N , y q P p , n ˆ p , n . ssumption 9. The system of ζ ℓ p¨q admits a unique-in-distribution weak solution with values in r , n ˆr , n {t u . We set T ℓ : “ t t ě | ζ ℓ p t q P O n u as the first hitting time of auxiliary process ζ ℓ p¨q to O n , the boundary of r , n . Proposition 3.3.
With the nondegeneracy condition of a ij , suppose that the functions ˆ σ ik p¨q are continuouslydifferentiable on p , n ; that the matrix ˆ a p¨q degenerates on O n ; and that the Fichera drifts for the process ζ ℓ p¨q can be extended by continuity on r , n . For an investor ℓ , if f i p¨q ě holds on each face of the orthant,then ˜ u ℓ p¨ , ¨q ” , and no arbitrage with respect to the market portfolio exists on any time-horizon. If f i p¨q ă on each face t x i “ u , i “ , . . . , n and t y i “ u , i “ n ` , . . . , n of the orthant, then ˜ u ℓ p¨ , ¨q ă and arbitragewith respect to the market portfolio exists, on every time-horizon r , T s with T P p , .Proof. With the nondegeneracy condition of covariance p a ij q ď i,j ď n , Theorem 2 in [7] suggests that˜ u ℓ p T, x N , y q “ Q r T ℓ ą T s , p T, x N , y q P r ,
8q ˆ r , n ˆ r , n . For the first claim, we only need to show the probability Q r T ℓ ą T s ”
1, for p T, x N , y q P r ,
8q ˆ r , n ˆr , n . Denote a bounded and connected C boundary G R : “ t z P R n , z i ă , || z || ă R u , and R can bearbitrarily large. Then the claim follows from Theorem 9.4.1 (or Corollary 9.4.2) of [14], since n ÿ i “ ˆ ˆ b i p x N , y q ´ n ÿ j “ D j ˆ a ij p x N , y q ˙ n i ď , in which n “ p n , . . . , n n q is the outward normal vector at p x N , y q to O n , the boundary O n is an obstaclefrom outside of G R , i.e., G : “ B R p q{ G R . The Fichera vector field points toward the domain interior at theboundary. Let R Ñ 8 , the boundary is not attainable almost surely for p x N , y q P r , n .If f i p¨q ă t z i “ u , i “ , . . . , n , then n ÿ i “ ˆ ˆ b i p x N , y q ´ n ÿ j “ D j ˆ a ij p x N , y q ˙ n i ě , and the Fichera drift at O n points toward the exterior of r , n . It is equivalent to show that Q r T ℓ ą T s ă p T, x N , y q P r ,
8q ˆ r , n ˆ r , n , we only need to show Q r T ℓ ă T s ą
0, i.e., the boundary t z i “ u , i “ , . . . , n , is attainable by ζ ℓ p¨q .From Chapter 11 and 13 in [14], every point in B G is a regular point, and thuslim z Ñ z ,z P G Q z p τ g ă 8 , || ζ ℓ p τ g q ´ z || ă δ q “ , where τ g is the exit time from ¯ G . Therefore, if z P Σ : “ Y ni “ t z P R n : z i “ u X G , for a fixed δ such that B ` δ p z q : “ X ni “ t z P R n : z i ą u X B δ p z q is a proper subset of G , we have • If || ζ ℓi ´ z || ď η , Q p τ g ă 8 , ζ ℓ p τ g q P Σ q ą • If || ζ ℓi ´ z || ą η , inf z P A Q z p ζ ℓ p τ g q P B δ p z q , τ g ă 8q ą , where A : “ n č i “ t z P R n : z i ą , || z ´ z || “ η u . Now take r P A and a continuous sample path ω ‹ such that ω ‹ p q “ z , ω ‹ p τ ‹ q “ r , and ω ‹ p s q R A for0 ď s ă τ ‹ , where τ ‹ : “ inf t t ą ζ ℓ p t q P A u . Consider an ǫ -neighborhood N ǫ,ω ‹ of ω ‹ P C p G q , N ǫ,ω ‹ “ t ω P C p G q : ω p q “ ζ ℓi , || ω ´ ω ‹ || ă ǫ, ω p τ ‹ q “ r u Ă t ω P Ω : ζ ℓ p τ ‹ , ω q P A u , Q ζ ℓi p N ǫ,ω ‹ q ą , where φ : r ,
8q Ñ R n is continuously differentiable, and || ¨ || sT is the supremum norm || ω ´ ω || “ sup ď s ď τ ‹ | ω ´ ω | , ω , ω P C p G q . Hence Q z p N ǫ,ω ‹ q ď Q z p τ ‹ ă 8 , ζ p τ ‹ q P A q . Therefore Q ζ ℓi p ζ ℓ p τ g q P Σ , τ g ă 8q ě Q ζ ℓi p ζ ℓ p τ g q P Σ , τ g ă 8qě E ζ ℓi r Q z p ζ ℓ p τ g q P Σ , τ g ă 8q ¨ p ζ ℓ p τ ‹ q , τ ‹ ă 8q| F τ ‹ s“ E ζ ℓi r Q ζ ℓ p τ ‹ q p ζ ℓ p τ g q P Σ , τ g ă 8q ¨ p ζ ℓ p τ ‹ q P A, τ ‹ ă 8qsě E ζ ℓi r inf z P A Q z p ζ ℓ p τ g q P Σ , τ g ă 8q ¨ p ζ ℓ p τ ‹ q P A, τ ‹ ă 8qsě Q ζ ℓi p ζ ℓ p τ ‹ q P A, τ ‹ ă 8q . The equality in the above expressions is from the strong Markov property of ζ ℓ p¨q .In conclusion, the process ζ ℓ p¨q attain the set Y ni “ t z i “ u with positive probability, so ˜ u ℓ p¨ , ¨q ă f i p¨q ă ℓ can find relative arbitrage opportunities with a unique ˜ u ℓ , the minimal solution of (28)given f i p¨q ă O n . As we have seen in the previous sections, the stock prices and investors’ wealth are coupled. Variation ofone investor’s strategies contributes to the change of the trading volume of each stock, and thus the changeof stock prices. Consequently, the wealth of others is affected by this investor. In addition, all the investorsconsidered here are competitive. They attempt to not only behave better than the market index but also beatthe performance of peers exploiting similar opportunities - everyone simultaneously wishes to optimize theirinitial wealth to achieve a relative arbitrage.Investors interact with each other, adopt a plan of actions after analyzing other people’s options, and finally,make decisions. This motivates us to model the investors as participants in a N -player game. The solution concept of this N -player game is Nash equilibrium. In this spirit, assuming that the others havealready chosen their own strategies, a typical player computes the best response to all the other players, whichamounts to the solution of an optimal control problem to minimize the expected cost ˜ u ℓ . Specifically, wheninvestor ℓ assumes the wealth of other players are fixed, they wish to take the solution of (25) and (26) as theirwealth to begin with so that V ℓ p T q ě e c ℓ V N p T q “ δ ¨ e c ℓ X N p T q ` p ´ δ q ¨ e c ℓ N N ÿ ℓ “ V ℓ p T q v ℓ . We articulate the definition of Nash equilibrium in this problem.
Definition 4.1 (Nash Equilibrium) . A vector π ℓ ˚ “ p π ℓ ˚ i , . . . , π ℓ ˚ n q of admissible strategies in Definition 2.1 isa Nash Equilibrium, if for all π ℓi P A and i “ , . . . , n , J ℓ p π ℓ ˚ i , π ´ ℓ ˚ i q ď J ℓ p π ℓi , π ´ ℓ ˚ i q , (29) where the cost to investor ℓ yields J ℓ p π q : “ inf " ω ℓ ą ˇˇ V ω ℓ e cℓ V N p q ,π ℓ p T q ě e c ℓ V N p T q * , here π p¨q “ p π p¨q , . . . , π N p¨qq . Hence, inf π ℓ P A J ℓ p π q “ u ℓ p T q . (30)Since v ℓ “ ω ℓ V N p q , the infimum is attained, and J ℓ p π ; 0 , x q “ e c ℓ V N p T q V N p q exp ´ " ż T π ℓ t p β t ´ α t π ℓt q dt ` ż T π ℓ t σ i p t q dW t * ď ω ℓ . (31)Each individual aims to minimize the relative amount of initial capital, beginning with which one can match orexceed the benchmark. Definition 4.2.
With the same conditions in Definition 2.2, we define empirical measures of the random vectors ` V ℓ p t q v ℓ ˘ ℓ “ ,...,N P R N ` , given the initial measure µ N P P p R ` q , µ N : “ N N ÿ ℓ “ δ p V ℓ { v ℓ q . Subsequently, we clarify the notion of unique Nash equilibrium we will apply in this paper. Investors paymore attention to the change of the wealth processes than the change of the strategies, since two different strategyprocesses may result in the same wealth at time T . Therefore we investigate the uniqueness in distribution ofwealth, and we use the strong uniqueness here because it satisfies the nature of the investment goal in thispaper. Definition 4.3.
We say that the uniqueness holds for Nash equilibrium if any two solutions µ Na , µ Nb , definedon p Ω , F , F , P q , with the same initial law µ N P P p R ` q , P r µ Na “ µ Nb s “ , where µ N is the empirical distribution of wealth processes as in definition 4.2. We construct the fixed point condition on the control space. Suppose we start from a control π , then solvethe equation of wealth processes (5) and trading volume (7) with the equation of optimal cost function (28).If the corresponding optimal strategy π ‹ agree with π , then the associated µ N is the Nash equilibrium. Wespecify the pathodology below. Searching Nash equilibrium in N -player game
1. Suppose we start with a given set of control processes π : “ p π , . . . , π N q . With the empirical distribution µ N and ν N , solve the N -particle system (9) and (10).2. We get J ℓ p¨q from µ N and ν N . Solve ˜ u ℓ p T q : “ inf π P A J ℓ p π q and the corresponding optimal control π ‹ . Wefind a function Φ so that π ‹ “ Φ p π q .3. If there exists ˆ π , such that ˆ π “ Φ p ˆ π q , then µ N ‹ : “ N ř Nℓ “ δ p V vℓ, ˆ πℓ { v ℓ q is the Nash equilibrium. We recall the information structure and the types of actions that players take in a game. It is an open loop Nashequilibrium if the admissible strategies satisfy the conditions of Definition 4.1, with the control π ℓ p t q given bythe form π ℓ p t q “ φ ℓ p t, v , W r ,t s q , (32)for every t ě v : “ p v , . . . , v N q , v ℓ “ ˜ u ℓ p T q V N p q , W r ,t s is the path of the Wiener process between time0 and time t deterministic functions φ ℓ : r , T s ˆ Ω Ñ A , ℓ “ , . . . , N . Here, π ´ ℓ is the process with thesame trajectories as the p π ‹ , . . . , π ℓ ‹ , . . . , π N ˚ q , even after player ℓ changes strategy from π ℓ ˚ to π ℓ . Thus thestrategies π k for k ‰ ℓ of the other players are not affected from the deviation of player ℓ .However, in closed loop equilibria, the trajectory of the state of the system enters the strategies, then when ℓ change π ℓ ˚ p t q to π ℓ p t q , other players is likely to be affected. Players at time t have complete information ofthe states of all the other players at time t , or in other words we allow feedback strategies. As a special case in14losed loop equilibria, a Markovian equilibrium is the admissible strategies profile π ‹ “ p π ‹ , . . . , π ℓ ‹ , . . . , π N ‹ q of the form π ℓ p s q “ φ ℓ p s, V t,xs q , (33)for each p t, x q , where φ ℓ : r , T s ˆ Ω ˆ R n ` Ñ A , V t,xs : “ p V p s q , . . . , V N p s qq t,x , and p V ℓ p s qq t ď s ď T is the uniquesolution of dV ℓ p s q V ℓ p s q “ n ÿ i “ π ℓi p s q dX Ni p s q X Ni p s q , V ℓ p t q “ v ℓt , t ď s ď T. We have the following result of Nash equilibrium strategies.
Theorem 4.1.
Nash equilibrium is attained when the strategies yield π ℓ ‹ i “ m i p t q ` X Ni p t q D i ¯ v N p t q ` τ i p t q σ ´ p t q D p i ¯ v N p t q . (34) for ℓ “ , . . . , N , where ¯ v N p t q “ log ˜ u ℓ p T ´ t, X N r ,t s , Y r ,t s q ` p ´ δδX Nt q N ř Nℓ “ V ℓ p t q v ℓ log ˜ u ℓ p T ´ t, X N r ,t s , Y r ,t s q . Thecorresponding Nash equilibrium µ N ‹ is unique in the sense of Definition 4.3 when the first exit time from thecompact set K is greater than T , i.e., τ K ą T where K “ „ , ` N ´ p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p T ´ t q ˘ N δ | ř Nℓ “ e c ℓ D m ˜ u ℓ p T ´ t q| , τ K “ inf t t ě X N p t q P K u . (35) Proof.
For a given choice of π P A , ˜ u ℓ : “ inf π P A J ℓ p π q is uniquely determined by the smallest nonnegativesolution of (28). For simplicity we denote ˜ u ℓ p T ´ t, X N r ,t s , Y r ,t s q . Assuming that all controls π k p¨q , k ‰ ℓ arechosen, player ℓ will choose the optimal strategy π ‹ that achieves V ℓ ‹ p¨q “ e c ℓ V N p t q ˜ u ℓ p T ´ t q . Suppose everyplayer ℓ follows V ℓ ‹ p¨q , we have a fixed point problem that yields V ℓ ‹ p t q “ e c ℓ ˜ u ℓ p T ´ t q δX N p t q ˆ ` p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p T ´ t q{ v ℓ N ´ p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p T ´ t q{ v ℓ ˙ , (36)equivalently,log V ℓ ‹ p t q “ log δx e c ℓ ˜ u ℓ p T q ´ p ´ δ q N ř Nℓ “ e c ℓ ˜ u ℓ p T q{ v ℓ ` ż t m s p β s ´ α s m s q ds ` ż t m t σ i p s q dW s ` log ˜ u ℓ p T ´ t q ´ log ` ´ p ´ δ q N N ÿ ℓ “ e c ℓ ˜ u ℓ p T ´ t q{ v ℓ ˘ . (37)With a fixed set of control processes π , we solve ˜ u ℓT ´ t , and expect that the π ‹ will coincide with the fixed π . Thus we can find the Nash equilibrium strategy by comparing V ℓ ‹ in (37) and V ℓ defined in (10). By Ito’sformula on ˜ u ℓ p¨q as a function of X N r ,t s and Y r ,t s , we obtain d ˜ u ℓ p T ´ t q “ p L ˜ u ℓ ´ B ˜ u ℓ Bp T ´ t q qp T ´ t q dt ` n ÿ k “ R ℓk p T ´ t, X N r ,t s , Y r ,t s q dW k p t q , where L is the infinitesimal generator of p x N , y q P p , n ˆ p , n , i.e., L ˜ u ℓ p τ q “ b p x N , y q ¨ B x ˜ u ℓ p τ q ` γ p x N , y q ¨ B y ˜ u ℓ p τ q`
12 tr “ a p x N , y q ¨ B xx ˜ u ℓ p τ q ` ψ p x N , y q ¨ B yy ˜ u ℓ p τ q ` p sτ ` τ s qp x N , y q ¨ B xy ˜ u ℓ p τ q ‰ and R ℓk p τ, x N , y q “ n ÿ i “ σ ik p x N , y q x i D i ˜ u ℓ p τ q ` n ÿ p “ τ pk p x N , y q D p ˜ u ℓ p τ q . Thus the volatility term in (37) is ż t m i p s q σ i p s q dW p s q ` ż t u ℓ p T ´ s q n ÿ k “ R ℓk p T ´ s q dW k p s q ` p ´ δ q V N p t q N δX N p t q ż t N ÿ ℓ “ e c ℓ v ℓ n ÿ k “ R ℓk p T ´ s q dW k p s q .
15y comparing the drift and volatility of (10) and (37), we arrive at (34).Next, to investigate the uniqueness of Nash equilibrium, we consider a fixed point mapping Φ : R ` Ñ R ` ofthe empirical mean of wealth m Nt from (36),Φ p m Nt q : “ δX N p t q ř Nℓ “ e c ℓ ˜ u ℓ p m Nt q{ v ℓ N ´ p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p m Nt q{ v ℓ . Denote D m as the partial derivative with respect to m Nt , then the derivative of Φ p m Nt q isΦ p m Nt q “ N δX N p t q ř Nℓ “ e c ℓ D m ˜ u ℓ p T ´ t q{ v ℓ ` N ´ p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p T ´ t q{ v ℓ ˘ . We denote A t “ N ´p ´ δ q ř Nℓ “ e c ℓ ˜ u ℓ p T ´ t q{ v ℓ . In the above derivative, D m ˜ u ℓ p T ´ t q “ ř Np “ D p ˜ u ℓ p T ´ t q φ ℓ foropen loop controls, and D m ˜ u ℓ p m Nt q “ ř Np “ D p ˜ u ℓ p T ´ t qp φ ℓ ` φ ℓ V ℓ q when π is of a closed loop form. In addition0 ă ˜ u ℓ p T ´ t q ď c ℓ ď ǫ for a positive ǫ , B φ ℓ {B V ℓ is bounded by Lipschitz coefficient L under Assumption 3.Hence | Φ p m Nt q| ă ď X N p t q ă A t N δ | ř Nℓ “ e c ℓ D m ˜ u ℓ p T ´ t q{ v ℓ | . (38)For simplicity, we set D t “ N δ | ř Nℓ “ e c ℓ D m ˜ u ℓ p T ´ t q{ v ℓ | , and K : “ r , A t D t q . By mean value theorem, Φ is acontraction of m Nt . The first exit time for the compact set K is τ K “ inf t t ě X N p t q P K u . If τ K ą T thenNash equilibrium generated by (34) is unique.By (12), log X N p t q is of the same distribution as log V m p t q . Thus, X N p t q is a log-normal distribution wherelog X N p t q ∼ N ˆ log x N ` ` m p t q β p t q ´ m p t q α p t q m p t q ˘ t, m p t q α p t q m p t q t ˙ . As a result, with the solution ˜ u of (28), the probability of attaining the unique Nash equilibrium is P p X N p t q P K q “ N ˆ log A t D t x N ´ ` m p t q β p t q ´ m p t q α p t q m p t q ˘ t m p t q α p t q m p t q t ˙ , where N is the cumulative distribution function of a standard Gaussian distribution.The end of Section 3.2 suggests that optimal strategies are linearly dependent on e c ℓ , ℓ “ , . . . , N . Toillustrate, the investors pursuing relative arbitrage end up with the terminal wealth V ℓ p T q proportional to e c ℓ if starting from a same initial wealth. However, at every time t , the information of every V ℓ p t q , ℓ “ , . . . , N is required to pinpoint the optimal strategy. Therefore, a mean field regime is discussed in the next chapter toresolve the complexity in N -player game.As a special case when investment decisions are based upon the current market environment only, we considerthe Markovian dynamics so that we write u ℓ p T ´ t, X N p t q , Y p t qq . We can obtain the optimal strategies in adifferent way. This approach will be useful when we derive the mean field equilibriums in the next section. Assumption 10.
In addtion to Assumption 6, we assume β p t q , σ p t q , γ p t q and τ p t q take values in R n ` ˆ R n ` ,and are Markovian, i.e., X Ni p t q β i p t q “ b i p X N p t q , Y p t qq , X Ni p t q σ ik p t q “ s ik p X N p t q , Y p t qq , n ÿ k “ s ik p t q s jk p t q “ a ij p X N p t q , Y p t qq ,γ i p t q “ γ i p X N p t q , Y p t qq , τ i p t q “ τ i p X N p t q , Y p t qq . Proposition 4.1.
Under Assumption 10, when controls of a closed loop Markovian form (33) , or an open loop φ p t, v , W t q are adopted, there is a Nash equilibrium π ‹ “ p π ‹ , . . . , π N ‹ q , where for ℓ “ , . . . , N , π ℓ ‹ follows π ℓ ‹ p t q “ m i p t q ` X Ni p t q D i ˜ v N p t q ` τ i p t q σ ´ p t q D p i ˜ v N p t q , here ˜ v N p t q “ log ˜ u ℓ p T ´ t, x N , y q ` ´ ´ δδX N p t q ¯ N N ÿ ℓ “ V ℓ p t q v ℓ log ˜ u ℓ p T ´ t, x N , y q , (39) and ˜ u ℓ p t q is the smallest nonnegative solution in (28) .Proof. The Markov property of V N p¨q gives E P “ V N p T q L p T q| F p t q ‰ V N p t q L p t q “ E P “ V N p T ´ t q L p T ´ t q ‰ V N p t q L p t q “ ˜ u ℓ p T ´ t, X N p t q , Y p t qq , where ˜ u ℓ p¨q is the minimal nonnegative solution of (28). Again we use the property for 0 ď t ď T that V ℓ p t q “ V N p t q ˜ u ℓ p T ´ t, X N p t q , Y p t qq , the deflated wealth processˆ V ℓ p t q : “ V ℓ p t q L p t q “ E P “ V N p T q L p T q| F t ‰ is a martingale. As a result, the dt terms in d ˆ V ℓ p t q will vanish, namely,ˆ V ℓ p t q “ ˆ V ℓ p q ` n ÿ k “ ż t ˆ V ℓ p s q B k p T ´ s, X p s q , Y p s qq dW k p s q , ď t ď T, (40)where B k p t, x, π q “ n ÿ i “ σ ik p x N , y q x i D i log ˜ u ℓ p T ´ t, x N , y q ` n ÿ m “ τ mk p x N , y q D m log ˜ u ℓ p T ´ t, x N , y q` n ÿ i “ δX N p t q V N p t q ˆ x i ř ni “ x i σ ik p t q ´ Θ k p x N , y q ˙ ` p ´ δ q{ N V N p t q n ÿ i “ N ÿ ℓ “ ˆ V ℓ p t q v ℓ π ℓi σ ik p t q ´ V ℓ p t q v ℓ Θ k p x N , y q ˙ . Thus we have the fixed point problem π ℓ ‹ i p t q “ X Ni p t q D i log ˜ u ℓ p T ´ t, x N , y q ` τ i p x N , y q σ ´ p x N , y q D k log ˜ u ℓ p T ´ t, x N , y q` δX N p t q V N p t q m i p t q ` p ´ δ q N V N p t q N ÿ ℓ “ V ℓ ‹ p t q v ℓ π ℓ ‹ i p t q , (41)where V ℓ ‹ p t q is generated from π ℓ ‹ p t q .Next, we check the consistency condition of π ‹ in (41) and π we start with. Define a map ρ : A Ñ A , wewant to find a fixed point so that ρ p π q “ π . By Brouwer’s fixed-point theorem, since A is a compact convex set,there exists a fixed point for the mapping ρ . In Nash equilibrium, we assume that all players follow the strategy π ‹ - if we multiply both sides by V ℓ and then summing over ℓ “ , . . . , n in (41), and after some computationswe conclude π ℓ ‹ i “ m i p t q ` X i p t q D i ˜ v N p t q ` τ i p t q σ ´ p t q D k ˜ v N p t q , (42)where ˜ v N p t q satisfies (39). I is the identity matrix of size n , and is a n -dimension column of ones. Now we want to show M containsstrong arbitrage opportunities relative to the performance benchmark, at least for sufficiently large real numbers T ą
0. We illustrate this path by example 4.1. We employ the idea of functional generated portfolios [9] to seekoptimal strategies. By doing so, we may reduce the intractability of the N -player game problem.The market portfolio follows the dynamic d m i p t q “ m i p t q „ γ mi dt ` n ÿ k “ τ mik p t q dW k p t q , i “ , . . . , n. (43)Here τ m p t q is the matrix with entries τ mik p t q : “ σ ik p t q ´ ř nj “ m j p t q σ jk p t q , e i is the i th unit vector in R n andthe vector γ m p t q is with the entries γ mi p t q : “ p e i ´ m p t qq p β p t q ´ α p t q m p t qq .17 heorem 4.2. Let G , G : U Ñ p , be positive C functions defined on a neighborhood U of A such thatfor all i , x i D i log G p x q , x i D i log G p x q are bounded on A . For t P r , T s , G , G generate the portfolio π ℓ p t q “ r G p t q ` r G p t q ` R p t q (44) where r G p t q “ p D i log G p m p t qq m i p t qq n p I ´ m p t qq ; r G p t q “ D log G p Y p t qq τ p t q σ ´ p t q ; R p t q “ δX Ni p t q ` p ´ δ q Y p t q V N p t q . The process d log V ℓ p t q e c ℓ V N p t q “ d log G p Y p t qq ` d log G p m p t qq ` d Ξ t , t P r , T s , a.s. (45) is with a drift process Ξ p¨q such that a.s., for t P r , T s , d Ξ p t q dt “ r G p t q α p t q m p t q ` r G p t q α p t q π ℓ p t q ´ ˆ || r G p t q σ || ` || r G p t q σ || ´ || π ℓ σ || ˙ ` G p m p t qq n ÿ i,j “ D ij G p m p t qq m i p t q m j p t qp n ÿ k “ τ mik p t q τ mjk p t qq ` G p Y p t qq n ÿ i,j “ D ij G p Y p t qq ψ ij p t q , More importantly, the notion of optimal strategies (42) can be treated through Theorem 4.2. Let G , G : “ U Ñ p , be positive C functions defined on a neighborhood U of A such that for all i , x i D i log G p x q , x i D i log G p x q are bounded on A . We write ˜ u ℓ p t, x , y q “ w ℓ p t, p m i q i “ ,...,n , p Y i q i “ ,...,n q , then by taking deriva-tives of X N p t q , Y p t q , it follows X Ni p t q D i log ˜ u ℓ p T ´ t, X N p t q , Y p t qq “ „ D i log G p m p t qq ´ n ÿ i “ D i log G p m p t qqq m i p t q m i p t q ; τ i p x N , y q σ ´ p x N , y q D p i log ˜ u ℓ p T ´ t, X N p t q , Y p t qq “ p D i log G p Y p t qqq n τ p t q σ ´ p t q e i . Furthermore, we can use portfolio generating functions to find conditions on investment strategies by ř ni “ π i p t q “ t P r , T s . We get 1 ´ δN δX N p t q N ÿ ℓ “ V ℓt w ℓt “ w ℓt , where w ℓt : “ X i p t q D i log ˜ u ℓ p t q ` τ i p t q σ ´ p t q D p i log ˜ u ℓ p t q . Hence ř Nℓ “ V ℓ p t q w ℓ p t q “ δX N p t q “ p ´ δ q N ř Nℓ “ V ℓ p t q . The latter indicates that the market is exactly consisted of the N investors we considered. If w ℓ p t q “
0, then every investor is the same, and their strategy follows the market portfolio. If w ℓ p t q ‰
0, then r G p t q “
0, and n ÿ j “ n ÿ i “ D i log G p t qp τ p t q σ ´ p t qq ji “ . (46) Example 4.1.
Suppose that M is nondegenerate, weakly diverse in [0,T], and has bounded variance, see Ap-pendix A for the definitions. We assume for t P r , T s , there exists constants c , N c , M π ą such that p V ℓ p t q{ p V ℓ p q ě c X N p t q L p t q ; Y i p t q{ m i p t q ď N c , i “ , . . . , n ; | n ÿ i “ γ i p t q| ď M π . Consider the function G and G are defined by G p x q “ n ź i “ x i , G p x q “ ´ n ÿ i “ x i . and G generate the portfolio π ℓi p t q “ ´ ´ n ` δX N p t q V N p t q ¯ m i p t q ` p ´ δ q Y i p t q V N p t q ` ˆ ´ Y p t q G p Y p t qq ˙ τ p t q σ ´ p t q e i , i “ , . . . , n. (47) Then π ℓ strictly dominates V N p t q in (13) if T ě nN ´ n ´ ´ ǫn ` M π n ´ n M ´ N n ` M n ´ n p n ´ q ` N ´ N λ p τ q λ p σ ´ q ¯ . The notations of constants and details of the proof can be found in Appendix C.
We have observed that it is unlikely to get a tractable equilibrium from N -player game, especially when N islarge. In this section, we study relative arbitrage for the infinite limit population of players. With propagationof chaos results provided, a player in a large game limit should feel the presence of other players through thestatistical distribution of states and actions. Then they make decisions through a modified objective involvesmean field as N Ñ 8 . For this reason, we expect MFG framework to be more tractable than N -player games. We formulate the model on p Ω , F , F “ p F t q t Pr ,T s , P q which support Brownian motion B , a n -dimensionalcommon noise B , equally distributed as W . The systemic effect of random noises towards the market might bedifferent when we consider a finite or infinite group of investors interacting with the market. B is adapted to the P -augmentation of F and can explain the limit random noises in the market M when N Ñ 8 . The admissiblestrategies π p¨q P A MF follow similiar conditions as (6) and is F B -progressively measurable.In general, the stock prices and state processes depend on the joint distribution of p V ℓ , π ℓi q , ℓ “ , . . . , N ,while the cost function is related to the empirical distribution of the private states. With a given initial condition µ P P p C pr , T s ; R ` qq as a degenerate distribution of value 1, we define the conditional law of V p t q{ v given F B as µ t : “ Law p V p t q v | F Bt q , (48)and the conditional law of p V p t q , π p t qq given F B , with a given initial condition ν P P p C pr , T s ; R ` ˆ A qq , is ν t : “ Law p V p t q , π p t q| F Bt q . Assumption 11.
Assume x P L p Ω , F , P ; R n ` q , and E r sup ď t ď T ||p V p t q , X p t qq|| s ď 8 . Under Assumption 11, the mean field game model contains McKean-Vlasov SDEs of stock prices and wealth d X p t q “ X p t q β p X p t q , ν t q dt ` X p t q σ p X p t q , ν t q dB t , X “ x ; dV p t q{ V p t q “ n ÿ i “ π i p t q dX i p t q{ X i p t q . (49)From Proposition D.1-D.3, we show that the above McKean-Vlasov problem admits a unique solution,where ν t : “ Law p V p t q , π p t q| F Bt q . Furthermore, the weak limit of ν N in Definition 2.2 is exactly ν t , V ℓ p t q isasymptotically identical independent copies given the common noise B when ℓ “ , . . . , N , N Ñ 8 . Hence weconsider a representative player which is randomly selected from the infinite number of investors in mean fieldset-up. Small deviations of a single player would not influence the entire system given the common noise B .The player competes with the market and the entire group with respect to the benchmark V p T q “ δ ¨ X p T q ` p ´ δ q ¨ µ T , and they try to minimize the relative amount of initial capital. The objective is J µ,ν p π ; 0 , x q : “ inf " ω ą ˇˇ V ωe c V p q ,π p T q ě e c V p T q * . (50)19 .2 Mean Field Equilibrium Specifically, if the mean field interaction is through the expected investments of an investor on assets - theconditional expectation of the product of wealth and controls, a representative player’s wealth is dZ t “ d E p V p t q π p t q| F Bt q “ γ p X p t q , Z t q dt ` τ p X p t q , Z t q dB t , Z “ z , (51)with dX i p t q “ X i p t q β p X p t q , Z t qq dt ` X i p t q σ p X p t q , Z t q dB t , X i p q “ x i, . Mean field equilibrium appears as a fixed point of best response function.
Definition 5.1. (Mean Field Equilibrium) Let π ‹ p¨q P A MF be an admissible strategy, then it gives mean fieldequilibrium (MFE) if J µ,ν in (50) satisfies J µ,ν p π ‹ q “ inf π P A MF J µ,ν p π q . In particular, ˆ A “ arg inf π P A J µ,ν p π q denotes the set of optimal controls. In the control problem, the flowof measure p m T , Z p T qq is frozen conditional on the common noise. p m T , Z p T qq is an equilibrium if there exists π ‹ P ˆ A such that the fixed point of the mean field measure exists, i.e., m T “ E r V ‹ T | F BT s ; Z p T q “ E r Z ‹ T | F BT s . Definition 5.2.
We say that uniqueness holds for the MFG equilibrium if any two solutions µ a , µ b , defined onfiltered probabilistic set-ups p Ω , F , F , P q , with the same initial law µ P P p R ` q , P r µ a “ µ b s “ , where µ is the distribution of wealth processes as in (48) . When F “ F X,Z “ F B , the representative agent’s optimal initial proportion to achieve relative arbitrage canbe characterized as u p T q : “ inf π P A MF J µ,ν p π q “ E r e c V p T q L p T qs{ V p q , (52)and it solves a single Cauchy problem as opposed to the N -dimensional PDEs system in N -player game, B ˜ u p τ, x , z , m qB τ ě A ˜ u p τ, x , z , m q , ˜ u p , x , z , m q “ e c , (53)where A ˜ u p τ, x , z , m q “ n ÿ i “ n ÿ j “ a ij p x , z q ˆ D ij ˜ u p τ, x , z , m q ` δD i ˜ u p τ, x , z , m q V p q ˙ ` n ÿ p “ n ÿ q “ ψ pq p x , z q ˆ D pq ˜ u p τ, x , z , m q ˙ ` L m ˜ u p τ, x , z , m q , for τ P p , , p x N , y q P p , n ˆ p , n .Note that Z ‹ “ E r V ‹ π ‹ | F B s is not expected to be unique. Moreover, since the diffusion process of Z p T q is given by Definition 2.1(3) and (51), we consider the fixed point over the control space when it comes to Z p T q “ E r Z ‹ T | F BT s . The steps of searching equilibrium for extended mean field game with joint measure ofstate and control is formulated in [4]. The paper [6] manifests an example of extended mean field games withapplication in price anarchy. They use two different measures as law of the state processes and the law of control.The equilibrium approaching steps we introduce is different in that a modified version of extended mean fieldgame is discussed, where the state processes and cost functional depend on different measures, and uniquenessof Nash equilibrium is specified here. In the following, we show the steps to attain a unique equilibrium in openloop or closed loop Markovian form. Steps of Solving Mean Field Game (i) Start with a fixed φ such that π “ p π p t qq ď t ď T “ φ p v , B r ,T s q or φ p V p t qq , the open loop and feedbackfunction respectively, and solve dV p t q “ π p t q β p X p t q , Z t q dt ` π p t q σ p X p t q , Z t q dB t , V p q “ ˜ u p T q V p q : “ v , X i p t q “ X i p t q β i p X p t q , Z t q dt ` X i p t q n ÿ k “ σ ik p X p t q , Z t q dB k p t q , i “ , . . . , n, where Z t “ E r V p t q π p t q| F Bt s for 0 ď t ď T .(ii) For each arbitrary stochastic process m “ p m t q ď t ď T on R ` adapted to the filtration generated by therandom measure B , solveinf π P A MF J m,Z p π q “ u p T q “ E “ e c ` δX p T q ` p ´ δ q m t ˘ L p T q ‰ { V p q , using X p T q from step (i). The corresponding φ ‹ : “ arg inf π P A MF J µ,ν p π q “ arg inf π P A MF J m,Z p π q . Definethe mapping φ ‹ “ Φ p φ q .(iii) If there exists ˆ φ such that ˆ φ ‹ “ Φ p ˆ φ q , find m so that for all 0 ď t ď T , m t “ E r V ‹ t | F Bt s , where V ‹ is theoptimal path with φ ‹ as a minimizer of J m,Z p φ q .Here the fixed point is formulated on both the control space and the flows of measure. Theorem 5.1.
Under Assumption 2,3, 10 and 11, there exists a unique Mean Field Equilibrium µ ‹ .The corre-sponding Nash equilibrium µ ‹ is unique in the sense of Definition 4.3 when the first exit time from the compactset ˜ K is greater than T , i.e., ˜ τ K ą T where ˜ K “ „ , ` ´ p ´ δ q E r e c ˜ u p T ´ t q| F Bt s ˘ δ ˇˇ E r e c D m ˜ u p T ´ t q| F Bt s ˇˇ , ˜ τ K “ inf t t ě X p t q P ˜ K u , (54) Proof.
For simplicity we denote ˜ u p T ´ t, x , z , m q “ ˜ u p T ´ t q . We fix the process m solve the optimal controlproblem for V ‹ . Suppose every player follows V ‹ p t q “ V ‹ p t q ˜ u p T ´ t q , we solve a fixed point problem whichyields V ‹ p t q “ e c δX ‹ t ˜ u T ´ t ´ p ´ δ q E r e c ˜ u p T ´ t q{ v | F Bt s . (55)As in Theorem 4.1, after comparing log V ‹ p t q in (49) and (55), this yields π ‹ i p t q “ m i p t q ` X i p t q D i ˜ v p t q ` τ i p t q σ ´ p t q D k ˜ v p t q , where ˜ v p t q “ log ˜ u T ´ t ` ´ δδX t E r V p t q v log ˜ u T ´ t | F Bt s , and ˜ u T ´ t is the smallest nonnegative solution in (53).We can further derive the expression of π ‹ when ˜ u is Markovian. We restrict m t in the form of E p V | F Bt q ,for each i . From now on, we use vol to represent the volatility of a process, as we are not given the explicitform of m t . By Ito’s formula we haveˆ V p t q “ ˆ V p q ` n ÿ k “ ż t ˆ V p s q B k p T ´ s, X p s q , Z p s qq dW k p s q , ď t ď T, (56)where B k p τ, x, z q “ n ÿ i “ σ ik p x , z q x i D i log ˜ u ℓ p τ, x , z , m q ` τ m p x , z q D m log ˜ u ℓ p τ, x , z , m q` n ÿ i “ δX p t q{ x V p t q ˆ x i ř ni “ x i σ ik p t q ´ Θ k p x , z q ˙ ` p ´ δ q V p t q vol p dL t m t q . By comparing (16) and (56), the strategy used for V ‹ should be π ‹ i p t q “ X ‹ i p t q D i log ˜ u p T ´ t q ` τ i p x , z q σ ´ p x , z q D k log ˜ u p T ´ t q` δX ‹ p t q V ‹ p t q m i p t q ` p ´ δ q V ‹ p t q vol p dL t m t q σ ´ . (57)The derivation of π ‹ ensures that it generates a wealth process V ‹ , thus π ‹ P A MF .21ext, we show the equilibrium is unique. Denote Φ p m t q : “ E r V p t q| B s , it is equivalent to show that there isthe unique fixed point mapping Φ p m t q “ m t . We have m t “ Φ p m t q “ δX ‹ p t q E r e c ˜ u p T ´ t, m q{ v | F Bt s ´ p ´ δ q E r e c ˜ u p T ´ t, m q{ v | F Bt s , Φ p m t q “ δX ‹ p t q E r e c D m ˜ u p T ´ t, m q{ v | F Bt s ` ´ p ´ δ q E r e c ˜ u p T ´ t, m q{ v | F Bt s ˘ , First, Φ : R ` Ñ R ` is a continuous function, since from Appendix D we have E r|| V ν ´ V p t q ν || t | F Bt s ď p t ` q L E r ż t W p ν , ν q dr s , (58)for any fixed m t P R ` , Φ p m t q “ E r V p t q| F Bt s P R ` . Furthermore, we set ˜ A t “ ´ p ´ δ q E r e c ˜ u p T ´ t, m q{ v | F Bt s ,˜ D t “ δ ˇˇ E r e c D m ˜ u p T ´ t, m q{ v | F Bt s ˇˇ . By mean value theorem, Φ is a contraction of m t if ˜ τ K ą T , where˜ τ K “ inf t t ě X p t q P ˜ K u , ˜ K : “ r , ˜ A t ˜ D t q . As a result, the mean field equilibrium generated by (57) is uniquewhen the first exit time from ˜ K is less than T . The probability of attaining the unique mean field equilibriumis P p X p t q P ˜ K q “ N ˆ log A t D t x ´ ` m p t q β p t q ´ m p t q α p t q m p t q ˘ t m p t q α p t q m p t q t ˙ , where N is the cumulative distribution function of a standard Gaussian distribution, x is initial value of totalcapitalization X p¨q .It is clear from (57) that the mean field strategy actually depends on p X p t q , Z p t q , m t q , which means theoptimal strategies are driven by stock prices, trading volumes and relative arbitrage benchmark. Similiarly to N -player game, π is independent of preference c , meaning that the representative player’s preference level c isnot a crucial factor when determining strategies. Next, we encompass a simplified class of market models to shed some light on mean field regimes, where themodels exhibit selected characteristics of real equity markets and provide a tractable mean field equilibrium.The class of models is inspired by the volatility-stabilized markets introduced in [10].
Example 5.1.
In real markets, the smaller stocks tend to have greater volatility than the larger stocks. Mean-while, the higher the trading volume of a stock, the larger the volatility of the trading. The parameters β , σ , γ , τ in M are set to the following specific forms which agree with these market behaviors. For ď i, j ď n , withinfinite number of investors, β i p t q “ p ` ζ q Z i p t q m i p t q , a ij “ X i p t q δ ij ; γ i p t q “ β i p t q ; ψ ij p t q “ Z i p t q δ ij . We can check that the Fichera drift f i p¨q ă . Similiarly to Proposition 3.3, we can get ˜ u p¨q ă .By Theorem 5.1, the optimal strategy π ℓ ‹ i of investor ℓ in a mean field game is π ‹ i p t q “ X i p t q D i log ˜ u ‹ T ´ t ` τ i p t q σ ´ i p t q D p i log ˜ u ‹ T ´ t ` vol p m t q σ ´ i p t q D m log ˜ u ‹ T ´ t ` δX p t q m i p t q δX p t q ` p ´ δ q m t ` vol p dL t m t q σ ´ p t q ´ δδX p t q ` p ´ δ q m t . We denote p t as the conditional density of V p t q given B t , which follows dp t “ ´p ` ζ qB v “ V p t q n ÿ i “ π i p t q z i p t q m i p t q p t ‰ dt ´ V p t q n ÿ i “ π i p t qp X i p t qq B v p t dB t . Next, plug π ‹ i p t q into the equation of p t , and let m t “ ş vp t p v q dv , i.e., the consistency condition, we get m t “ ´p ` ζ q ż m t ` X ´ i p t q z ´ i p t qB v vol p m t q m t ´ X ´ i p t q z ´ i p t q m t vol p m t qB v m t ´ δ p ´ δ q X p t q z i p t qp δX p t q ` p ´ δ q m t q B v m t ˘ dvdt ´ ż m t ` B v vol p m t q m t ´ m t vol p m t qB v m t ´ δ p ´ δ q X i p t q z i p t qp δX p t q ` p ´ δ q m t q B v m t ˘ dvdB t ´ ż m t L t ` p ´ δ qp Θ t B v m t ` B v vol p m t qq δX p t q ` p ´ δ q m t ` p ´ δ q p Θ t m t ` vol p m t qqp δX p t q ` p ´ δ q m t q B v m t ˘ dvdt, where vol p m t q “ ż δ p ´ δ q X i p t q z i p t qp δX p t q ` p ´ δ q m t q B v m t dv „ ş m t B v m t dv p ´ ş m t B v m t dv q ` . With the explicit m t expression, we can obtain closed form solution of π ‹ p t q in terms of X p t q , Z p t q , ˜ u T ´ t .If δ “ , meaning a investor intend to achieve relative arbitrage with respect to peer competitors, and B ˜ u p τ, x , z qB τ “ ÿ i “ X i p τ q D ii ˜ u p x, z q ` ÿ p “ Z p p τ q D pp ˜ u p x, z q . (59) We separate the variables τ , x , x , z , and z . We denote S k p x q as the function for x , x , z , and z ,when k “ , , , , repsectively. T p τ q “ e c ` ξτ . A general solution can be found as S k p x q “ x k r c J p p´ C k x k q q ` c Y p p´ C k x k q qs , where ř k “ C k “ ξ , J and Y are order 1 Bessel function of first and second kind, respectively. It concludes ˜ u “ e c ` ξτ ś k “ S k p x k q , which is the smallest amount of initial capital proportion that a generic investor needto outperform the others, given it is the minimal nonnegative solution of (59) . Thus we can get an explicitequilibrium if we have the information on X p t q and Z p t q and initial condition x , z . A numerical scheme forthe relative arbitrage problem will be considered elsewhere. N -player game and mean field game In this last section we justify if mean field game is an appropriate generalization of N -player relative arbitrageproblem. We conclude in the following proposition that the MFE we obtain agrees with the limit of the finite equilibrium,and that the optimal arbitrage in the sense of (19) strongly converges to optimal arbitrage in the mean fieldgame setting (52).
Proposition 6.1.
Suppose p β, σ, γ, τ qp t, x , z q take values in R n ˆ GL p n qˆ R n ˆ GL p n q is bounded and continuousin every variable. If π p¨q is Markovian, and min t τ K , ˜ τ K u ą T , then u p T q “ lim N Ñ8 u ℓ p T q a.s, for T P p , .Proof. It follows from Appendix D, P ˝ p X N , V , ν N , W q is tight on the space C pr , T s ; R n ` q ˆ C pr , T s ; R N ` q ˆ P p C pr , T s ; R ` ˆ A qq ˆ C pr , T s ; R n q and the weak limit exists. We proved in Proposition D.2 that the equilib-rium µ is the weak limit of µ N conditional on B . What left here to show is that the optimal cost in finite gameconverges to the mean field optimal cost, since ˜ u and π are both bounded. By using the Markovian property of π p¨q , b p¨q and σ p¨q , we would have ˜ u ℓ p T ´ t q : “ E P “ e c ℓ V N p T ´ t q L p T ´ t q ‰ V N p t q L p t q . Then by the bounded convergence theorem and Proposition D.2, the deflator L p X p t q , Z p t qq “ lim N Ñ8 L p X N p t q , Y p t qq a.s., and V p T q “ lim N Ñ8 V N p T q in the weak sense. c ℓ is i.i.d samples from Law p c q .23herefore as N Ñ 8 , u ℓ p T q : “ inf π P A J ℓ p π ℓ ‹ q Ñ inf π P A MF J µ,ν p π ‹ q “ u p T q almost surely, and u p T ´ t q is the weak limit of u ℓ p T ´ t q when t ą We show here that MFE can be used to construct an approximate Nash equilibrium for the N -player game.Since we derive strong equilibrium in both N -player and mean field game, µ N and µ are measurable with respectto the information generated by W and B , respectively.From (57), the mean field control in general is of the form π ‹ i p t q “ φ p X i p t q , X p t q , µ t , ν t , ˜ u T ´ t q . (60) Definition 6.1.
For ǫ N ě , an open-loop ǫ N -equilibrium is a tuple of admissible controls φ N : “ p φ N, p t q , . . . , φ N,N p t qq ď t ď T , φ N,ℓ p t q P A Ă ∆ n , for every ℓ , such that J ℓ p φ N q ď inf p P A J ℓ p p, φ N, ´ ℓ q ` ǫ N , where p P A is an open loop control, and φ is of the form in (60) . An closed-loop ǫ N -equilibrium is a tuple φ N such that J ℓ p φ N q ď inf p P A J ℓ p p N q ` ǫ N , where each component in φ N is defined in (60) ; p N : “ p p p U r ,t s q , φ N, ´ ℓ p U r ,t s qq , in which U t is the N -vectorof wealth processes generated by this strategy, p : r , T s ˆ C pr , T s ; R N ` q Ñ A is of the form p p p t, U r ,t s qq ď t ď T , φ N, ´ ℓ is defined in (60) . For any ℓ “ , . . . , N , φ N,ℓ and p are F -progressively measurable functions. Proposition 6.2.
Under Assumption 2, 3, and 12, assume ˜ u T ´ t is Lipschitz in p X p t q , µ t , ν t q , there exists asequence of positive real numbers p ǫ N q N ě converging to 0, such that any admissible strategy π ℓ “ p π ℓt q t Pr ,T s for the first player J N,ℓ p p , π ‹ , . . . , π N, ‹ q ě J ´ ǫ N , ℓ “ , . . . , N. Proof.
We look into the approximate open and closed loop Nash equilibrium. Without loss of generality, by thesymmetry of the game, we focus on player 1. For a fixed number of players N , each player utilizes the optimalstrategy π ‹ from the associated mean field game, i.e., the strategy set is π N : “ p π ‹ t , . . . , π ‹ t q as in (60). Therest part of the proof is mainly adapted to the general method on [4]. We articulate the different part fromgeneral method: when π N deviates to p p, π ´ q , the state processes are V p t q and V ℓ p t q , ℓ ‰
1, and the empiricalmeasures are µ Nt “ N ˆ δ V p t q ` N ÿ ℓ “ δ V ‹ ℓ p t q ˙ , ν Nt “ N ˆ δ p V p t q ,p q ` N ÿ ℓ “ δ p V ‹ ℓ p t q ,π ‹ q ˙ . We can show p µ Nt , ν Nt q Ñ p µ t , ν t q in open loop, and p U t , µ Nt , ν Nt q Ñ p V pt , µ t , ν t q in closed loop in the similiarvein of Proposition D.1 and D.2. µ N d “ µ , x N d “ x , By Ito’s isometry and L convergence we getlim N Ñ8 J N pp p, π ´ q ; 0 , x N q “ J µ,ν p p ; 0 , x q . eferences [1] E. Bayraktar, Y.-J. Huang, Q. Song, Outperforming the market portfolio with a given probability . Ann.Appl. Probab. 22(4), 1465-1494, 2012.[2] P. Billingsley,
Convergence of Probability Measures . New York, NY: John Wiley & Sons, Inc. (1999) ISBN0-471-19745-9.[3] P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions,
The master equation and the convergenceproblem in mean field games . arXiv preprint arXiv:1509.02505, 2015.[4] R. Carmona, F. Delarue,
Probabilistic Theory of Mean Field Games with Applications I-II: Mean FieldGames with Common Noise and Master Equations . Volume 84 of Probability Theory and StochasticModelling, Springer, 2018.[5] R. Carmona, F. Delarue, D. Lacker,
Mean field games with common noise . The Annals of Probability 44(6), 3740-3803[6] R. Carmona, C. Graves, and Z. Tan,
Price of Anarchy for Mean Field Games . ESAIM: Proceedings andSurveys, 65:349-383, 2019[7] D. Fernholz, I. Karatzas,
On Optimal Arbitrage . Ann. Appl. Probab. 20 1179-1204.[8] D. Fernholz, I. Karatzas,
Optimal Arbitrage under model uncertainty . Ann. Appl. Probab, 2011, Vol. 21,No.6, 2191-2225.[9] R. Fernholz,
Stochastic Portfolio Theory, volume 48 of Applications of Mathematics (New York) . Springer-Verlag, New York, 2002. Stochastic Modelling and Applied Probability.[10] R. Fernholz, I. Karatzas,
Relative arbitrage in volatility-stabilized markets . Ann. Finance 1, 149-177 (2005).[11] R. Fernholz, I. Karatzas, C. Kardaras,
Diversity and relative arbitrage in equity market . Finance & Stochas-tics 9, 1-27 (2005).[12] R. Fernholz, I. Karatzas, J. Ruf,
Volatility and arbitrage . Ann. Appl. Probab. 28 (1) (2018) 378-417.[13] H. F¨ollmer,
The exit measure of a supermartingale . Zeitschrift fu¨r Wahrscheinlichkeitstheorie und ver-wandte Gebiete 21, 154-166 (1972).[14] A. Friedman,
Stochastic Differential Equations and Applications . Vol. I, Vol. 28 of Probability and Math-ematical Statistics, Academic Press, New York (1975).[15] O. Gu´eant, J.M. Lasry, and P.L. Lions,
Mean field games and applications . In R. Carmona et al., editor,Paris Princeton Lectures in Mathematical Finance IV, volume 2003 of Lecture Notes in Mathematics.Springer Verlag, 2010.[16] M. Huang, R.P. Malham´e, P.E. Caines,
Large Population Stochastic Dynamic Games: Closed LoopMcKean-Vlasov systems and the Nash certainty equivalence principle . Communications in Informationand Systems 6 (2006), no. 3, 221-252.[17] N. Ikeda, S. Watanabe,
Stochastic Differential Equations and Diffusion Processes . 2nd ed. North-Holland,Amsterdam (1989).[18] D. Lacker,
A general characterization of the mean field limit for stochastic differential games . ProbabilityTheory and Related Fields 165 (2016), no. 3, 581-648.[19] D. Lacker,
Limit theory for controlled McKean-Vlasov dynamics . SIAM J. Control Optim., 55 (2017), pp.1641-1672, https://doi.org/10.1137/16M1095895.[20] D. Lacker and T. Zariphopoulou,
Mean field and n -agent games for optimal investment under relativeperformance criteria . Math. Finance 29 (4) (2019) 1003-1038.2521] J.M. Lasry, and P.L. Lions, Mean field games . Japanese Journal of Mathematics 2 (2007), 229-260.[22] D. Majerek, and W. Nowak, W. Ziba,
Conditional strong law of large numbers . International Journal ofPure and Applied Mathematics, 20(2), January 2005[23] B. K. Øksendal,
Stochastic Differential Equations: An Introduction with Applications . Springer, Berlin.(2010). 6th edition. ISBN 9783642143946.[24] S. Pal, T.K.L. Wong,
The geometry of relative arbitrage
Math. Financ. Econ. 10, 263-293 (2016).[25] K. R. Parthasarathy,
Probability Measures on Metric Spaces
Ann. Math. Statist. 40 (1969), no. 1, 328.doi:10.1214/aoms/1177697834.[26] J. Ruf,
Optimal Trading Strategies Under Arbitrage . PhD thesis, Columbia University, New York, USA(2011).[27] J. Ruf,
Hedging under Arbitrage . Math. Financ. 23, 297-317 (2013)[28] D. W. Stroock, S. R. S. Varadhan,
Multidimensional Diffusion Processes . Classics in Mathematics,Springer - Verlag, Berlin, 2006.[29] T.K.L. Wong,
Information geometry in portfolio theory
In Frank Nielsen (Ed.), Geometric Structures ofInformation, Springer (2019).
Appendices
A Market dynamics and conditions
This section recalls some properties of the market which are used to show the existence of relative arbitrage.
Definition A.1 (Non-degeneracy and bounded variance) . A market is a family M “ t X , . . . , X n u of n stocks,each of which is defined as in (3) , such that the matrix α p t q is nonsingular for every t P r , , a.s. The market M is called nondegenerate if there exists a number ǫ ą such that for x P R n P p xα p t q x T ě ǫ || x || , @ t P r , , The market M has bounded variance from above, if there exists a number M ą such that for x P R n P p xα p t q x T ď M || x || , @ t P r , , Remark 2.
Let π be a portfolio in a nondegenerate market. Then there exists an ǫ ą such that for i “ , . . . , n,τ πii p t q ě ǫ p ´ π max p t qq , @ t P r , (61) almost surely. Indeed, this is directly from definition A.1, and τ πii p t q “ α ii p t q ´ α iπ p t q ` α ππ p t q , where α ππ p t q “ π p t q α p t q π p t q . Details of the proof can be found in [9]. Intuitively, no single company can ever be allowed to dominate the entire market in terms of relative capi-talization.
Definition A.2 (Diversity of market) . The model M of (3) , (4) is diverse on the time-horizon r , T s , with T ą a given real number, if there exists a number η P p , q such that max ď i ď n m i : “ m p q ă ´ η, @ ď t ď T (62) almost surely and M is weakly diverse if there exists a number η P p , q such that T ż T m p q p t q dt ă ´ η, @ ď t ď T (63) almost surely. roof of Proposition 3.1. For (2), Since everyone follows V ℓ p T q v ℓ ě e cℓ v ℓ V N p T q , we sum up this expression for ℓ “ , . . . , N to get an inequality of ř Nk “ V k p t q{ N . (17) follows immediately. Next, (3) can be easily derivedfrom Definition 3.1 that c ℓ ď log ˆ V ℓ p T q δX N p t q ` p ´ δ q N ř Nℓ “ V ℓ p T q v ℓ ˙ . B Relative arbitrage and Cauchy problem
Proof of Proposition 3.2.
From Ito’s formula, discounted process p V ℓ p¨q admits d p V ℓ p t q “ p V ℓ p t q π ℓ p t q ` β p t q ´ σ p t q Θ p t q ˘ dt ` p V ℓ p t q ` π ℓ p t q σ p t q ´ Θ p t q ˘ dW p t q ; p V ℓ p q “ p v ℓ , and p V ℓ p¨q is a supermartingale. For this reason, we get from (19) that for an arbitrary ω ℓ , ω ℓ V N p q ě E “ p V ℓ ‰ ě E „ p X p T q δe c ℓ ` L p T qp ´ δ q e c ℓ N N ÿ ℓ “ V ℓ p T q v ℓ : “ p ℓ . Hence, u ℓ p T q ě p ℓ .To prove the opposite direction u ℓ p T q ď p ℓ , we use martingale representation theorem (Theorem 4.3.4, [23])to find U ℓ p t q : “ E “ e c ℓ V N p T q L p T q| F t ‰ “ ż t ˜ p p s q dW s ` p ℓ , ď t ď T, (64)where ˜ p : r , T s ˆ Ω Ñ R k is F -progressively measurable and almost surely square integrable. Next, construct awealth process V ˚ p¨q “ U ℓ p¨q{ L p¨q , it satisfies V ˚ p q “ p ℓ , V ˚ p T q “ e c ℓ V N p T q . If we plug a trading strategy h ˚ p¨q “ L p¨q V ℓ p¨q α ´ p¨q σ p¨qr ˜ p p¨q ` U ℓ p¨q Θ p¨qs , into (16), further calculations imply V ˚ p¨q ” V p,h ˚ p¨q ě V p,h ˚ p¨q is the wealth process from h ˚ p¨q .Therefore, h ˚ p¨q P A with exact replication property V p,h ˚ p T q “ e c ℓ V N p T q a.s. Consequently, p ℓ ě u ℓ p T q for p ℓ V N p q P ω ą |D π ℓ P A , given π ´ ℓ p¨q P A N ´ , s.t. V ω V N p q ,π ℓ ě e c ℓ V N p T q ( . Thus, we proved u ℓ p T q “ E “ e c ℓ V N p T q L p T q ‰ { V N p q . Proof of Theorem 3.1.
Suppose a solution of (28) and (26) is ˜ w ℓ : C pp ,
8q ˆ p , n ˆ p , n q Ñ p , .Define ˜ N p t q : “ ˜ w ℓ p T ´ t, X N r ,t s , Y r ,t s q e c ℓ V N p t q L p t q , 0 ď t ď T .By calculating d ˜ N p t q{ ˜ N p t q and using the inequality (28), we get that the dt terms in d ˜ N p t q{ ˜ N p t q is alwaysno greater than 0. ˜ N p t q is a local supermartingale. And since ˜ N p t q “ ˜ w ℓ p T ´ t, X r ,t s , Y r ,t s q e c ℓ V N p t q L p t q ě N p t q is a supermartingale.Hence ˜ N p q “ ˜ w ℓ p T, x , y q V N p q ě E P r ˜ N p t qs “ E P r e c ℓ V N p T q L p T qs holds for every p T, x , y q P p ,
8q ˆp , n ˆ p , n . Then ˜ w ℓ p T, x , y q ě E P “ e c ℓ V N p T q L p T q ‰ { V N p q “ ˜ u ℓ p T, x , y q . C Function Generated Portfolios
Proof of Theorem 4.2.
By Ito’s lemma, d log V ℓ p t q e c ℓ V N p t q “ “ π ℓ p t qp β p t q ´ α p t q π ℓ p t qq ´ R p t q β p t q ` || R p t q σ p t q|| ‰ dt ` “ π ℓ p t q ´ R p t q ‰ σ p t q dW p t q . (65)27ince G and G are twice continuously differentiable function, it follows D ij log G p m p t qq “ D ij G p m p t qq G p m p t qq ´ D i log G p m p t qq D j log G p m p t qq ,D ij log G p Y p t qq “ D ij G p Y p t qq G p Y p t qq ´ D i log G p Y p t qq D j log G p Y p t qq (66)Then using (66) and Ito’s lemma, the right hand side of (45) becomes d log G p m p t qq ` d log G p Y p t qq “ n ÿ i “ D i log G p m p t qq d m i p t q` G p m p t qq n ÿ i,j “ D ij G p m p t qq m i p t q m j p t qp n ÿ k “ τ mik p t q τ mjk p t qq dt ´ n ÿ i,j “ D i log G p m p t qq D j log G p m p t qq m i p t q m j p t qp n ÿ k “ τ mik p t q τ mjk p t qq dt ` n ÿ i “ D i log G p Y p t qq d Y i p t q ` G p Y p t qq n ÿ i,j “ D ij G p Y p t qq ψ ij p t q dt ´ n ÿ i,j “ D i log G p Y p t qq D j log G p Y p t qq ψ ij p t q dt, (67)The local martingale part of (65) and (67) are the same, and this leads to π ℓ p t q “ “ p D i log G p m p t qq m i p t qq n p I ´ m p t qq ` p D i log G p Y p t qqq n τ p t q σ ´ p t q ‰ ` R p t q , for t P r , T s , and for each k . Substitute this result into (65), d log V ℓ e c ℓ V p t q “ d log G p Y p t qq ` d log G p m p t qq´ ! r G p t qp´ α p t q m p t qq ` r G p t qp´ α p t q π ℓ p t qq ` ` || r G p t q σ || ` || r G p t q σ || ´ || π ℓ σ || ˘ ´ G p m p t qq n ÿ i,j “ D ij G p m p t qq m i p t q m j p t qp n ÿ k “ τ mik p t q τ mjk p t qq´ G p Y p t qq n ÿ i,j “ D ij G p Y p t qq ψ ij p t q ) dt. Lemma C.1.
A matrix A is semi-definite if and only if p xAy q ď p xAx qp yAy q for all x , y in R n . The equalityholds if and only if xA and yA are linearly dependent. Lemma C.2. If A “ p a ij q is positive semi-definite matrix, then there is an index k such that a kk ě a ij , forany i and j . In other words, the largest entry of the matrix A appears on the diagonal. We show here the derivation in Example 4.1.
Proof of Example 4.1.
Let M be a market without dividends. Suppose that M is nondegerate and has boundedvariance. Suppose M is weakly diverse in [0,T]. Consider the function G and G are defined as in example 4.1. aś ni “ m i ď ř ni “ m i n ď b ř ni “ m i n implies that0 ď G p m q ď n , ´ N ď G p Y p t qq ď ´ N n then 1 ´ N ď log G p m q ` log G p Y p t qq ď n ` ´ N n . G and G implies π ℓi ą max t , ´ p n ` δX N p t q{ V N p t qqp ´ η q ` ˆ ´ Y p t q G p Y p t qq ˙ τ p t q σ ´ p t q e i u ; (68) π ℓi ă min t ` ´ δN p V ℓ p t qq N p π ℓi p t qq N { V N p t q ` ˆ ´ Y p t q G p Y p t qq ˙ τ p t q σ ´ p t q e i , u . (69)Denote max i “ ,...,n m i “ m p q , min i “ ,...,n m i “ m p n q , max i “ ,...,n π i “ π p q , min i “ ,...,n π i “ π p n q , and theeigenvalues of α p t q : max i “ ,...,n λ i “ λ p q , min i “ ,...,n λ i “ λ p n q . m max : “ p m p q , m p q , . . . , m p q q .We’ll use the following results to simplify Ξ p T q :(i) M is nondegenerate, weakly diverse and has bounded variation;(ii) n ď ř ni “ m i ď ď ||p ´ n m q|| ď a n p n ´ q ; ř ni “ p π ℓi q ď || ř Nℓ “ π ℓ p t q τ σ ´ || ď || ř Nℓ “ π ℓ p t q|| ¨ || τ || ¨ || σ ´ || ď N b λ max p ψ q λ min p α q , where thenorm for τ and σ ´ is matrix induced norm. For a matrix A P R m ˆ n , a Trace p AA q “ || A || F ď ? n || A || ,where || ¨ || is the matrix induced norm. Trace p τ τ π q “ ř ni “ ř Nℓ “ τ ℓii ě nǫ ř Nℓ “ p ´ π ℓ p q q , then || τ || ě ǫ ř Nℓ “ p ´ π ℓ p q q ;(iii) | β i | and | α ij | for any i and j is bounded by lemma C.2, thus we could easily get Y p t q τ p t q σ ´ p t q β p t q ą M ;By lemma C.1, e i α p t q m p t q ď p e i α p t q e i qp m p t q α p t q m p t qq ď M M || m p t q|| ď M M , where e i α p t q e i ď M || e i || , m p t q α p t q m p t q ď M || m p t q|| .Ξ p T q “ ż T p e i ´ m p t qq α p t q m p t q ` ´ Y p t q G p Y p t qq ` γ π p t q ´ τ σ ´ p t q β p t q ˘ ` d ℓ p t q α p t q d ℓ p t q + dt (i) ď ż T e i α p t q m p q p t q ´ ǫ || m || ` G p Y p t qq N ÿ ℓ “ π m p t q ` γ π p t q ´ τ σ ´ p t q β p t q ˘ ` M ” || ´ n m || ` G p Y p t qq || N ÿ ℓ “ π ℓ p t q τ ℓ σ ´ || ı ´ ǫ || π ℓ || + dt (ii,iii) ď T ” M M ´ ǫn ` M π ´ N ´ M ´ N n ` M ´ n p n ´ q ` N p ´ N q λ p q p τ q λ p n q p σ ´ q ¯ı ´ ǫ ż T max i | π ℓi | dt where d ℓ p t q : “ ´ n m ´ π ℓ p t q ` ´ Y p t q G p Y p t qq τ σ ´ , max i | π ℓi | ą “ max t , ´ p n ` c ℓ δ X N p t q V p t q qp ´ η q ´ Y i p t q G p Y p t qq τ p t q σ ´ p t q e i u ‰ . Hence, for t P r , T s ,log V ℓ e c ℓ V N p t q “ log G p Y p t qq ` log G p m p t qq ` Ξ t ď ` n ´ N n ` T ” M M ´ ǫn ` M π ´ N ´ M ´ N n ` M ´ n p n ´ q ` N ´ N λ p τ q λ p σ ´ q ¯ı . Then π strictly dominates the weighted average V N p t q if T ě nN ´ n ´ ´ ǫn ` M π n ´ n M ´ N n ` M n ´ n p n ´ q ` N ´ N λ p τ q λ p σ ´ q ¯ . Limiting behavior of finite games vs mean field games
Differentiating from the usual Mckean-Vlasov SDEs of the form that the coefficients of the diffusion dependon the distribution of the solution itself, we here consider the joint distribution of the state processes and thecontrol, and show the propagation of chaos holds.In this section we attempt to show that in the limit N Ñ 8 , a vector of stock prices X p t q : “ p X p t q , . . . , X n p t qq and the wealth of a representative player will satisfy Mckean-Vlasov SDEs. Namely, d X p t q “ X p t q β p X p t q , ν t q dt ` X p t q σ p X p t q , ν t q dB t , X “ x ; (70) dV p t q “ π p t q β p X p t q , ν t q dt ` π p t q σ p X p t q , ν t q dB t , V p q “ v , (71)where B t “ p B , . . . , B n q is n -dimensional Brownian motion, ν : “ Law p V, π | F Bt q . v is with the same law as v ℓ , and it is supported on p Ω , F , F , P q . Remark 3.
In this section, we analyze a Mckean-Vlasov system with initial states given, but we shall see inmean field games sections that it is given in the form as v “ ˜ u p T q V p q . The following proposition shows that ν N has a weak limit ν P P p C pr , T s ; R n ˆ A qq with W distance. Wedenote C n,N “ C pr , T s ; R n ˆ A q for simplicity. Proposition D.1.
Under Assumption 2, 3, and 11, there exists a unique strong solution of the Mckean-Vlasovsystem (70) - (71) .Proof. Define the truncated supremum norm || x || t and the truncated Wasserstein distance on P p C n,N q as in[19]. || x || t : “ sup ď s ď t | x s | , d t p µ, ν q “ inf π P Π p µ,ν q ż C n ˆ C n || x ´ y || t π p dx, dy q . Define a map Φ : P p C n,N q Ñ P p C n,N q so that Φ p ν q “ Law p V ν , π ν | F B q . Fix ν , solve (70) and (71). Sincesolutions of (70) and (71) are equivalent to fixed points of Φ, we begin by proving that Φ is a contractionmapping in a complete space C n,N .We take two arbitrary measures ν a , ν b P P p C n,N q , and denote the wealth involving measure ν as V ν , andstock price vector involving ν as X ν . By Cauchy-Schwartz and Jensen’s inequality, Lipschitz conditions inAssumption 2 and 3, it follows E “ ||p V ν a , X ν a q ´ p V ν b , X ν b q|| t | F Bt ‰ ď t E „ ż t | V ν a p r q π ν a p r q β p x , ν a q ´ V ν b p r q π ν b p r q β p x , ν b q| ` | b p x , ν a q ´ b p x , ν a q| dr ˇˇˇˇ F Bt ` E „ sup ď s ď t | ż s V ν a p r q π ν a p r q σ p x , ν a q ´ V ν b p r q π ν b p r q σ p x , ν b q dB r | ` sup ď s ď t | ż s X ν a r σ p x , ν a q ´ X ν b r σ p x , ν b q dB r | ˇˇˇˇ F Bt ďp t ` q L E „ ż t p| V ν a r ´ V ν b r | ` | X ν a r ´ X ν b r | ` W p ν ar , ν br qq dr ˇˇˇˇ F Bt By Gronwall’s inequality, E r|| V ν a ´ V ν b || t | F Bt s ď E r||p V ν a , X ν a q ´ p V ν b , X ν b q|| t | F Bt s ď p t ` q L E r ż t W p ν ar , ν br q dr s . (72)If π is open loop control, i.e., π p t q “ φ p v , ν t , B r ,T s q , E r|| π ν a ´ π ν b || t | F Bt s ď L E r| v ν a ´ v ν b | ` | B ν a r ,t s ´ B ν b r ,t s | ` W p ν a , ν b q| F Bt s ď L E r W p ν at , ν bt qs . (73)If π is closed loop Markovian, i.e., π p t q “ φ p t, V p t q , ν t q , E r|| π ν a ´ π ν b || t | F Bt s ď L E r|| V ν a ´ V ν b || t ` W p ν at , ν bt qs ď p t ` q L E r ż t W p ν ar , ν br q dr s . (74)30hen the coupling of Φ p ν q , Φ p ν q gives the following inequality d t p Φ p ν a q , Φ p ν b qq ď E r||p V ν a , π ν a q ´ p V ν b , π ν b q|| T | F BT sď C T E r ż T W p ν ar , ν br q dr sď ż T d r p ν ar , ν br q dr, (75)where C T “ p T ` q L ` p T ` q L for closed loop Markovian controls, and C T “ p T ` q L for open loopcontrols.Following Picard iteration scheme, choose an arbitrary ν P P p C n,N q , ν ℓ ` “ Φ p ν ℓ q , Φ has been proved as an contraction mapping when 0 ă T ă L ´ L ă and thus Φ has a unique fixed pointand ν ℓ converges to ν by the contraction mapping principle. For T ą L ´
2, we prove the above argument for r T, T s , r T, T s , etc.Subsequently, we show in the following proposition that MFE strategies coincide with the limit of optimalempirical measure in the weak sense. Proposition D.2.
There exists limits for measure flows ν N P P p C n,N q , µ N P P p C pr , T s ; R ` qq , i.e., thelimits ν t “ lim N Ñ8 ν Nt , µ t “ lim N Ñ8 µ Nt exist in the weak sense for t P r , T s with respect to the 2-Wassersteindistance.Proof. Let p U ℓ q be the solution of closed loop Markovian dynamics φ ℓ : r , T s ˆ R ` Ñ A , dU ℓ p t q “ U ℓ p t q φ ℓ p t, U ℓ p t qq β p X t , ν t q dt ` U ℓ p t q φ ℓ p t, U ℓ p t qq σ p X t , ν t q dB t , U ℓ p q “ v ℓ , or of open loop dynamics dU ℓ p t q “ U ℓ p t q φ ℓ p v ℓ , B r ,T s q β p X t , ν t q dt ` U ℓ p t q φ ℓ p v ℓ , B r ,T s q σ p X t , ν t q dB t , U ℓ p q “ v ℓ . for ℓ “ , . . . , N . The initial states v ℓ are i.i.d copies of v . We assume the initial value of U ℓ p q is of the samelaw with V ℓ p q . E r||p V ℓ , φ ℓ p V ℓ qq ´ p U ℓ , φ ℓ p U ℓ qq|| t s ď C T E “ ż t W p ν Nr , ν r q dr ‰ ď C T E “ ż t d r p ν n , ν q dr ‰ (76)for t P r , T s , C T is defined in Proposition D.1. For simplicity, let us discuss in the case of closed loop dynamics,the result of which can be generalized to open loop dynamics.˜ ν N are the empirical measure of N i.i.d samples U ℓ . We follow the coupling arguments in [4], the empiricalmeasure of p V ℓ , U ℓ q is a coupling of the N -player empirical measure ν N defined in Definition 2.2 and ˜ ν N . d t p ν N , ˜ ν N q ď N N ÿ ℓ “ ||p V ℓ , φ ℓ p V ℓ qq ´ p U ℓ , φ ℓ p U ℓ qq|| t , a.s. . (77)By the triangle inequality and (76), (77), E r d t p ν N , ν q dr s ď E r d t p ˜ ν N , ν qs ` C T E r ż t d r p ν N , ν q dr s , and then by Gronwall’s inequality and set t “ T , it follows E r W p ν N , ν qs ď e C T T E r W p ˜ ν N , ν qs . Since p U ℓ , π ℓ q , ℓ “ , . . . , N is independent given the noise B , use conditional law of large numbers (Theorem3.5 in [22]), P ` lim n Ñ8 N ÿ ℓ “ f p U ℓ , π ℓ q ´ E r f p U ℓ , π ℓ q| F Bt sq “ , for every f P C b p R n ˘
31e then use Theorem 6.6 in [25], which states that on a separable metric space, ν N Ñ ν weakly.lim N Ñ8 ż R N d p x, x q ν N p dx q “ ż R N d p x, x q ν p dx q a.s. , which lead us to E r W p ˜ ν N , ν qs Ñ . Therefore E r W p ν N , ν qs Ñ
0. We can use similiar methods to derive E r W p µ N , µ qs Ñ Assumption 12.
There are the following bounds on β and σ : ż ts | β i p r, ω q| dr ď η p ω, ν q| t ´ s | ` βα , ż ts | σ ij p r, ω q| dr ď ξ p ω, ν q| t ´ s | ` βα , where t, s P r , T s , α and β are positive constants, and η , ξ being F -measurable random variables with values in p ,
8q ˆ Ω ˆ C n,N such that there is ǫ ą with E r η p ω, ν q s ă 8 , E r ξ p ω, ν q s ă 8 . Proposition D.3.
If Assumption 12 holds, then there exist n dimensional continuous process X defined on theprobability space p Ω , F , P q , such that X p t q “ lim N Ñ8 X N p t q exists a.s. for all t P r , T s .Proof. First we show that t P X N u is tight. By [17], a sequence of measures µ N on P p C pr , T s ; R ` qq is tight ifand only if • there exist positive constants M x and γ such that E t| x N | γ u ď M x for every N “ , , . . . , • there exist positive constants M k and δ , δ such that E t| X N p t q ´ X N p s q| δ u ď M k | t ´ s | ` δ for every N “ , , . . . , t, s P r , T s .Apparently, the first condition holds. Then, | X N p t q ´ X N p s q| α ď N α { p| X N p t q ´ X N p s q| α ` . . . ` | X Nn p t q ´ X Nn p s q| α q , | X Ni p t q´ X Ni p s q| α “ | ż ts X Ni p r q β i p r q dr ` n ÿ k “ ż ts X Ni p r q σ ik p r q dW k p r q| α ď p n ` q α p η p ω, ν q α | t ´ s | ` β ` n ÿ k “ | ż ts σ ik p r q dW k p r q| α q . Then let α “ E r| X N p t q ´ X N p s q| s ď N α { p n ` q α ˆ E r η p ω, ν q s h ` β ` n ÿ i “ n ÿ k “ E “ ż ts | σ ik p r q| dW k p r q ‰˙ ď N p n ` q ˆ E r η p ω, ν q s ` E r ξ p ω, ν q s ˙ h ` β , where E r η p ω, ν q s ` E r ξ p ω, ν q s ă 8 , h P p , T s . Thus the second condition follows.By Prokhorov theorem [2], tightness implies relative compactness, which means here that each subsequenceof X N contains a further subsequence converging weakly on the space C pr , T s ; R n ` q . As a result, a subsequenceexists such that X p t q “ lim N Ñ8 X N p t q a.s.. Then if every finite dimensional distribution of t P X N u converges,then the limit of t P X N u is unique and hence t P X N u converges weakly to P as N Ñ 8 . Proposition D.4.
Under Assumption 2, X p t q “ lim N Ñ8 X N p t q exists in the weak sense, and the limit X p t q match the solution of the Mckean-Vlasov SDE dX i p t q “ X i p t q β i p X p t q , ν t q dt ` X i p t q σ i p X p t q , ν t q dB t roof. Since ν t “ lim N Ñ8 ν Nt , it is equivalent to show that the drift and volatility of ν t matches the weak limitof that of ν Nt , i.e., β p X p t q , ν t q “ lim N Ñ8 β p X N p t q , ν N p t qq , σ p X p t q , ν t q “ lim N Ñ8 σ p X N p t q , ν N p t qq . in the weak sense.By Lebesgue dominated convergence theorem || ż t β p X Ns , ν Ns q ´ β p X s , ν s q ds || L ď ż t || β p X Ns , ν Ns q ´ β p X s , ν s q|| L ds ď L E r ż t | X Ns ´ X s | ds ` ż t W p ν Ns , ν s q ds s By Ito’s isometry and Assumption 2, we have || ż t σ p X Ns , ν Ns q dW s ´ ż t σ p X s , ν s q dB s || L “ E r ż t | σ p X Ns , ν Ns q ´ σ p X s , ν s q| ds sď L E r ż t | X Ns ´ X s | ` W p ν Ns , ν s q ds s . Hence it follows from the fact that X p t q “ lim N Ñ8 X N p t q a.s., ν t weakly convergent to ν N p t q with W , weget X p t q satisfies d X p t q “ X p t q β p X p t q , ν t q dt ` X p t q σ p X p t q , ν t q dB t Finally, under Assumption 6 we conclude that when N Ñ 8 , the limiting system is driven by X t and ν t : “ Law p V p t q , π p t qq . The stock market follows d X t “ X p t q β p X t , ν t | F Bt qq dt ` X p t q σ p X t , ν t | F Bt q dB t , X “ x , and a generic player’s wealth is dV p t q “ π p t q β p X p t q , ν t | F Bt qq dt ` π p t q σ p X p t q , ν t | F Bt q dB t , V p q “ v . (78)With the notations in Definition 2.1 (3), if we consider the mean Z p t q of the measure Law p V p t q , π p t q| F Bt q , wecan get Z p t q “ lim N Ñ8 Y p t q exists in the weak sense, and the limit Z p t q match the solution of the Mckean-VlasovSDE dZ p t q “ γ p X t , Z p t qq dt ` τ p X t , Z p t qq dB t ..