Capital growth and survival strategies in a market with endogenous prices
aa r X i v : . [ q -f i n . M F ] J a n Capital growth and survival strategiesin a market with endogenous prices
Mikhail Zhitlukhin ∗
24 January 2021
Abstract
We call an investment strategy survival, if an agent who uses it maintains anon-vanishing share of market wealth over the infinite time horizon. In a discrete-time multi-agent model with endogenous asset prices determined through a short-run equilibrium of supply and demand, we show that a survival strategy can beconstructed as follows: an agent should assume that only their actions determinethe prices and use a growth optimal (log-optimal) strategy with respect to theseprices, disregarding the actual prices. Then any survival strategy turns out to beclose to this strategy asymptotically. The main results are obtained under theassumption that the assets are short-lived.
Keywords: survival strategies, capital growth, relative growth optimal strategies,endogenous prices, evolutionary finance, martingale convergence.
MSC 2010:
JEL Classification:
C73, G11.
1. Introduction
The main object of study of this paper is asymptotic performance of investment strate-gies in stochastic market models. The mathematical theory of optimal capital growthoriginated with the works of Kelly (1956), Latan´e (1959), Breiman (1961), and one ofits central results consists in that an agent who maximizes the expected logarithm ofwealth achieves the fastest asymptotic growth of wealth over the infinite time horizon(see, e.g., Algoet and Cover (1988)). The standard assumption made in this theory isthat an agent has negligible impact on a market, and hence asset prices can be specifiedby exogenous random processes not depending on agents’ strategies. The aim of thispaper is to extend these results and describe analogues of growth optimal strategies ina multi-agent market model which may contain assets with endogenously determinedprices.We consider a discrete-time model of a market with two type of assets. Assets of thefirst type, further called exogenous , have prices and dividends represented by exogenousrandom sequences (without loss of generality, we will assume that the dividends areincluded in the prices). Agents get profit or loss when the prices of these assets change.Assets of the second type, further called endogenous , have exogenous dividends, but theirprices are determined endogenously via a short-run equilibrium of supply and demand. ∗ Steklov Mathematical Institute of the Russian Academy of Sciences. 8 Gubkina St., Moscow, Russia.Email: [email protected] research was supported by the Russian Science Foundation, project no. 18-71-10097. survival strategy. Such a strategyallows an agent to keep the relative wealth strictly bounded away from zero over theinfinite time horizon. Our goal is to construct a survival strategy in an explicit formand to find what effect the presence of this strategy has on the asymptotic distributionof wealth between market agents. In particular, we are interested in conditions underwhich a strategy is asymptotically dominating , i.e. an agent using it becomes the singlesurvivor in a market with the relative wealth converging to 1. In order to find a survivalstrategy, the notion of a relative growth optimal strategy will be useful. This is astrategy with the logarithm of its relative wealth being a submartingale. The fact thata non-positive submartingale converges implies that a relative growth optimal strategyis survival. The convergence of the compensator of this submartingale allows to obtaina sufficient condition for a survival strategy to be also dominating.Note that, in contrast to the optimal growth theory for markets with exogenousprices, which deals with absolute wealth of agents, we focus on relative wealth, whichturns out to be more amenable to asymptotic analysis in the case of endogenous prices.Drokin and Zhitlukhin (2020, Section 6) show that the goals of maximization of relativeand absolute wealth in a model with endogenous prices may be incompatible.Our first main result consists in showing that a relative growth optimal strategy canbe constructed as a growth optimal strategy in a market with exogenous prices equal tothe endogenous prices induced by this strategy when all the agents in the market use it.We find such a strategy in a tractable form, as a solution of a two-stage optimizationproblem. On the first stage, an agent determines the portfolio of exogenous assets bymaximizing the expected log-return (with some adjustments if it is not integrable); onthe second stage the portfolio of endogenous assets is found via a solution of anothermaximization problem. We show that this strategy is relative growth optimal in anystrategy profile, irrespectively of the strategies used by the other agents. Another itsfeature, which can be attractive for possible applications, is that it needs to know littleinformation about the market: only the current total market wealth and the probabilitydistribution of returns of the exogenous assets and payoffs of the endogenous assets, butdoes not require the knowledge of the other agents’ individual wealth or their strategies.It also does not depend on the spot prices of the endogenous assets, and so is not affectedby the impact which an agent may have on the market.Our second main result shows that the obtained strategy becomes the single survivingstrategy in a market if the representative strategy of the other agents is asymptoticallydifferent from it in a certain sense. Consequently, if some agent uses this strategy, thenany other agent who wants to survive in the market must use an asymptotically similarstrategy. As a corollary, we show that this strategy asymptotically determines the prices2f the endogenous assets.The results we obtain are tightly related to and generalize the main results ofAmir et al. (2013) and Drokin and Zhitlukhin (2020). Those papers also studied sur-vival and growth optimal strategies in markets with short-lived assets and endogenousprices, however the models were less general. In the former paper it was assumed thatthere are only assets with endogenous prices; the latter paper also included a risk-freebank account with an exogenous interest rate. Another extension consists in that weallow the model to include constraints on agents’ portfolios specified by random convexsets. Among other recent papers related to this setting, let us mention the paper ofBelkov et al. (2020), which builds another model that includes assets with endogenousprices and a risk-free asset. A difference with our model is that they assume asset payoffsdepend linearly on the amount of money invested in the risk-free asset, which allows toreduce the model to previously known results for models without a risk-free asset.Let us mention how this paper is related to other results in the literature. In modelswith exogenous prices, the asymptotic growth optimality of the log-optimal strategy(also called the Kelly strategy, after Kelly (1956)) was proved for a general discrete-timemodel by Algoet and Cover (1988); a review of other related results in discrete timecan be found in, e.g., Cover and Thomas (2012, Chapter 16) or Hakansson and Ziemba(1995). For a treatment of a general model with continuous time and portfolio con-straints, and a connection of growth optimal portfolios (num´eraire portfolios) with ab-sence of arbitrage, see, e.g., Karatzas and Kardaras (2007).Among various lines of research on markets with endogenous prices, our paper ismost closely related to works in evolutionary finance on stability and survival of in-vestment strategies, which focus on evolutionary dynamics and properties like survival,extinction, dominance, and how they affect the structure of a market. Central to thisdirection are strategies that perform well irrespectively of competitors’ actions. Oneof the main results consists in that the strategy which splits its investment budget be-tween risky assets proportionally to their expected dividends (often also called the Kellystrategy) survives in a market provided that the agent’s beliefs about the dividends arecorrect. See, for example, the papers of Amir et al. (2005, 2011); Blume and Easley(1992); Evstigneev et al. (2002, 2006); Hens and Schenk-Hopp´e (2005), which establishthis fact for different models and under different assumptions. Reviews of this directioncan be found in Evstigneev et al. (2016) or Amir et al. (2020). Typically, the Kellystrategy turns out to be the only surviving strategy in a market, i.e. it dominates allother asymptotically different strategies. For results on market wealth evolution whenagents use strategies different from the Kelly strategy, which may result in survival ofseveral strategies, see, e.g., Bottazzi and Dindo (2014); Bottazzi et al. (2018).Most of the above mentioned papers (including the present paper) consider agent-based models, where agents’ strategies are specified directly as functions of a mar-ket state. Another large body of literature consists of results on market selectionof investment strategies in the framework of general equilibrium, where agents max-imize utility from consumption. Among those results one can mention, for example,Blume and Easley (2006); Boroviˇcka (2020); Sandroni (2000); Yan (2008). Holtfort(2019) provides a detailed survey of the literature in evolutionary finance over the lastthree decades, including also some earlier results.The paper is organized as follows. Section 2 describes the model. The main resultsof the paper are stated in the three theorems included in Section 3. Section 4 containstheir proofs. 3 . The model
For vectors x, y ∈ R N , we will denote by h x, y i their scalar product, and by | x | = P n | x n | , k x k = p h x, x i the L and L norms. If f : R → R is a scalar function and x is a vector,then f ( x ) = ( f ( x ) , . . . , f ( x N )) denotes the coordinatewise application of f to x .By e we will denote the vector consisting of all unit coordinates, e = (1 , . . . , h e, x i is equal to thesum of coordinates of a vector x .All equalities and inequalities for random variables are assumed to hold with prob-ability 1 (almost surely), unless else is stated. Let (Ω , F , P) be a probability space with a discrete-time filtration F = ( F t ) ∞ t =0 on whichall random variables will be defined. Without loss of generality, we will assume that F is P-complete and F contains all P-null events.The market in the model consists of M agents (investors) and N = N + N assetsof two types. The assets of the first type are available in unlimited supply and haveexogenous prices; they are treated as in standard models of mathematical finance. Theassets of the second type are in limited supply; they yield payoffs which are definedexogenously, but their prices are determined endogenously from an equilibrium of supplyand demand in each time period. These assets are short-lived in the sense that they canbe purchased by agents at time t , yield payoffs at t + 1, and then get replaced with newassets; agents cannot sell them, and, in particular, short sales are not allowed (addingshort sales would lead to conceptual difficulties which we prefer to avoid). We will callthe assets of the first and the second type, respectively, exogenous and endogenous.The prices of the exogenous assets are represented by positive random sequences( S nt ) ∞ t =0 , n = 1 , . . . , N , which are F -adapted (i.e. S nt is F t -measurable). We assume thatdividends, if there are any, are already included in the prices. By X nt = S nt /S nt − > Y nt ) ∞ t =1 , n = 1 , . . . , N .Without loss of generality, we assume that the supply of each endogenous asset is equalto 1, so Y nt is the total payoff of an asset. Their prices will be defined later, as we firstneed to define agents’ strategies, on which they will depend.The agents enter the market at time t = 0 with non-random initial wealth v m > m = 1 , . . . , M . Actions of an agent at time t ≥ h t = ( α t , β t ), where α t ∈ R N , β t ∈ R N + specify in what proportions this agent allocatesthe current wealth between the assets of the two types (the wealth sequences are yet tobe defined), i.e. the proportion α nt (respectively, β nt ) of wealth is allocated to asset n . Since α t , β t are proportions, we require that h e, α t i + h e, β t i = 1. The componentsof β t are non-negative, because short sales of the endogenous assets are not allowed.Additionally, we will assume that it is not possible to buy the endogenous assets onborrowed funds, i.e. h e, α i ∈ [0 , h e, β i ∈ [0 , h t assumes In the literature, time indices are often shifted by 1 forward (so h t represents actions at time t − h t specify actions at time t . H = { ( α, β ) ∈ R N × R N + : h e, α i ∈ [0 , , h e, β i = 1 − h e, α i} . In order to emphasize that a pair h t is selected by agent m we will use the superscript m ,e.g. h mt = ( α mt , β mt ).A strategy of an agent consists of investment proportions h mt selected at consecutivemoments of time. It may (and, usually, does) depend on a random outcome and markethistory. In order to specify this dependence, introduce the measurable space (Θ , G ) withΘ = Ω × R M + × ( H M ) ∞ , G = F ⊗ B ( R M + × ( H M ) ∞ ) , where an element χ = ( ω, v , h , h , . . . ) ∈ Θ consists of a random outcome ω , a vectorof initial wealth v = ( v , . . . , v M ) ∈ R M + , and vectors of investment proportions h t =( h t , . . . , h Mt ) selected by the agents at each moment of time. Let G = ( G t ) t ≥ be thefiltration on Θ defined by G t = F t ⊗ B ( R M + × ( H M ) t +1 ) , i.e. G t is generated by sets Γ × V × H × . . . × H t × ( H M ) ∞ with Γ ∈ F t and Borel sets V ⊆ R M + , H s ⊆ H M . We define a strategy of an agent as a sequence of G t -measurablefunctions h t ( χ ) : Θ → H , t ≥ . Basically, h t can be thought of as a function h t ( ω, v , h , . . . , h t ), but the notation h t ( χ )will be more convenient for us because we will deal with functions depending on markethistories of different length appearing in one formula, see, e.g., (3) below. Note thedependence of h t on the argument h t , i.e. an agent may use information (partial orwhole) about actions of other agents at the same moment of time t . This informationmay be available to an agent, for example, from asset prices.We call a vector of initial wealth v and a strategy profile ( h , . . . , h M ) feasible ifthere exists a sequence of F t -measurable functions h t ( ω ) = ( h t ( ω ) , . . . , h Mt ( ω )) ∈ H M such that for all ω, t, m h mt ( χ ( ω )) = h mt ( ω ) , where χ ( ω ) = ( ω, v , h ( ω ) , h ( ω ) , . . . ) . (1)Such a sequence h ( ω ) will be called a realization of the agents’ strategies correspondingto the given strategy profile and initial wealth. We do not require the uniqueness of arealization, i.e. equation (1) may have several solutions. The main results of the paperwill hold for any chosen realization (however, the uniqueness may be desirable for otherapplications). Remark 1 (On notation) . By the bold font we denote functions which depend on χ ,i.e. on a random outcome and market history, while functions which depend only on arandom outcome ω (e.g. realizations of strategies) are denoted by the normal font. Inparticular, if ζ is a function of χ , then, given a vector of initial wealth and a strategyprofile, we denote by ζ ( ω ) its realization ζ ( χ ( ω )), where χ ( ω ) is as in (1).If ξ is a random variable, i.e. a function of ω only, we will sometimes use the sameletter to denote the function ξ ( χ ) which just ignores the values of v and h s , i.e. ξ ( χ ) = ξ ( ω ) at an element χ = ( ω, v , h , h , . . . ). 5ufficient conditions for a vector of initial wealth and a strategy profile to be feasible,in general, can be formulated in terms of assumptions of fixed-point theorems, but wedo not investigate this question in details – our main goal is to find an optimal strategy,and the strategy which we find will be optimal in any feasible profile. Nevertheless, it iseasy to see that a simple sufficient condition for the feasibility is that the functions h mt do not depend on the argument h t , i.e. adapted to the filtration G − = ( G − t ) t ≥ , where G − t = F t ⊗ B ( R M + × ( H M ) t ) . This condition can be interpreted as that at each moment of time the agents decideupon their actions simultaneously and independently of each other.Now we can define the prices of the endogenous assets and the wealth sequences v mt ( χ ) inductively in t , beginning with v m ( χ ) = v m . Denote the prices at time t by p nt ( χ ), n = 1 , . . . , N . Suppose for some χ ∈ Θ the wealth sequences are defined up to amoment of time t , and v mt ( χ ) ≥ m . Then agent m can purchase y m,nt ( χ ) unitsof asset n at this moment, where y m,nt = β m,nt v mt p nt , and β m,nt (also α m,nt below) are taken from the component h t entering χ . In order toclear the market (recall that the supply of each asset is 1), the prices should be equal to p nt = M X m =1 β m,nt v mt . (2)Essentially, we employ the principle of moving equilibrium, which operates with eco-nomic variables changing with different speeds. In our model, the endogenous assetprices move fast, while the investment proportions selected by the agents move slow;the proportions can be considered fixed while the prices rapidly adjust to clear the mar-ket. The mechanics of this adjustment process is not important to us (as long as itdoes not inflict transaction costs) and it can be modeled by various approaches, e.g.limit order books, auctions, etc. Note that we do not require the agents to agree uponfuture asset prices at each random outcome. For a discussion of this moving equilibriumapproach in a similar model, see Section 4 in Evstigneev et al. (2020).If P m β m,nt ( χ ) = 0 in formula (2) for some n , i.e. no one invests in asset n , we put y m,nt ( χ ) = 0 for all m ; in this case the price p nt ( χ ) can be defined in an arbitrary waywith no effect on the agents’ wealth, so we will put p nt ( χ ) = 0 in accordance with (2).Thus, the portfolio of agent m between moments of time t and t + 1 consists of y m,nt units of endogenous asset n , and x m,nt units of exogenous asset n , where x m,nt = α m,nt v mt S nt . Consequently, the wealth of this agent at t + 1 is determined by the relation v mt +1 = N X n =1 x m,nt S nt +1 + N X n =1 y m,nt Y nt +1 = (cid:18) N X n =1 α m,nt X nt +1 + N X n =1 β m,nt Y nt +1 P k β k,nt v kt (cid:19) v mt (3)(with 0 / v mt +1 may become negative, which willmake the right-hand side of the equation meaningless for the next time period. However,below we will introduce portfolio constraints which prohibit strategies that may lead tonegative wealth. In view of this, we will restrict the domain of the functions v mt anddefine them on sets smaller than Θ. Namely, introduce inductively the setsΘ t = { χ ∈ Θ : v ms ( χ ) ≥ s ≤ t, m = 1 , . . . , M } , t ≥ , where v ms ( χ ) are computed by (3). Note that Θ = Θ, Θ t ⊇ Θ t +1 , and Θ t ∈ G − t . Fromnow on, we will assume that the functions v mt are defined only for χ ∈ Θ t .It will be also convenient to introduce the setsΘ ′ t = { χ ∈ Θ t : v t ( χ ) = 0 } , t ≥ . Observe that, essentially, components ( α t , β t ) of an agent’s strategy need to be definedonly on Θ ′ t , since elements from Ω \ Θ t do not correspond to any realization, and on theset { χ : v t ( χ ) = 0 } they can be defined in an arbitrary way without any effect on (zero)wealth. Portfolio constraints in the model are specified by a sequence of G − t -measurable random non-empty closed convex sets C t ( χ ) ⊆ H , t ≥
0. The constraints are the same for eachagent.We say that a strategy h satisfies the portfolio constraints if h t ( χ ) ∈ C t ( χ ) for all t ≥ χ ∈ Θ . From now on, when writing “a strategy”, we will always mean a strategy satisfying theportfolio constraints.Notice that the sets C t are essentially needed to be defined only for elements χ ∈ Θ ′ t .Thus it may be convenient to put, for example, C t = R N + on Θ \ Θ ′ t , without any effecton realizations of the agents’ wealth in the model.We will consider portfolio constraints only of the following particular form: they areimposed on the exogenous and endogenous assets separately, and an agent can freelychoose what proportion of wealth to invest in the assets of each of the two types. Namely,it will be assumed that C t = ( A t × B t ) ∩ H , (4)where A t and B t are G − t -measurable closed convex sets in R N and R N + such that h e, α i ∈ [0 , h e, β i ∈ [0 ,
1] for any α ∈ A t ( χ ), β ∈ B t ( χ ). We also require thatif α ∈ A t ( χ ) , then λα ∈ A t ( χ ) for any λ ∈ [0 , / h e, α i ] , (5)if β ∈ B t ( χ ) , then λβ ∈ B t ( χ ) for any λ ∈ [0 , / h e, β i ] (6)(or λ ∈ [0 , ∞ ) if h e, α i = 0 or h e, β i = 0); i.e. A t and B t can be represented asintersections of some convex cones with the sets { α ∈ R N : h e, α i ∈ [0 , } and { β ∈ R N + : h e, β i ∈ [0 , } respectively. Note that relation (4) implies that the sets A t , B t cannot simultaneously (for the same t, χ ) consist of only elements α or, respectively, β with zero sum of coordinates, since then the set C t would be empty. See Section 4.1 for details on random sets.
7e will need to further restrict the class of portfolio constraints by introducingseveral assumptions on the structure of the sets A t , B t . In what follows, let K t ( ω, d e ω )denote some fixed version of the regular conditional distribution with respect to F t . ByP t and E t we will denote, respectively, the regular probability and expectation computedwith respect to K t , i.e. for a random event Γ ∈ F and a random variable ξ we putP t (Γ)( ω ) = K t ( ω, Γ) , E t ( ξ )( ω ) = Z Ω ξ ( e ω ) K t ( ω, d e ω ) . When ξ depends also on market history, i.e. ξ = ξ ( χ ) is G -measurable, we putE t ( ξ )( χ ) = Z Ω ξ ( e ω, v , h , . . . ) K t ( ω, d e ω ) , χ = ( ω, v , h , . . . ) , provided that the integral is well-defined.Let us introduce several random sets which will be needed to formulate the assump-tions on the sets A t , B t : • the sets of portfolios of exogenous assets which have non-negative values at the nextmoment of time: D t ( ω ) = { α ∈ R N : P t ( h α, X t +1 i ≥ ω ) = 1 } ; • the linear spaces of null investments (portfolios of exogenous assets with zero currentand next value): L t ( ω ) = { α ∈ R N : h e, α i = 0 , P t ( h α, X t +1 i = 0)( ω ) = 1 } ; • the projection of A t on the orthogonal space L ⊥ t : A p t ( χ ) = { α ∈ L ⊥ t ( ω ) : ∃ u ∈ L t ( ω ) such that α + u ∈ A t ( χ ) } . Observe that the sets D t , L t are F t -measurable, and A p t are G − t -measurable. Indeed,we can represent D t ( ω ) = { α : f ( ω, α ) = 0 } with the function f ( ω, α ) = E t ( h α, X t i − ∧ ω ), which is a Carath´eodory function, so D t is measurable by Filippov’s theorem (seeProposition 4 in Section 4.1). The set L t is measurable since it is the intersection of D t , − D t and { α : h α, e i = 0 } . The measurability of A p t follows from Proposition 6.Now we are ready to formulate the assumptions on the portfolio constraints. In theremaining part of the paper we always assume that they are satisfied. Assumptions.
For all t ≥ χ = ( ω, v , h , . . . ) it holds that (A.1) A t ( χ ) ⊆ D t ( ω ); (A.2) there exists ( α, β ) ∈ C t ( χ ) such that P t ( h α, X t +1 i + h β, Y t +1 i > ω ) = 1; (A.3) A p t ( χ ) ⊆ A t ( χ ); (A.4) A p t ( χ ) is a compact set.Let us comment on interpretation of these assumptions. (A.1) is imposed to en-sure that any strategy which satisfies the portfolio constraints generates a non-negativewealth sequence. As a consequence, for the realization of any profile of strategies satis-fying the portfolio constraints we have χ ( ω ) = ( ω, v , h ( ω ) , h ( ω ) , . . . ) ∈ Θ t a.s. for all t ≥ . ω ∈ Ω, if necessary modifying the strategies on aset of zero probability.Assumption (A.2) implies that there exists a strategy with a strictly positive wealthsequence. Such a strategy can be found via a standard measurable selection argument,using that C t are measurable sets. Observe that (A.2) is a very mild assumption. Forexample, it holds if there is a non-zero vector α ∈ A t with all non-negative coordinates(recall that X nt > n ), since then ( α/ | α | , ∈ C t by (5).Assumption (A.3) means that the agents can remove null investments from theirportfolios. Note that in the literature it is sometimes required that L t ⊆ A t (i.e. anyinvestment that leads to no profit or loss is allowed). It is not difficult to see that in ourmodel this requirement implies (A.3).Assumption (A.4) will allow to reduce the optimal strategy selection problem to anoptimization problem on a compact set. Actually, it is equivalent to the no arbitragecondition for the exogenous assets – or, more precisely, no unbounded arbitrage condition– as we show in the next section. Let U t ( ω ) denote the cone of arbitrage opportunities in the exogenous assets at time t ≥
0, which consists of all u ∈ R N such that h e, u i = 0 , P t ( h u, X t +1 i ≥ ω ) = 1 , P t ( h u, X t +1 i > ω ) > . We say that there are no unbounded arbitrage opportunities in the model if for all χ = ( ω, v , h , . . . ) ∈ Θ and t ≥ (A.5) there is no u ∈ U t ( ω ) such that λu ∈ A t ( χ ) for any λ > A t may contain some of them. This condition is analogous to theno unbounded increasing profit condition (NUIP), known in connection with num´eraireportfolios, see Karatzas and Kardaras (2007, Proposition 3.10). If there are no con-straints on the exogenous assets (i.e. A t = { α ∈ R N : h e, α i ∈ [0 , } ), then (A.5) isequivalent to the usual no-arbitrage condition U t = ∅ . Proposition 1.
Suppose the model satisfies assumptions (A.1), (A.3). Then assump-tions (A.4) and (A.5) are equivalent.Proof.
It is easy to see that (A.4) implies (A.5). Let us prove the converse implication.Suppose (A.5) holds. The closedness of A p t follows from that A t is closed and assumption(A.3).To prove that A p t is bounded, fix χ = ( ω, v , h , . . . ) and suppose, by way of contra-diction, that there is a sequence u n ∈ A p t ( χ ) such that | u n | → ∞ . The sequence u n / | u n | is bounded, so there exists a convergent subsequence u n k / | u n k | → u . It is easy to seethat h e, u i = 0 (because h e, u n i ∈ [0 , | u | = 1, u ∈ L ⊥ t ( ω ). The last two propertiesimply that u / ∈ L t ( ω ). Moreover, since A t ⊆ D t , we have P t ( h u n , X t +1 i ≥ ω ) = 1,and hence P t ( h u, X t +1 i ≥ ω ) = 1. Consequently, u ∈ U t ( ω ).However, for any λ > k such that | u n k | ≥ λ , we have λ | u n k | u n k ∈ A t ( χ ) , λu ∈ A t ( χ ), so u is an unbounded arbitrage opportunity, which is acontradiction. Examples.
Arbitrage opportunities may be eliminated by imposing appropriate port-folio constraints, even if the unconstrained model with the same exogenous prices S t has arbitrage. As an example, observe that assumption (A.5) automatically holds whenportfolio constraints limit portfolio leverage in the sense that A t ⊆ { α ∈ R N : c t | a + | ≥ | a − |} , (7)where 0 ≤ c t < α ± =(( α ) ± , . . . , ( α N ) ± ) are the vectors consisting of the positive and negative parts of thecoordinates of α . In this case, if h e, α i = 0 for α ∈ A t , then α = 0, so U t ∩ A t = ∅ and(A.5) holds.Constraint (7) means that the long positions of a portfolio should cover the shortpositions with some margin, which is determined by the constant c t . If α = 0, this isequivalent to that | α + | ≥ | α − | and | α − || α + | − | α − | ≤ c ′ t , where c ′ t = c t / (1 − c t ), which can be interpreted as that the ratio of the debt to thevalue of a portfolio (the leverage) is bounded by c ′ t . If c t = 0, then (7) prohibits shortsales of the exogenous assets. For details on this leverage constraint and how it canbe used in problems of hedging and optimal growth, see e.g. Babaei et al. (2020a,b);Evstigneev and Zhitlukhin (2013).Constraint (7) can be relaxed if one requires A t ⊆ { α ∈ R N : d t + c t | a + | ≥ | a − |} , where d t ≥
0. In this case, A t may include some portfolios with h e, α i = 0 (besides α = 0), in particular arbitrage opportunities, but the set A t ∩ { α : h e, α i = 0 } remainsbounded, so there are still no unbounded arbitrage opportunities.
3. Main results
We will be interested in long-run behavior of relative wealth of agents, i.e. their sharesin total market wealth. We define the total market wealth and the relative wealth ofagent m as, respectively, W t = M X m =1 v mt , r mt = v mt W t , where r mt = 0 if W t = 0. Recall that v t is defined on the set Θ t , hence we will assumethat W t and r mt are defined only on this set as well.For a given feasible strategy profile and a vector of initial wealth, by W t ( ω ) = W t ( χ ( ω )), r mt ( ω ) = r mt ( χ ( ω )) we will denote the corresponding realizations defined asin Remark 1. The realizations of the agents’ wealth sequences will be denoted by v mt ( ω ).10 efinition 1. In a feasible strategy profile ( h , . . . , h M ) with initial wealth v ∈ R M + such that v m >
0, we call a strategy h m survival ifinf t ≥ r mt > , and call it dominating if lim t →∞ r mt = 1 a.s.Our main goal will be to show that the strategy b h which we construct in the nextsection is survival in any strategy profile and dominating in a strategy profile if thestrategies of the other agents are, in a certain sense, different from it asymptotically.Consequently, if some agents use b h , then any other survival strategy should be asymp-totically close to it.Note that any survival strategy is asymptotically unbeatable in the following sense:if agent m uses a survival strategy then there exists a (finite-valued) random variable γ such that r kt ≤ γr mt , k = 1 , . . . , M, t ≥ , which expresses the fact that the wealth of any other agent cannot grow asymptoticallyfaster than the wealth of an agent who uses a survival strategy. For a discussion ofunbeatable strategies as a game solution concept in related evolutionary finance models,see e.g. Amir et al. (2013).At the same time, we would like to emphasize that we do not insist on that allagents should use only survival strategies, as they may have other economic goals ormake systematic errors. We only investigate what happens with a market if someagents use such strategies.For construction of a survival strategy, the following notion will be useful. Definition 2.
For a given feasible strategy profile and initial wealth, we call a strategy h m relative growth optimal if v mt > t ≥ r mt is a submartingale . Since any non-positive submartingale has a finite limit with probability 1 (see,e.g., Shiryaev (2019, Chapter 7.4)), for a relative growth optimal strategy we havelim t →∞ ln r mt > −∞ , and therefore r m ∞ = lim t →∞ r mt >
0. This implies the followingresult.
Proposition 2.
A relative growth optimal strategy is survival.
Note that if the relative wealth of an agent is “infinitesimal” (so the strategy of thisagent does not affect the prices of the endogenous assets), then a relative growth optimalstrategy for this agent, which depends on the current endogenous prices p t , can be foundas a growth optimal portfolio in a market with N = N + N exogenous assets, consider-ing p t as exogenous prices. In particular, if the asset returns are sufficiently integrable,then such a strategy maximizes the one-period expected logarithmic return, see, e.g.,Algoet and Cover (1988) or Cover and Thomas (2012, Chapter 16). The important fea-ture of the strategy that we construct in the next section is that it essentially depends only on the current total market wealth W t , but not on the current endogenous prices,and hence will be a survival strategy for an agent with any relative wealth. We use the terminology of Amir et al. (2013). Note that often a strategy is called survival iflim sup t →∞ r mt >
0, see, e.g., Blume and Easley (1992). Strictly speaking, this strategy may also depend on some additional information contained in themarket history χ , but only through the dependence of the portfolio constraints on such information. .2. Construction of a relative growth optimal strategy In this section we find one relative growth optimal strategy in an explicit form. The ideabehind the construction of this strategy consists in that we find it as a growth optimalportfolio in a market with endogenous prices induced by it (see Theorem 2). We beginwith a lemma which defines the components b α , b β of this strategy. Its statement issomewhat involved, but clarifying comments will be provided in Remark 2 below.Recall that we need to define b α t , b β t only on the set Θ ′ t , while on its complementthese functions can be defined in an arbitrary way (respecting the G t -measurability andthe portfolio constraints), since this will not have any effect on realizations of wealthsequences.In the statement of the lemma and subsequent results, we will use the followingagreement to treat indeterminacies: 0 / · ln 0 = 0, a · ln 0 = −∞ if a > Lemma 1.
The following statements hold true for each t ≥ .(a) Consider the G − t +1 -measurable vectors e Y t +1 in R N with the components e Y nt +1 ( χ ) = Y nt +1 ( ω ) I( ∃ β ∈ B t ( χ ) : β n > , and the functions g i ( x ) = 1 i + i arctan (cid:16) xi (cid:17) , x ∈ R + , i = 1 , , . . . Then there exist G − t -measurable functions b α t,i such that for all χ ∈ Θ ′ t b α t,i ∈ argmax α ∈ A p t (cid:8) E t ln g i ( h α, X t +1 i W t + | e Y t +1 | ) − h e, α i (cid:9) . (8) (b) There exists an increasing sequence of G − t -measurable functions i j ( χ ) , j ≥ ,with positive integer values, and a G − t -measurable function b α t with values in A p t , suchthat on the set Θ ′ t b α t = lim j →∞ b α t,i j . (c) The set e B t = { β ∈ B t ( χ ) : | β | = 1 − h e, b α t ( χ ) i} is non-empty for χ ∈ Θ ′ t andthere exists a G − t -measurable function b β t with values in B t such that for any χ ∈ Θ ′ t b β t ∈ argmax β ∈ e B t (cid:26) E t h ln β, e Y t +1 ih b α t , X t +1 i W t + | e Y t +1 | (cid:27) . (9) Theorem 1.
In every feasible strategy profile, any strategy b h = ( b α , b β ) constructed asin Lemma 1 is relative growth optimal. Note that Lemma 1 defines b α t , b β t not necessarily in a unique way (hence, we write“any strategy b h ” in the theorem). This may be so if, for example, some of the vectors X t have linearly dependent components.Let us show that the strategy b h can be found as an equilibrium strategy of therepresentative agent who holds a growth optimal portfolio in a market with N + N exogenous assets, where the first N assets are the same as in the original market, and theremaining N assets are treated as exogenous with the prices being equal to the prices ofthe endogenous assets induced by b h in the original market. This notion of equilibrium12s conceptually similar to the one in the Lucas model of an exchange economy (Lucas,1978) with the logarithmic utility, though we do not consider consumption.Recall that in a market with exogenous prices a strategy with value b v t > v t ≥ v t / b v t is a supermartingale. If z t = ( v t − v t − ) /v t − and b z t = ( b v t − b v t − ) / b v t − denote the one-period returns on the strategies’ portfolios, thenthis supermartingality condition is equivalent to that for each t ≥ t z t +1 b z t +1 ≤ . (10)Let b p t denote the endogenous prices that would clear the market if all the agentsused the strategy b h , i.e. b p nt = b β nt W t . Let Z t denote the returns on the endogenous assets in this case, Z nt +1 = Y nt +1 b p nt . Consequently, the return on a portfolio ( α t , β t ) would be h α t , X t +1 i − h β t , Z t +1 i . (11) Theorem 2.
For any t ≥ , χ ∈ Θ ′ t , and ( α, β ) ∈ C t ( χ ) , we have ( cf. (10) , (11))E t h α, X t +1 i + h β, Z t +1 ih b α t , X t +1 i + h b β t , Z t +1 i ≤ . (12) If E t | ln( h b α t , X t +1 i + h b β t , Z t +1 i ) | < ∞ , then ( b α t , b β t ) ∈ argmax ( α,β ) ∈ C t E t ln( h α, X t +1 i + h β, Z t +1 i ) . (13)Relation (12) expresses the above-mentioned idea of equilibrium, and relation (13)is an analogue of the well-known fact that a num´eraire portfolio maximizes one-periodexpected log-returns, under the respective integrability condition. Remark 2.
Let us comment on technical aspects of the above results. Why in Lemma 1do we introduce the functions g i and consider maximization problem (8)? Actually, wewould like to find a strategy ( b α , b β ) such that b α t maximizes E t ln( h α, X t +1 i W t + | Y t +1 | ) − h e, α i over α ∈ A t , (14)and, for this b α t , to define the component b β t as in (9). This strategy would satisfyinequalities (19) and (26), which play the key role in the proofs.But it may be not possible to define b α t in this way, since problem (14) may haveno solution. For this reason, we find the solutions b α t,i of the maximization problemstruncated by the functions g i and select a convergent subsequence. Then inequalities(19), (26) still remain satisfied. To ensure that such a subsequence exists, we use theobservation that it is possible to maximize not over the whole set A t but over its compactsubset A p t . We also replace Y t with e Y t to avoid the situation when an asset yields apositive payoff with positive conditional probability, but it is not possible to invest in it.13ote that when no portfolio constraints are imposed on the endogenous assets, i.e. B t = { β ∈ R N + : | β | ≤ } and hence e Y t +1 = Y t +1 , we can find b β t explicitly: b β nt = E t Y nt +1 h b α t , X t +1 i W t + | Y t +1 | (15)(this formula will be used in Section 3.3). Indeed, for b β t defined by (15), we have | b β t | = 1 − h e, b α t i as follows from equality (20) below, and for any β ∈ R N + with | β | = 1 − h e, b α t i we haveE t h ln β, Y t +1 ih b α t , X t +1 i W t + | Y t +1 | = h ln β, b β t i ≤ h ln b β t , b β t i , so b β t indeed delivers the maximum in (9). The inequality here follows from Gibb’sinequality (see Proposition 8 below).To conclude this section, let us show how the above theorems generalize knownresults on asymptotically optimal strategies. An immediate corollary from Theorem 2is that in a market with only exogenous assets the strategy b α t is a num´eraire portfolio.Note that in this case b α t depends only on ω , but not on market history (assuming thatthe constraints set C t also depend only on ω ).In a market with only endogenous assets and no portfolio constraints, as follows from(15), the optimal strategy is given by b β nt = E t Y nt +1 | Y t +1 | (note that again we have the dependence on ω only). This strategy was obtained byAmir et al. (2013); see also the earlier results of Amir et al. (2005); Evstigneev et al.(2002); Hens and Schenk-Hopp´e (2005) for models with short-lived assets which imposeadditional assumptions on admissible strategies or on asset payoffs.Finally, suppose that there is only one exogenous asset, short sales of this asset arenot allowed, and there are no other portfolio constraints, i.e. C t = R + × R N + . Then b α t ( χ ) is defined as follows: if E t ( X t +1 W t / | Y t +1 | ) ≤
1, then b α t = 0; otherwise b α t is theunique solution of the equation E t X t +1 W t α + | Y t +1 | = 1 . This can be seen from relations (19)–(20) below. Indeed, if E t ( X t +1 W t / | Y t +1 | ) ≤ b α t = 0. In the case E t ( X t +1 W t / | Y t +1 | ) > b α t = 0, then(19) cannot hold true for α >
0, hence we are left only with the non-zero solution.After b α t has been defined as above, the component b β t can be found from (15), whichgives b β nt = E t Y nt +1 b α t X t +1 W t + | Y t +1 | . This strategy was obtained by Drokin and Zhitlukhin (2020) in the case when the se-quence X t is predictable (e.g. the exogenous asset is a risk-free bond or cash); seeZhitlukhin (2019, 2020) for its extensions to continuous time.14 .3. Asymptotic proximity of survival strategies In this section we investigate evolution of relative wealth of strategies different from b h .The theorem below will be stated for the case when there are no portfolio constraintson the endogenous assets ( B t = { β ∈ R N + : | β | ≤ } ). This assumption is necessarybecause the proof relies on the explicit form of b β t given by (15).Given a feasible strategy profile and a vector of initial wealth, by ¯ h = ( ¯ α, ¯ β ) we willdenote the realization of the representative strategy of all the agents, which we defineas the weighted sum of their strategies with r mt as the weights:¯ α t = M X m =1 r mt α mt , ¯ β t = M X m =1 r mt β mt , where α t , β t , r t are the corresponding realizations. In a similar way, by e h = ( e α, e β ) we willdenote the realization of the representative strategy of agents m = 2 , . . . , M weightedwith their relative wealths excluding agent 1: e α t = M X m =2 r mt − r t α mt , e β t = M X m =2 r mt − r t β mt , where 0 / Theorem 3.
Suppose B t = { β ∈ R N + : | β | ≤ } , and agent 1 uses the strategy h = b h .Considering the realizations of the strategies, the wealth sequences, and the constraintssets, let Q t +1 ( ω ) = max α ∈ A t ( ω ) h α, X t +1 ( ω ) i + | Y t +1 ( ω ) | W t ( ω ) . Then, with probability 1, ∞ X t =0 (cid:18) h α t − ¯ α t , X t +1 i Q t +1 (cid:19) + k β t − ¯ β t k < ∞ , (16) and lim t →∞ r t = 1 on the set (cid:26) ω : ∞ X t =0 (cid:18) h α t − e α t , X t +1 i Q t +1 (cid:19) + k β t − e β t k = ∞ (cid:27) . (17)Note that the maximum in the definition of Q t +1 is attained because, according toProposition 1, it can be taken over the compact set A p t ( ω ). Furthermore, Q t +1 > b h then this agentasymptotically determines the representative strategy of the market so that ¯ h t becomesclose to b h t in the sense that the series in (16) converges, and, consequently, h α t − ¯ α t , X t +1 i Q t +1 → , β t − ¯ β t → t → ∞ . Relation (17) provides a sufficient condition for an agent using the strategy b h todominate in the market, which happens when the realization of the representative strat-egy of the other agents is asymptotically different from the realization of b h in the sense15hat the series in (17) diverges. From here, we also get a necessary condition for astrategy to be survival. Indeed, a survival strategy must survive against b h , so if agents m = 1 , . . . , M − b h , and hence can be considered as a single agent, theremaining agent m = M has to use a strategy with a realization close to b h in the senseof (17).Another corollary from Theorem 3 is that the presence of an agent who uses thestrategy b h asymptotically determines the relative prices ρ nt = p nt /W t of the endogenousassets. It is not difficult to see that ρ nt = ¯ β nt , and hence (16) implies that for each n and t → ∞ we have b β nt − ρ nt →
4. Proofs of the main results
In this section we provide several results from the theory of random sets which will beused in the proofs for dealing with portfolio constraints.By a random set (or a measurable correspondence ) in R N defined on a measurablespace ( S, S ) we call a set-valued function φ : S → R N such that for any open set A ⊆ R N it holds that φ − ( A ) ∈ S , where φ − ( A ) = { s : φ ( s ) ∩ A = ∅} is the lower inverse of A . An equivalent definition is that the distance function d ( x, φ ( s )) is S -measurable forany x ∈ R N (where d ( x, ∅ ) = ∞ ). In what follows, the role of ( S, S ) will be played by(Ω , F t ), (Θ , G t ), or (Θ , G − t ).A random set is called closed (respectively, compact, non-empty) if φ ( s ) is closed(compact, non-empty) for any s ∈ S . A measurable selector is an S -measurable func-tion ξ such that ξ ( s ) ∈ φ ( s ) for any s . A function f ( s, x ) : S × R N → R is called a Carath´eodory function if it is measurable in s and continuous in x .The following results are known for random sets in R N . Proposition 3. If φ n , n = 1 , , . . . , are random sets, then ∪ n φ n is a random set; if φ n are also closed, then ∩ n φ n is a closed random set. Proposition 4 (Filippov’s theorem) . Suppose φ is a non-empty compact random set, f is a Carath´eodory function, and π is a measurable function. Then the correspondence ψ ( s ) = { x ∈ φ ( s ) : f ( s, x ) = π ( s ) } is measurable and compact. Moreover, if ψ is non-empty, then it has a measurableselector ξ , and hence f ( s, ξ ( s )) = π ( s ) . Proposition 5 (Measurable maximum theorem) . For a non-empty compact random set φ and a Carath´eodory function f , let µ be the maximum function and ψ be the argmaxcorrespondence defined by µ ( s ) = max x ∈ φ ( s ) f ( s, x ) , ψ ( s ) = argmax x ∈ φ ( s ) f ( s, x ) . Then µ is measurable, and ψ is non-empty, compact, measurable, and has a measurableselector. Proofs of the above results can be found in the book of Aliprantis and Border (2006,Chapter 18) for random sets in general metric spaces, except the result about ∩ n φ n ,16hich holds (in a metric space) if φ n are compact. For R N , it can be extended to closedsets using that R N is σ -compact.For the reader’s convenience, the following results are provided with proofs (they arenot included in the above-mentioned book). Proposition 6.
Let L be a random linear subspace of R N (i.e. for each s the set L ( s ) is a linear space and the correspondence L is measurable), L ⊥ be the orthogonal space,and φ be a closed random set in R N . Then the projection correspondence pr L φ ( s ) = { x ∈ L ( s ) : ∃ y ∈ L ⊥ ( s ) such that x + y ∈ φ ( s ) } is measurable.Proof. By Castaing’s theorem (see Corollary 18.14 in Aliprantis and Border (2006)), anon-empty closed correspondence is measurable if and only if it can be represented asthe closure of a countable family of measurable selectors from it. Hence, we can findmeasurable ξ i such that φ ( s ) = cl { ξ i ( s ) , i ≥ } on the set { s : φ ( s ) = ∅} . Using thatcl(pr L φ ( s )) = ( cl { pr L ξ i ( s ) , i ≥ } , if φ ( s ) = ∅ , ∅ , if φ ( s ) = ∅ , one can see that cl(pr L φ ) is measurable. Since the measurability of a correspondence isequivalent to the measurability of its closure (Aliprantis and Border, 2006, Lemma 18.3),pr L φ is measurable. Proposition 7.
Let φ be a non-empty compact random set and ξ n be a sequence ofmeasurable selectors from it. Then there exists a measurable selector ξ from φ and asequence of measurable functions ≤ i ( s ) < i ( s ) < . . . with integer values such that lim j →∞ ξ i j ( s ) ( s ) = ξ ( s ) for all s .Proof. The set ψ ( s ) = ∩ n cl { ξ k ( s ) , k ≥ n } is measurable, non-empty, and closed, sothere exists a measurable selector ξ ∈ ψ (by Castaing’s theorem mentioned above).Then the sequence i j can be constructed by induction as follows. Put i = 1. If i j isdefined, consider the random set η j ( s ) = { k > i j ( s ) : | ξ k ( s ) − ξ ( s ) | ≤ j − } ⊂ N , whichis measurable, non-empty, and closed. Let i j +1 be a measurable selector from η j . Then | ξ i j +1 − ξ | < j − , which gives the desired convergence. Proof of claim (a) . Fix any t ≥
0. Let f i ( χ, α ) be the function which is maximized inthe definition of b α t,i , i.e. on the set Θ ′ t put f i = E t ln g i ( h α, X t +1 i W t + | e Y t +1 | ) − h e, α i , while on the set Θ \ Θ ′ t put f i = 0. The function f i is a Carath´eodory function, and theset A p t , over which it is maximized, is compact by Proposition 1. Hence the measurablemaximum theorem implies the existence of a measurable selector b α t,i from the argmaxin (8). Proof of claim (b) readily follows from Proposition 7. Before we continue with theproof of claim (c), let us show that b α t satisfies a number of relations that will be usedin its proof, as well as in the proof of Theorem 1.17 emma 2. For any t ≥ , χ ∈ Θ ′ t , and α ∈ A t ( χ ) we have P t ( h b α t , X t +1 i W t + | e Y t +1 | >
0) = 1 , (18)E t (cid:18) h b α t − α, X t +1 i W t h b α t , X t +1 i W t + | e Y t +1 | (cid:19) ≥ h e, b α t − α i , (19)E t (cid:18) h b α t , X t +1 i W t h b α t , X t +1 i W t + | e Y t +1 | (cid:19) = h e, b α t i . (20) Proof.
Fix t, χ, α, i , and consider the function u ( ε ) = f i ((1 − ε ) b α t + εα ), ε ∈ [0 , u ( ε ) = E t ln g i (cid:0) h (1 − ε ) b α t,i + εα, X t +1 i W t + | e Y t +1 | (cid:1) ( χ ) − h e, (1 − ε ) b α t,i ( χ ) + εα i . (21)Since ln g i ( x ) is concave for x ≥
0, the function u ( ε ) is also concave. As it attainsthe maximum value at ε = 0, the right derivative u ′ (0) ≤
0. We want to interchangethe order of differentiation and taking the expectation. The expectation in (21) can bewritten as E t ln g i ( q ( e ω, ε )) = R Ω ln g i ( q ( e ω, ε )) K t ( ω, d e ω ) with q ( e ω, ε ) = h (1 − ε ) b α t,i ( e χ ) + εα, X t +1 ( e ω ) i W t ( e χ ) + | e Y t +1 ( e χ ) | , where e χ = ( e ω, v , h , . . . ) and v , ( h s ) s ≥ are taken from χ = ( ω, v , h , . . . ). By applyingFatou’s lemma, we obtain ( e ω is omitted for brevity)(E t ln g i ( q ( ε ))) ′ ε =0 ≥ E t g ′ i ( q (0)) g i ( q (0)) h α − b α t,i , X t +1 i W t . (22)Fatou’s lemma can be applied since for ε ∈ [0 ,
1) we have the lower bound (P t -a.s. in e ω )ln g i ( q ( ε )) − ln g i ( q (0)) ε ≥ (ln g i ( q ( ε ))) ′ = g ′ i ( q ( ε )) g i ( q ( ε )) h α − b α t,i , X t +1 i W t ≥ − ig ′ i ((1 − ε ) h b α t,i , X t +1 i W t ) h b α t,i , X t +1 i W t ≥ − i − ε . Here in the first inequality we used the concavity of ln g i ( q ( ε )). In the second inequalitywe used the relation P t ( h α, X t +1 i ≥
0) = 1, the bound g i ( x ) ≥ /i , and that g ′ i ( x ) isnon-increasing for x ≥
0. The last last inequality holds because g ′ i ( x ) x ≤ i .Therefore, from (21) and (22) we obtain0 ≥ u ′ (0) ≥ E t ( ξ i h α, X t +1 i W t ) − E t ( ξ i h b α t,i , X t +1 i W t ) − h e, α − b α t,i i , (23)where ξ i = g ′ i ( q (0)) g i ( q (0)) = g ′ i ( h b α t,i , X t +1 i W t + | e Y t +1 | ) g i ( h b α t,i , X t +1 i W t + | e Y t +1 | ) . One can see that for all x ≥ ≤ xg ′ i ( x ) g i ( x ) ≤ , lim i →∞ xg ′ i ( x ) g i ( x ) = I( x > . (24)The above inequality can be obtained by using that arctan( x/i ) ≥ x/ (2 i ) if x ≤ i andarctan( x/i ) ≥ π/ x ≥ i ; the computation of the limit is straightforward. Relations1824) allow to apply the dominated convergence theorem to the second expectation in(23), which giveslim i →∞ E t ( ξ i h b α t,i , X t +1 i W t ) = E t (cid:18) h b α t , X t +1 i W t h b α t , X t +1 i W t + | e Y t +1 | (cid:19) , (25)where at this point we assume 0 / e α such thatP t ( h e α, X t +1 i W t + | e Y t +1 | >
0) = 1. Applying Fatou’s lemma to the first expectation in(23) with α = e α , we find that (18) must hold, since otherwise we would havelim inf i →∞ E t ( ξ i h e α, X t +1 i W t ) = + ∞ , which contradicts (23). Consequently, for any α ∈ A t we havelim inf i →∞ E t ( ξ i h α, X t +1 i W t ) ≥ E t (cid:18) h α, X t +1 i W t h b α t , X t +1 i W t + | e Y t +1 | (cid:19) , which together with (23) and (25) implies (19).Let us prove (20). If h e, b α t ( χ ) i = 1, it clearly follows from (19) with α = 0. If h e, b α t ( χ ) i <
1, we can consider small ε > α in (19) α ( ± ε ) := (1 ± ε ) b α t ( χ ) ∈ A t ( χ ) , which gives (20) after simple transformations. Proof of claim (c) of Lemma 1 . If B t ( χ ) = { } , then e B t ( χ ) = ∅ in view of (6). If B t ( χ ) = { } , then e Y t ( χ ) = 0, and (20) implies that h e, b α t ( χ ) i = 1, so e B t ( χ ) = { } isnon-empty again.Let f ( χ, β ) denote the function being maximized in (9): f ( χ, β ) = N X n =1 ln β n E t (cid:18) e Y nt +1 h b α t , X t +1 i W t + | e Y t +1 | (cid:19) ( χ ) . The function f may be discontinuous in β . In order to apply the measurable maximumtheorem, let us take G t − -measurable e β ( χ ) ∈ B t ( χ ) such that | e β ( χ ) | = 1 − b α t ( χ ) and e β n ( χ ) > t ( e Y nt +1 > ω ) >
0. Then we can consider the function e f ( χ, β ) = max( f ( χ, β ) , f ( χ, e β ( χ ))) , which is a Carath´eodory function and satisfies the relationargmax β ∈ e B t f ( χ, β ) = argmax β ∈ e B t e f ( χ, β ) . Hence the measurable maximum theorem can be applied to e f , giving b β t which alsomaximizes f . 19 .3. Proofs of Theorems 1 and 2 Let us prove two more inequalities which together with (19) will be used in the proofs.
Lemma 3.
For any t ≥ , χ ∈ Θ ′ t , and β ∈ B t ( χ ) we have E t (cid:18) h ln b β t − ln β, e Y t +1 ih b α t , X t +1 i W t + | e Y t +1 | (cid:19) ≥ | b β t | − | β | , (26)E t | e Y t +1 | − P n β n e Y nt +1 / b β nt h b α t , X t +1 i W t + | e Y t +1 | ≥ | b β t | − | β | , (27) where in (27) we let β n e Y nt +1 ( χ ) / b β nt ( χ ) = 0 if b β nt ( χ ) = 0 ( then P t ( e Y nt +1 = 0)( χ ) = 1 asfollows from (9)) .Proof. Clearly, (26) holds if | β | = | b β t ( χ ) | , as follows from the definition of b β t . If | β | 6 = | b β t ( χ ) | , we haveE t (cid:18) h ln b β t − ln β, e Y t +1 ih b α t , X t +1 i W t + | e Y t +1 | (cid:19) ≥ E t (cid:18) | e Y t +1 | ln( | b β t | / | β | ) h b α t , X t +1 i W t + | e Y t +1 | (cid:19) ≥ E t (cid:18) | e Y t +1 |h b α t , X t +1 i W t + | e Y t +1 | (cid:19) | b β t | − | β || b β t | = | b β t | − | β | , where in the first inequality we represented ln β = ln( β | b β t | / | β | ) − ln( | b β t | / | β | ) and applied(26) to β | b β t | / | β | instead of β ; in the second inequality we used the estimate ln a ≥ − a − ;and in the equality applied (20). This proves (26).To prove (27), observe that the function f ( ε ) = E t (cid:18) h ln((1 − ε ) b β t + εβ ) , e Y t +1 ih b α t , X t +1 i W t + | e Y t +1 | (cid:19) − | (1 − ε ) b β t + εβ t | , ε ∈ [0 , , attains its maximum at ε = 0 and is differentiable on [0 , f ′ (0) = N X n =1 ( β n − b β nt ) e Y nt +1 b β nt ( h b α t , X t +1 i W t + | e Y t +1 | ) + | b β t | − | β | should be non-positive, which gives (27) (here, the n -th term in the sum is treated aszero when b β nt ( χ ) = 0, and hence e Y nt +1 = 0). Proof of Theorem 1.
Assume that the strategy b h is used by agent m = 1. Let us fix theinitial wealth and the strategies of the other agents, and pass on to a realization of thestrategies h mt = ( α mt , β mt ), wealth v mt , and relative wealth r mt as functions of ω only. Innotation for agent 1, we will also use the hat instead of the superscript “1”, i.e. b α = α , b β = β , etc.Introduce the predictable sequence of random vectors F t ∈ R N + with the components F nt = b β nt P m r mt β m,nt , where 0 / b v t +1 = (cid:18) h b α t , X t +1 i + h F t , e Y t +1 i W t (cid:19)b v t , W t +1 = (cid:18) M X m =1 r mt h α mt , X t +1 i + | e Y t +1 | W t (cid:19) W t . b r t +1 − ln b r t = f t ( X t +1 , e Y t +1 ), where f t = f t ( ω, x, y ) is the F t ⊗ B ( R N )-measurable function f t ( x, y ) = ln (cid:18) h b α t , x i W t + h F t , y i W t P m r mt h α mt , x i + | y | (cid:19) (the argument ω is omitted for brevity).We need to show that E t f t ( X t +1 , e Y t +1 ) ≥
0. Rewrite the function f t ( x, y ) as f t ( x, y ) = ln (cid:18) h b α t , x i W t + | y | W t P m r mt h α mt , x i + | y | (cid:19) + ln (cid:18) h b α t , x i W t + h F t , y ih b α t , x i W t + | y | (cid:19) := f (1) t ( x, y ) + f (2) t ( x, y ) . (28)For the first term, we can use the inequality ln x ≥ − x − and apply (19), which givesE t f (1) t ( X t +1 , e Y t +1 ) ≥ E t h b α t − P m r mt α mt , X t +1 i W t h b α t , X t +1 i W t + | e Y t +1 | ≥ (cid:28) e, b α t − M X m =1 r mt α mt (cid:29) . (29)For the second term in (28), we haveE t f (2) t ( X t +1 , e Y t +1 ) ≥ E t h ln F t , e Y t +1 ih b α t , X t +1 i W t + | e Y t +1 | ≥ | b β t | − M X m =1 r mt | β mt | , (30)where the first inequality follows from the concavity of the logarithm, and the secondone follows from that ln F t = ln b β t − ln P m r mt β mt and inequality (26).Using that | β mt | + h e, α mt i = 1, we see that E t f t ( X t +1 , e Y t +1 ) ≥
0, hence E t ln b r t +1 ≥ ln b r t . Since ln b r t is a non-positive sequence, this inequality also implies the integrabilityof ln b r t (by induction, beginning with ln b r ), so it is a submartingale. Proof of Theorem 2.
When all the agents use b h , from (2) we find p nt = b β nt W t , andhence h b β t , Z t +1 i = e Y t +1 / W t . Adding (19) and (27), we obtain (12). Then (13) followsby Jensen’s inequality. We will need the following proposition which provides two inequalities of a generalnature.
Proposition 8.
1) For any a, b ∈ (0 , a + b − ln a + ln b ≥ ( a − b ) . (31)
2) Suppose x, y ∈ R N + are two vectors such that | x | ≤ , | y | ≤ , and for each n it holdsthat if y n = 0 , then also x n = 0 . Then h x, ln x − ln y i ≥ k x − y k | x | − | y | . (32) Proof.
1) Assume a ≤ b . The inequality clearly holds if a = b . Let f ( a ) be the differenceof its left-hand side and right-hand side, with b fixed. It is enough to show that f ′ ( a ) ≤ a ∈ (0 , b ]. After differentiation, this becomes equivalent to a ( a + b ) ≤
2. The latterinequality is clearly true, provided that a, b ∈ (0 , x/ | x | and y/ | y | are considered as probability distributions on a set of N elements.Its short direct proof can be found in Drokin and Zhitlukhin (2020, Lemma 2).21 roof of Theorem 3. We will use the same notation for realizations of strategies as inthe proof of Theorem 1. It was shown that ln b r t is a submartingale. Let c t be its com-pensator, i.e. the predictable non-decreasing sequence such that ln b r t − c t is a martingale;in the explicit form c t = X s ≤ t (E s − ln b r s − ln b r s − ) . As was shown in the proof of Theorem 1, c t +1 − c t = E t f t ( X t +1 , Y t +1 ) = E t ( f (1) t ( X t +1 , Y t +1 ) + f (2) t ( X t +1 , Y t +1 ))with f (1) , f (2) defined in (28). Since ln b r t is non-positive and converges, we have c ∞ < ∞ with probability 1. Let us consider again inequalities (29)–(30) and strengthen themusing Proposition 8. Fix t ≥ a = h b α t , X t +1 i + | Y t +1 | /W t Q t +1 , b = h ¯ α t , X t +1 i + | Y t +1 | /W t Q t +1 . Note that a, b ∈ (0 , t f (1) t ( X t +1 , Y t +1 ) = 2 E t (cid:18) ln a − ln a + ln b (cid:19) ≥ t (cid:18) ln a + b − ln a + ln b (cid:19) + h e, b α t − ¯ α t i≥ (cid:18) h b α − ¯ α, X t +1 i Q t +1 (cid:19) + h e, b α t − ¯ α t i . (33)Here, in the first inequality we used the estimateE t (cid:18) ln a − ln a + b (cid:19) ≥
12 E t h b α t − ¯ α t , X t +1 i W t h b α t , X t +1 i W t + | Y t +1 | ≥ h e, b α t − ¯ α t i , which is obtained similarly to (29). In the second inequality of (33) we applied (31).For the function f (2) , using that there are no portfolio constraints on the endogenousassets, so b β t is given by (15), we findE t f (2) t ( X t +1 , Y t +1 ) ≥ E t h ln F t , Y t +1 ih b α t , X t +1 i W t + | Y t +1 | = h ln F t , b β t i = h b β t , ln b β t − ln ¯ β t i≥ k b β t − ¯ β t k | b β t | − | ¯ β t | , (34)where the first inequality is obtained similarly to (30), and in the second one we ap-plied (32). Consequently, from (33), (34), we obtain c t +1 − c t ≥ (cid:18) h b α t − ¯ α t , X t +1 i Q t +1 (cid:19) + k b β t − ¯ β t k . From here, using that c ∞ < ∞ , we get (16). Moreover, b α t − ¯ α t = (1 − b r t )( b α t − e α t ) and b β t − ¯ β t = (1 − b r t )( b β t − e β t ), so on the set (17) we necessarily have lim t →∞ b r t = 1.22 eferences Algoet, P. H. and Cover, T. M. (1988). Asymptotic optimality and asymptotic equipartitionproperties of log-optimum investment.
The Annals of Probability , 16(2):876–898.Aliprantis, C. D. and Border, K. C. (2006).
Infinite Dimensional Analysis: A Hitchhiker’s Guide .Springer, 3rd edition.Amir, R., Evstigneev, I. V., Hens, T., Potapova, V., and Schenk-Hopp´e, K. R. (2020). Evolutionin pecunia.
Swiss Finance Institute Research Paper , pages 20–44.Amir, R., Evstigneev, I. V., Hens, T., and Schenk-Hopp´e, K. R. (2005). Market selection andsurvival of investment strategies.
Journal of Mathematical Economics , 41(1-2):105–122.Amir, R., Evstigneev, I. V., Hens, T., and Xu, L. (2011). Evolutionary finance and dynamicgames.
Mathematics and Financial Economics , 5(3):161–184.Amir, R., Evstigneev, I. V., and Schenk-Hopp´e, K. R. (2013). Asset market games of survival:a synthesis of evolutionary and dynamic games.
Annals of Finance , 9(2):121–144.Babaei, E., Evstigneev, I. V., Schenk-Hopp´e, K. R., and Zhitlukhin, M. (2020a). Von Neumann–Gale dynamics and capital growth in financial markets with frictions.
Mathematics and Fi-nancial Economics , 14(2):283–305.Babaei, E., Evstigneev, I. V., Schenk-Hopp´e, K. R., and Zhitlukhin, M. (2020b). Von neumann–gale model, market frictions and capital growth.
Stochastics , pages 1–32 (published online).Belkov, S., Evstigneev, I. V., and Hens, T. (2020). An evolutionary finance model with a risk-freeasset.
Annals of Finance , pages 1–15.Blume, L. and Easley, D. (1992). Evolution and market behavior.
Journal of Economic Theory ,58(1):9–40.Blume, L. and Easley, D. (2006). If you’re so smart, why aren’t you rich? Belief selection incomplete and incomplete markets.
Econometrica , 74(4):929–966.Boroviˇcka, J. (2020). Survival and long-run dynamics with heterogeneous beliefs under recursivepreferences.
Journal of Political Economy , 128(1):206–251.Bottazzi, G. and Dindo, P. (2014). Evolution and market behavior with endogenous investmentrules.
Journal of Economic Dynamics and Control , 48:121–146.Bottazzi, G., Dindo, P., and Giachini, D. (2018). Long-run heterogeneity in an exchange economywith fixed-mix traders.
Economic Theory , 66(2):407–447.Breiman, L. (1961). Optimal gambling systems for favorable games. In
Proceedings of the 4thBerkeley Symposium on Mathematical Statistics and Probability , volume 1, pages 63–68.Cover, T. M. and Thomas, J. A. (2012).
Elements of Information Theory . John Wiley & Sons,2nd edition.Drokin, Y. and Zhitlukhin, M. (2020). Relative growth optimal strategies in an asset marketgame.
Annals of Finance , 16:529–546.Evstigneev, I., Hens, T., and Schenk-Hopp´e, K. R. (2016). Evolutionary behavioral finance. InHaven, E. et al., editors,
The Handbook of Post Crisis Financial Modelling , pages 214–234.Palgrave Macmillan UK.Evstigneev, I. V., Hens, T., Potapova, V., and Schenk-Hopp´e, K. R. (2020). Behavioral equi-librium and evolutionary dynamics in asset markets.
Journal of Mathematical Economics ,91:121–135.Evstigneev, I. V., Hens, T., and Schenk-Hopp´e, K. R. (2002). Market selection of financialtrading strategies: Global stability.
Mathematical Finance , 12(4):329–339.Evstigneev, I. V., Hens, T., and Schenk-Hopp´e, K. R. (2006). Evolutionary stable stock markets.
Economic Theory , 27(2):449–468.Evstigneev, I. V. and Zhitlukhin, M. V. (2013). Controlled random fields, von neumann–galedynamics and multimarket hedging with risk.
Stochastics , 85(4):652–666.Hakansson, N. H. and Ziemba, W. T. (1995). Capital growth theory. In
Finance , volume 9 of
Handbooks in Operations Research and Management Science , pages 65–86. Elsevier. ens, T. and Schenk-Hopp´e, K. R. (2005). Evolutionary stability of portfolio rules in incompletemarkets. Journal of Mathematical Economics , 41(1-2):43–66.Holtfort, T. (2019). From standard to evolutionary finance: a literature survey.
ManagementReview Quarterly , 69(2):207–232.Karatzas, I. and Kardaras, C. (2007). The num´eraire portfolio in semimartingale financialmodels.
Finance and Stochastics , 11(4):447–493.Kelly, Jr, J. L. (1956). A new interpretation of information rate.
Bell System Technical Journal ,35(4):917–926.Latan´e, H. A. (1959). Criteria for choice among risky ventures.
Journal of Political Economy ,67(2):144–155.Lucas, Jr, R. E. (1978). Asset prices in an exchange economy.
Econometrica , 46(6):1429–1445.Sandroni, A. (2000). Do markets favor agents able to make accurate predictions?
Econometrica ,68(6):1303–1341.Shiryaev, A. N. (2019).
Probability–2 . Springer-Verlag, New York, 3rd edition.Yan, H. (2008). Natural selection in financial markets: Does it work?
Management Science ,54(11):1935–1950.Zhitlukhin, M. (2019). Survival investment strategies in a continuous-time market model withcompetition. arXiv:1811.12491v2, to appear in International Journal of Theoretical and Ap-plied Finance (2021) .Zhitlukhin, M. (2020). A continuous-time asset market game with short-lived assets. arXiv:2008.13230 ..