A continuous-time asset market game with short-lived assets
aa r X i v : . [ q -f i n . M F ] A ug A continuous-time asset market game with short-lived assets
Mikhail Zhitlukhin ∗
30 August 2020
Abstract
We consider a continuous-time game-theoretic model of an investment marketwith short-lived assets and endogenous asset prices. The first goal of the paper isto formulate a stochastic equation which determines wealth processes of investorsand to provide conditions for the existence of its solution. The second goal is toshow that there exists a strategy such that the logarithm of the relative wealth ofan investor who uses it is a submartingale regardless of the strategies of the otherinvestors, and the relative wealth of any other essentially different strategy vanishesasymptotically. This strategy can be considered as an optimal growth portfolio inthe model.
Keywords: asset market game, relative growth optimal strategy, martingale con-vergence, evolutionary finance.
MSC 2010:
JEL Classification:
C73, G11.
1. Introduction
This paper proposes a dynamic game-theoretic model of an investment market – an assetmarket game – and study strategies that allow an investor to achieve faster growth ofwealth compared to rival market participants. The model provides an outlook on growthoptimal portfolios different from the well-known theory in a single-investor setting, whichoriginated with Kelly [18] and Breiman [7] (see also [1, 16, 22] for a modern expositionof the subject). Our results belong to the strand of research on evolutionary finance –the field which studies financial markets from a point of view of evolutionary dynamicsand investigates properties of investment strategies like survival, extinction, dominance,and how they affect the structure of a market. Reviews of recent results in this directioncan be found in, e.g., [10, 13]. While the majority of models in evolutionary financeare discrete-time, the novelty and one of the goals of this paper consists in developing acontinuous-time model.The model considered here describes a market consisting of several assets and in-vestors. The assets yield random payoffs which are divided between the investors pro-portionally to the number of shares of each asset held by an investor. One feature ofthe model, which makes it different from the classical optimal growth theory, is thatthe asset prices are determined endogenously by a short-run equilibrium of supply anddemand and depend on the investors’ strategies. As a result, the investor’s wealth de-pends not only on their own strategies and realized assets’ payoffs but also on strategiesof the other investors in the market. ∗ Steklov Mathematical Institute of the Russian Academy of Sciences. 8 Gubkina St., Moscow, Russia.Email: [email protected]. The research was supported by the Russian Science Foundation, projectno. 18-71-10097. short-lived assets. Such assets can be viewed as short-term investment projectsrather than, e.g., common stock – they are traded at time t , yield payoffs at the “nextinfinitesimal” moment of time, and then get replaced with new assets. Short-livedassets have no liquidation value, so investors can get profit (or loss) only from receivingasset payoffs and paying for buying new assets. Despite being a simplification of realstock markets, models with short-lived assets have been widely studied in the literaturebecause they are more amenable to mathematical analysis and ideas developed for themmay be transferred to advanced models (see a discussion in [10]).This paper is tightly connected with paper [9], which considers the same model indiscrete time. Regarding mathematical methods, both of the papers are based on theapproach proposed by Amir et al. [4], which directly shows (in discrete time) that thelogarithm of wealth of an investor who uses the optimal strategy is a submartingaleregardless of the other investors’ strategies (see also the paper of Amir et al. [3] wheresimilar but technically more involved ideas were used for a model with long-lived assets).Then, using martingale convergence theorems, we can obtain results about the asymp-totic structure of a market. This martingale approach is more general compared tomethods used in earlier works, which were based on assumptions that payoff sequencesand/or strategies are stationary (as in, e.g., [11, 12]). An essential difference of ourmodel and the model of [4] (in addition to that we consider a continuous-time model)2s that Amir et al. assume that market agents spend their whole wealth for purchasingassets in each time period, so the total market wealth is always equal the most recenttotal payoff of the assets. On the other hand, our model includes a risk-free asset (cashor a bank account with zero interest rate) that can be used by investors to store capi-tal. This leads to more complicated wealth dynamics, but is necessary for considerationof a continuous-time model, where asset payoffs can be infinitesimal but yielded in acontinuous way. Moreover, adding the possibility to store capital in cash opens inter-esting questions about the asymptotics of the total market wealth, which do not arisein models where the whole wealth is spend for purchasing assets with random payoffs.For example, as was observed in [9], greater uncertainty in asset payoffs may result infaster growth of investors’ wealth – a fact which at first may seem counter-intuitive. Inthe present paper, we consider similar questions for the continuous-time model.In evolutionary finance, there are few models with continuous time. One can mentionthe papers of Palczewski and Schenk-Hopp´e [20, 21], in which a continuous-time modelwith long-lived assets is constructed. The paper [20] proves that the model can be ob-tained as a limit of discrete-time models, and [21] investigates questions of survival of in-vestments strategies in it. However, their results are obtained only for time-independentstrategies and under the assumption that cumulative dividend processes are pathwiseabsolutely continuous. In the present paper, we allow strategies to be time-dependentand asset payoffs to be represented by arbitrary processes. A continuous-time modelwith short-lived assets was also constructed in [24]. An essential limitation of that paperconsists in the assumption that all investors spend the same proportion of their wealthfor purchasing assets which is specified exogenously. This makes the mathematical anal-ysis of the model considerably simpler compared to the present paper, both in showingthat the wealth process is well-defined, and in construction of the optimal strategy.The paper is organized as follows. In Section 2, we briefly describe a discrete-time model, which helps to explain the main ideas of the paper. The general model isformulated in Section 3. In Section 4, we define the notion of optimality of a strategyand construct a candidate optimal strategy. In Section 5, we formulate the main results,which state that this strategy is indeed optimal, and investigate some of its properties.Section 6 contains the proofs of the results. In the appendix, we formulate and proveseveral auxiliary facts about the Lebesgue decomposition and Lebesgue derivatives ofnon-decreasing random functions which are used in the paper.
2. Preliminary consideration: a discrete-time model
In this section, we describe the main ideas of the paper using a simple model withdiscrete time which avoids technical details of continuous time. Based on the discrete-time model, in Section 3 a general continuous-time model will be formulated. The modelpresented here is a slightly simplified version of the model from [9].Let (Ω , F , P) be a probability space with a filtration F = ( F t ) ∞ t =0 . The model includes M ≥ N ≥ t = 1 , , . . . The assets live for one period: they are purchased by the investors attime t , yield payoffs at t + 1, and then the cycle repeats. The asset prices are determinedendogenously by a short-run equilibrium of supply and demand. The supply of each assetis normalized to 1, and the demand depends on actions of the investors. The payoffs arespecified in an exogenous way, i.e. do not depend on the investor’s actions. Each investorreceives a part of a payoff yielded by an asset which is proportional to the owned shareof this asset. 3he asset payoffs are specified by random sequences A nt ≥ m is described by an adapted random sequence Y mt ≥
0. Theinitial wealth Y m of each investor is non-random and strictly positive. The wealth Y mt at subsequent moments of time t ≥ m is a plan according to which this investor allocates theavailable budget Y mt towards a purchase of assets. Such an allocation is specified bya sequence of vectors l mt = ( l m, t , . . . , l m,Nt ), where l m,nt is a budget allocated towards apurchase of asset n at time t −
1. At each moment of time, the vectors l mt are selected bythe investors simultaneously and independently, so the model represents a simultaneous-move N -person dynamic game, and l mt represent the investors’ actions. These actionsmay depend on a random outcome ω and current and past wealth of the investors, so wedefine a strategy l m of investor m as a sequence of F t − ⊗ B ( R tM + )-measurable functions l mt ( ω, y , . . . , y t − ) : Ω × R tM + → R N + , t = 1 , , . . . . (We will use boldface letters to distinguish between strategies and their realizations, seebelow.) The arguments y s = ( y s , . . . , y Ms ) ∈ R M + , s ≤ t −
1, correspond to the wealthof the investors at the past moments of time. It is assumed that short sales are notallowed, so l m,nt ≥
0, and it is not possible to borrow money, so P n l m,nt ≤ y mt − . Theamount of wealth y mt − − P n l m,nt is held in cash and carried forward to the next timeperiod.After selection of investment budgets l mt by the investors, the equilibrium asset prices p nt − are determined from the market clearing condition that the aggregate demand foreach asset is equal to the aggregate supply, which is assumed to be 1. At time t − m can buy x m,nt = l m,nt /p nt − units of asset n , so its price at time t − p nt − = P m l m,nt . If P m l m,nt = 0, i.e. no one buys asset n , we put p nt − = 0 and x m,nt = 0 for all m .Thus, investor m ’s portfolio between moments of time t − t consists of x m,nt units of asset n and c mt := y mt − − P n l m,nt units of cash. At a moment of time t , the totalpayoff received by this investor from the assets in the portfolio is equal to P n x m,nt A nt .In our model, the assets have no liquidation value, so the budgets used at time t − m ’s wealthis described by the adapted sequence Y mt which is defined by the recursive relation Y mt ( ω ) = Y mt − ( ω ) − N X n =1 l m,nt ( ω ) + N X n =1 l m,nt ( ω ) P k l k,nt ( ω ) A nt ( ω ) , t ≥ , (1)where l m,nt ( ω ) = l m,nt ( ω, Y , Y ( ω ) , . . . , Y t − ( ω )) are the realizations of the investors’strategies, with 0 / m , we define4he relative wealth as the adapted sequence r mt = Y mt P k Y kt . Our goal will be to identify a strategy such that the relative wealth of an investorwho uses it grows in the following sense: for any strategies of the other investors andany initial wealth, the sequence ln r mt is a submartingale (as a consequence, r mt will bea submartingale as well). Such a strategy will exhibit several asymptotic optimalityproperties, which we will consider in Sections 4 and 5.
3. The general model
In order to formulate a continuous-time counterpart of equation (1), observe that it canbe written in the following form:∆ Y mt ( ω ) = − N X n =1 ∆ L m,nt ( ω ) + N X n =1 ∆ L m,nt ( ω ) P k ∆ L k,nt ( ω ) ∆ X nt ( ω ) , (2)where L m,nt ( ω ) = t X s =1 l m,ns ( ω ) , X nt ( ω ) = t X s =1 A ns ( ω )are, respectively, the process of the cumulative wealth invested by investor m in as-set n and the cumulative payoff process of asset n . The symbol ∆ denotes a one-stepincrement, e.g. ∆ Y mt = Y mt − Y mt − .The form of equation (2) suggests that an analogous model with continuous time canbe obtained by considering continuous-time processes X t , Y t , L t and replacing one-stepincrements with infinitesimal increments, e.g. ∆ X t with dX t . Our next goal will be todefine such a model properly. The model we are about to formulate includes the abovediscrete-time model as a particular case, but we do not investigate convergence of thediscrete-time model to the general model. Notation.
We will work on a filtered probability space (Ω , F , F , P) with a continuous-time filtration F = ( F t ) t ∈ R + satisfying the usual assumptions. By P we will denote thepredictable σ -algebra on Ω × R + .As usual, equalities and inequalities for random variables are assumed to hold withprobability one. For random processes, an equality X = Y is understood to hold up toP-indistinguishability, i.e. P( ∃ t : X t = Y t ) = 0; in the same way we treat inequalities.Pathwise properties (continuity, monotonicity, etc.) are assumed to hold for all ω .For vectors x, y ∈ R N , by xy = P n x n y n we denote the scalar product, by | x | = P n | x n | the l -norm of a vector, and by k x k = √ xx the l -norm. For a scalar function f : R → R and a vector x the notation f ( x ) means the application of the function toeach coordinate of the vector, f ( x ) = ( f ( x ) , . . . , f ( x N )). If x ∈ R MN , we denote by x m the vector ( x m, , . . . , x m,N ) ∈ R N and by x · ,n the vector ( x ,n , . . . , x M,n ) ∈ R M . Themaximum of two numbers a, b is written as a ∨ b , and the minimum as a ∧ b .The notation ξ · G t is used for the integral of a process ξ with respect to a process G .In what follows, all the integrators are non-decreasing c`adl`ag processes, so the integralsare understood in the pathwise Lebesgue-Stieltjes sense ( f · G t ( ω ) = R t f s ( ω ) dG s ( ω )).If f, G are vector-valued, then f · G t = P n f n · G nt .5 .1. Payoff processes and investment strategies There are N ≥ M ≥ X t with values in R N + . Without loss of generality X = 0.A strategy of investor m is identified with a function L which represents the cumu-lative wealth invested in each asset and assuming values in R N + . In order to specify howa strategy may depend on the past history of the market, let ( D, D , ( D t ) t ≥ ) denotethe filtered measurable space consisting of the space D of non-negative c`adl`ag functions y : R + → R M + , the filtration D t = σ ( d u , u ≤ t ), where d u is the mapping d u ( y ) = y u for y ∈ D , and D = W t ≥ D t . Elements y of the space D represent possible paths ofthe wealth processes of the investors (which are yet to be defined) on the whole timeaxis R + . The wealth of each investor cannot become negative (this assumption will beimposed on a solution of the wealth equation in the next section), hence y assume valuesin R N + .Let ( E, E , ( E t ) t ≥ ) be the filtered measurable space with E = Ω × D, E t = F t ⊗ D t , E = _ t ≥ E t . Let P E denote the predictable σ -algebra on E × R + , i.e. P E is generated by all mea-surable functions ξ ( ω, y, t ) : E × R + → R which are left-continuous in t for any fixed( ω, y ) and E t -measurable for any fixed t . In what follows, functions ξ ( ω, y, t ) will beoften written as ξ t ( ω, y ), or ξ t ( y ) when omitting ω does not lead to confusion. Definition 1.
A strategy of an investor is a P E -measurable function L t ( ω, y ) withvalues in R N + and L ( ω, y ) = 0, which is non-decreasing and c`adl`ag in t .The following lemma will be used further in the construction of the model. Lemma 1.
Let L t ( y ) be a P E -measurable function, and Y an adapted c`adl`ag processwith values in R M + . Then the process L t ( ω ) = L t ( ω, Y ( ω )) is predictable ( P -measurable).Proof. The σ -algebra P E is generated by sets C × [ s, ∞ ), where s ≥ C ∈ E s − (as usual, E s − = W u We call a strategy profile ( L , . . . , L M ) and a vector of initial wealth y ∈ R M ++ feasible if there exists a unique (up to P-indistinguishability) non-negativec`adl`ag adapted process Y , called the wealth process , which assumes values in R M + andsatisfies the following conditions:1) Y solves the wealth equation dY mt = − d | L mt | + N X n =1 l m,nt | l · ,nt | dX nt , Y m = y m (4)for m = 1 , . . . , M , where L mt ( ω ) = L mt ( ω, Y ( ω )), and l is any P ⊗ H -version of the R MN + -valued process of predictable Lebesgue derivatives (see the appendix for detailson Lebesgue derivatives; the measure P ⊗ H is defined as in (65) there) l m,nt = dL m,nt dH t ; (5)2) if Y mt ( ω ) = 0 or Y mt − ( ω ) = 0, then L ms ( ω ) = L mt − ( ω ) and Y s ( ω ) = 0 for all s ≥ t .When in (4) we have | l · ,nt ( ω ) | = 0 for some ω, t, n , we put l m,nt ( ω ) / | l · ,nt ( ω ) | = 0.Observe that the derivatives l are well-defined, since if Y is an adapted c`adl`ag process,then L m,n is a predictable process according to Lemma 1.7s usual, equation (4) should be understood in the integral sense (a.s. for all t ): Y mt = Y m − | L mt | + N X n =1 Z t l m,ns | l · ,ns | dX ns , (6)where the integral is understood as a pathwise Lebesgue–Stieltjes integral. It is well-defined since the process X is c`adl`ag and non-decreasing, and the integrand is non-negative and bounded.Let us clarify that we use Lebesgue derivatives in the wealth equation and notRadon-Nikodym derivatives (e.g. dL m,nt /d | L · ,nt | ) for two reasons. First, this allows todifferentiate with respect to a process H not depending on the solution of the equation,which is yet to be found. Second, it is natural to require that the solution should notdepend on what particular version of the derivatives is used. This is so if G ≪ H (seeProposition 1 below). Thus, if one would like to use Radon–Nikodym derivatives, theprocess H should dominate both the processes | L | and G , which would make formulasrather cumbersome.Sufficient conditions for the existence and uniqueness of a solution of equation (4)will be provided in the next section. But now let us prove a result which shows that thesolution, if it exists, does not depend on the choice of the process H and the versions ofthe derivatives l . Proposition 1. Suppose Y is a solution of (4) , where the derivative process l is definedas in (5) with respect to some c`adl`ag non-decreasing predictable process H such that G ≪ H . Then for any c`adl`ag non-decreasing predictable process e H such that G ≪ e H and any P ⊗ e H -version of the derivative e l = dL/d e H , the process Y also solves (4) with e l in place of l .Proof. Let F : R MN → R MN denote the function which specifies the distribution ofpayoffs in (4): F ( l ) m,n = l m,n | l · ,n | , (7)where F ( l ) m,n = 0 if | l · ,n | = 0. As follows from (6), we have to show that for each m, nF ( l ) m,n · X n = F ( e l ) m,n · X n , (8)where F ( l ) denotes the process F ( l t ( ω )), and F ( e l ) denotes F ( e l t ( ω )).One can see that if f, f ′ ≥ f = f ′ P ⊗ G -a.s.,then f · X n = f ′ · X n . We have l m,n = dL m,n dH = dL m,n d e H d e HdH = e l m,n d e HdH P ⊗ G -a.s. , where the second equality holds in view of claim (b) of Proposition 4 from the appendix.Since d e H/dH > ⊗ G -a.s. by claim (c) of Proposition 4, we have F ( l ) m,n = F ( e l ) m,n ,so (8) holds, which finishes the proof. The following theorem provides a sufficient condition for the existence and uniquenessof a solution of equation (4). Note that the main results of our paper, formulated inSection 5, do not require this condition to hold (they only require a unique solution toexist), and may be valid under less strict assumptions.8 heorem 1. Suppose that for each m a strategy L m of investor m satisfies the followingtwo conditions.(C1) There exists a P ⊗ B ( R M + ) -measurable function v mt ( ω, z ) with values in R N + suchthat for all ω ∈ Ω , y ∈ D , t ∈ R + , n = 1 , . . . , N , L m,nt ( ω, y ) = Z t v m,ns ( ω, y s − ) I( inf u For a given payoff process X , we call a strategy L relative growth optimal for investor m , if for any feasible initial wealth and a strategy profile where investor m uses this strategy, it holds that Y mt > t ≥ r m is a submartingale.Observe that if a strategy is relative growth optimal, then also r m is a submartingaleby Jensen’s inequality. Another corollary from the relative growth optimality is thatsuch a strategy is a survival strategy in the sense that the relative wealth of an investorwho uses it always stays bounded away from zero,inf t ≥ r mt > , (15)(we use the terminology of [4]; note that, for example, in [5], the term “survival” hasa somewhat different meaning). This follows from the fact that ln r m is a non-positivesubmartingale, and hence it has a finite limit z = lim t →∞ ln r mt . Therefore, lim t →∞ r mt = e z > k lim sup t →∞ t ln Y mt ≥ lim sup t →∞ t ln Y kt a.s. (16)This property is analogous to the notion of asymptotic growth optimality in single-investor market models (see, e.g., Section 3.10 in [17]). The validity of (16) followsfrom that sup t ≥ | Y t | /Y mt < ∞ by (15), so sup t ≥ Y kt /Y mt < ∞ for any k . Hencelim sup t →∞ t − ln( Y kt /Y mt ) ≤ 0, from which one can obtain (16). Denote by ν { t } the predictable random measure on B ( R N + ) defined by ν { t } ( ω, A ) = ν ( ω, { t } × A ) , A ∈ B ( R N + ) , and introduce the predictable process¯ ν t = ν { t } ( R N + ) . One can see that ¯ ν t is the conditional probability of a jump of the process X t given the σ -algebra F t − [14, Proposition II.1.17], i.e. ¯ ν t = P(∆ X t = 0 | F t − ). We will always10ssume that a “good” version of the compensator is chosen – such that ¯ ν t ( ω ) ∈ [0 , 1] forall ω, t .The candidate relative growth optimal strategy, which we define below, will behavedifferently at points t where ¯ ν t = 0 and where ¯ ν t > 0. To deal with them, let us partitionΩ × R + × (0 , ∞ ) into the following three sets belonging to P ⊗ B ( R + ):Γ = { ( ω, t, c ) : ¯ ν t ( ω ) = 0 } , Γ = (cid:26) ( ω, t, c ) : 0 < ¯ ν t ( ω ) < , or ¯ ν t ( ω ) = 1 and Z R N + c | x | ν { t } ( ω, dx ) > (cid:27) , Γ = (cid:26) ( ω, t, c ) : ¯ ν t ( ω ) = 1 and Z R N + c | x | ν { t } ( ω, dx ) ≤ (cid:27) . In the definition of the optimal strategy, the argument c in the triple ( ω, t, c ) will corre-spond to the value of the total wealth of all the investors right before time t , i.e. | Y t − | (points ( ω, t, c ) with c = 0 are not included in any of the sets; it will be easier to dealwith them separately). Roughly speaking, the sets Γ i differ in the conditional size ofpossible jumps of the payoff process X . On Γ , the conditional probability of a jump iszero. On Γ , only “large” jumps of X occur (large relatively to the current total wealth),and Γ is the remaining set where both “small” and “large” jumps can occur.The next lemma defines an auxiliary function ζ which will be needed to specify whatproportion of wealth the optimal strategy keeps in cash. Lemma 2. For each ( ω, t, c ) ∈ Γ , there exists a unique solution z ∗ ( ω, t, c ) ∈ (0 , c ) ofthe equation Z R N + cz + | x | ν { t } ( ω, dx ) = 1 − cz (1 − ¯ ν t ( ω )) . (17) The function ζ ( ω, t, c ) defined on Ω × R + × R + by ζ = c I(Γ ) + z ∗ I(Γ ) (18) is P ⊗ B ( R + ) -measurable.Proof. For ( ω, t, c ) ∈ Γ , the left-hand side of (17) is a strictly decreasing continuousfunction of z , while the right-hand side is a non-decreasing continuous function of z .The existence and uniqueness of the solution z ∗ then follows from comparison of theirvalues at z = c and z → ζ , consider the function f defined on Ω × R → R by f ( ω, t, c, z ) = (cid:18)Z R N + cz + | x | ν { t } ( ω, dx ) − cz (1 − ¯ ν t ( ω )) (cid:19) I(( ω, t, c ) ∈ Γ ) ∧ . Observe that f is a Carath´eodory function, i.e. P ⊗ B ( R + )-measurable in ( ω, t, c ) andcontinuous in z . Then by Filippov’s implicit function theorem (see, e.g., [2, Theo-rem 18.17]), the set-valued function φ ( ω, t, c ) = { z ∈ [0 , c ] : f ( ω, t, c, z ) = 0 } admits a measurable selector. Since φ on Γ is single-valued ( φ ( ω, t, c ) = { z ∗ ( ω, t, c ) } ),this implies the P ⊗ B ( R + )-measurability of ζ .11t is known that there exists a predictable process b t with values in R N + and a tran-sition kernel K ω,t ( dx ) from (Ω × R + , P ) to ( R N + , B ( R N + )) such that up to P-indistingui-shability X ct ( ω ) = b · G t ( ω ) , ν ( ω, dt, dx ) = K ω,t ( dx ) dG t ( ω ) . (19)Since the filtration is complete, we can assume (19) holds for all ω, t . Also, it will beconvenient to select “good” versions of b and K , which satisfy the following conditionsfor all ( ω, t ) (it is always possible to select such versions, see, e.g., [14, PropositionII.2.9]): | b t ( ω ) | = 0 if ∆ G t ( ω ) > , K ω,t ( { } ) = 0 , | b t ( ω ) | + Z R N + (1 ∧ | x | ) K ω,t ( dx ) = 1 . (20)Define the P ⊗ B ( R + )-measurable function b λ ( ω, t, c ) with values in R N + : b λ t (0) = 0 , b λ t ( c ) = b t c + Z R N + xζ t ( c ) + | x | K t ( dx ) for c > ω is omitted for brevity). Now we are in a position to introduce thestrategy, which will be shown to be relative growth optimal. When used by investor m ,its cumulative investment process is defined by b L t ( y ) = Z t y ms − b λ s ( | y s − | ) dG s (22)(for s = 0, put y − = y ). When it is necessary to emphasize that this strategy, as afunction of y , depends on which investor uses it, we will use the notation b L mt ( y ).Generally speaking, the strategy b L resembles optimal strategies in other models inevolutionary finance, as they all split investment budget between assets proportionallyto expected asset payoffs (but quantitatively they differ in how these proportions arecalculated). In the particular case when the payoff process X t is discrete-time (as inSection 2), we obtain the same strategy that was found in [9]. Formally, the discrete-time case can be included in the general model by taking a process X t such that X t = P ⌊ t ⌋ s =0 ∆ X s ; then b = 0 and K t ( dx ) is the conditional distribution of the jump ∆ X t forinteger t .To conclude this section, we state a proposition which provides sufficient conditionsof feasibility of a strategy profile where one or several investors use the strategy b L . It isbased on Theorem 1, but we show that the conditions of that theorem hold automaticallyfor b L under some mild additional assumptions on the payoff process. In particular, ifthese assumptions hold, then a strategy profile where all the investors use the strategy b L is feasible (we will consider such profiles in Theorem 4 in the next section).Define the predictable process with values in R N + h t = b t + Z R N + x | x | K t ( dx ) , (23)and define the scalar predictable process p t = Z R N + ν { t } ( dx )(1 + | x | ) . roposition 2. Suppose the process ( p t ∆ G t ) − I(∆ G t > is locally bounded and foreach n the process ( h nt ) − I( h nt > is locally bounded (where / for these pro-cesses). Then any strategy profile, in which every investor uses either the strategy b L ora strategy which satisfies the conditions of Theorem 1, is feasible for any initial wealth y ∈ R M ++ . The proof is given in Section 6. 5. The main results The following three theorems are the main results on relative growth optimal strategies.For convenience, we divide this section into three parts, each containing a theorem andcomments. The proofs are in Section 6. The first result establishes the existence of a relative growth optimal strategy ( b L issuch a strategy) and shows that it is, in a certain sense, unique. Theorem 2. 1. The strategy b L is relative growth optimal.2. Suppose L is a strategy of investor M such that the profile ( b L , . . . , b L M − , L ) and a vector of initial wealth y ∈ R M ++ are feasible and r M is a submartingale. Then L t ( Y ) = b L Mt ( Y ) for all t ≥ , where Y is the solution of the wealth equation for thisstrategy profile and initial wealth. Let us comment on the second part of the theorem. It can be regarded as a uniquenessresult for a relative growth optimal strategy: if M − b L ,then the remaining investor, who wants the relative wealth to be a submartingale, hasnothing to do but to act as using the strategy b L as well. Here, “to act” means thatthe realization of the strategy of this investor, i.e. the cumulative investment process L t ( ω ) = L t ( ω, Y ( ω )) coincides (up to P-indistinguishability) with the process b L Mt ( ω ) = b L Mt ( ω, Y ( ω )). As a consequence, the relative wealth of each investor will stay constant.However, note that the strategy L t ( ω, y ), as a function on Ω × D × R + , may bedifferent from b L Mt ( ω, y ). Let us provide an example. Suppose there is only one asset withthe non-random payoff process X t = t and two investors with initial wealth y = y = 1.In this case, G t = t and the strategy b L , if used by investor 2, has the form b L t ( y ) = Z t y s − y s − + y s − ds. On the other hand, consider the strategy L for investor 2 defined as L t ( y ) = Z t (cid:18) 13 I( y u = 1 for all u < s ) + y s − y s − + y s − I( y u = 1 for some u < s ) (cid:19) ds. It is not hard to see that L is also relative growth optimal. However it leads to a differentwealth process of investor 2 compared to b L if, for example, L t ≡ The second result shows that the strategy b L asymptotically determines the structureof the market in the sense that if there is an investor who uses it, then the representativestrategy of all the investors is asymptotically close to b L . (By the representative strategywe call the weighted sum of the investors’ strategies with their relative wealth as the13eights; see below.) Moreover, if the representative strategy of the other investors isasymptotically different from b L , they will be driven out of the market – their relativewealth will vanish as t → ∞ .In order to state the theorem, let us introduce auxiliary processes. Suppose a uniquesolution of the wealth equation exists. Let L mt ( ω ) = L mt ( ω, Y ( ω )) be the realizationsof the investors’ strategies, and, as above, l mt = dL mt /dG t . For each m , define thepredictable process L ( s ) ,mt = L mt − l m · G t , which is the singular part of the Lebesguedecomposition of L mt with respect to G t (hence the superscript “( s )”).Define the proportion λ mt of wealth invested in the assets by investor m as thepredictable process with values in R N + and the components λ m,nt = l m,nt Y mt − , where 0 / l m,nt = 0 on theset { ( ω, t ) : Y mt − ( ω ) = 0 } (P ⊗ G -a.s.). Introduce also the processes of “cumulativeproportions” of invested wealth and their singular parts:Λ mt = 1 Y m − · L mt , Λ ( s ) ,mt = Λ mt − λ m · G t = 1 Y m − · L ( s ) ,mt , which are non-decreasing, predictable, c`adl`ag, and with values in [0 , + ∞ ] N .For a set of investors M ⊆ { , . . . , M } , let us denote their total wealth by Y M t = P m ∈ M Y mt , their relative wealth by r M t = P m ∈ M r mt , and the processes associated withthe realization of their representative strategy by L M t = P m ∈ M L mt , l M t = dL M t /dG t = P m ∈ M l mt , L ( s ) , M t = P m ∈ M L ( s ) ,mt , and λ M t = l M ,nt Y M t − = X m ∈ M r mt − r M t − λ mt , Λ M t = 1 Y M − · L M t = X m ∈ M r m − r M − · Λ mt , Λ ( s ) , M t = 1 Y M − · L ( s ) , M t = X m ∈ M r m − r M − · Λ ( s ) ,mt . To shorten the notation, for the set of all investors M = { , . . . , M } we will write¯ λ nt = λ M ,nt , and for the set M = { , . . . , M } write e λ nt = λ M ,nt , and similarly for theother processes. Theorem 3. Suppose investor 1 uses the strategy b L , the other investors use arbitrarystrategies L m , and the strategy profile ( b L , L , . . . , L M ) is feasible for some initial wealth y ∈ R M ++ . Then k λ − ¯ λ k · G ∞ + | ¯Λ ( s ) ∞ | < ∞ a.s. , (24) and, as t → ∞ , r t → a.s. on { ω : k λ − e λ k · G ∞ ( ω ) = ∞ or | e Λ ( s ) ∞ ( ω ) | = ∞} . (25)Equation (24) expresses the idea that the investment proportions ¯ λ of the representa-tive strategy of all the investors are close to λ = b λ asymptotically in the sense that the14ntegral R t k b λ s − ¯ λ s k dG s converges as t → ∞ and the singular part ¯Λ ( s ) t stays bounded.If G ∞ = ∞ , this, roughly speaking, means that k b λ t − ¯ λ t k is small asymptotically.Equation (25) shows that the strategy b L drives other strategies out of the market ifthey are asymptotically different from it. This result can be also regarded as asymptoticuniqueness of a survival strategy: if investors m = 2 , . . . , M want to survive againstinvestor 1 who uses the strategy b L , they should also use (at least, collectively) a strategyasymptotically close to b L . Theorems 2, 3 lead to the natural question: since the strategy b L is s good, what willhappen if all the investors decide to use it? Obviously, in this case their relative wealthwill remain the same. However, it is interesting to look at the asymptotic behavior ofthe absolute wealth W t := | Y t | . A priori, it is even not obvious whether it will grow.Our third result partly answers this question: we prove that W does not decrease “onaverage” and provide a condition for W t → ∞ as t → ∞ . Theorem 4. Suppose all the investors use the strategy b L , and the initial wealth y ∈ R M ++ and the strategy profile ( b L , . . . , b L ) are feasible.Then the process V t := 1 /W t is a supermartingale and there exists the limit W ∞ :=lim t →∞ W t ∈ (0 , ∞ ] a.s. Moreover, if X is quasi-continuous (i.e. ¯ ν ≡ ), then { W ∞ = ∞} = { (1 ∧ | x | ) ∗ ν ∞ = ∞} a.s. If E | X t | < ∞ for all t , then also E W t < ∞ (since W t ≤ | y | + | X t | ), and the process W t will be a submartingale by Jensen’s inequality. This is what we mean by the phrasethat the total wealth does not decrease on average.It is interesting to note that if one investor uses the strategy b L and the other investorsuse arbitrary strategies, then it does not necessarily hold that the wealth of such aninvestor will grow. In particular, it may happen that W t → t → ∞ , which isremarkable because an investor always has a trivial strategy which guarantees that thewealth will not vanish – just keep all the wealth in cash. An example can be found in[9]. Another fact worth mentioning is that, as will become clear from the proof of thetheorem, the continuous part of the payoff process X does not affect the process W ifall the investors use the strategy b L , i.e. W will be the same for any payoff processes X and X ′ such that X − X ′ is a continuous process. For example, if X is continuous, then W t = W for all t ≥ X is a strictly increasing process. In particular, observethat in the second claim of the theorem, the continuous part of X does not enter thecondition for having W ∞ = ∞ . 6. Proofs Without loss of generality we will assume that the functions C m and the processes δ m are the same for all the investors, since otherwise one can take C ( a ) = max m C m ( a )and δ t = max m δ mt . Moreover, we can assume that δ is a non-decreasing process or,otherwise, take δ ′ t = sup s ≤ t δ t ( δ ′ t will be finite-valued since δ t is predictable and c`adl`ag,and, hence, locally bounded; see, e.g., VII.32 in [8]). Proposition 1 implies that it isenough to prove the existence and uniqueness of a solution for some particular choice ofthe process H such that G ≪ H . We will do this for H = G .15e are going to construct the process Y by induction on stochastic intervals [0 , τ i,j ]with appropriately chosen stopping times τ i,j ( i ∈ { , , . . . , M } , j ∈ Z + ) such that τ i,j ≤ τ i ′ ,j ′ if ( i, j ) ≤ ( i ′ , j ′ ) lexicographically (i.e. i < i ′ , or i = i ′ and j ≤ j ′ ), andsup i,j τ i,j = ∞ . Here, “by induction” means that we will construct processes Y i,j suchthat on the set { ( ω, t ) : t ≤ τ i,j ( ω ) } they satisfy equation (6) and on this set Y i,j = Y i ′ ,j ′ for any ( i ′ , j ′ ) ≥ ( i, j ). From these processes we can form the single process Y satisfying(6) on the whole set Ω × R + .Before providing an explicit construction, let us briefly explain the role that τ i,j willplay. The stopping times τ i, for i ≥ m we have Y mt > t < τ i, but Y mτ i, − = 0). The index i will correspond to the i -th such event. Notnecessarily all the investors will eventually have zero wealth; in that case we will put τ i,j ( ω ) = ∞ for i starting from some i ′ and all j .Between τ i, and τ i +1 , we will construct a sequence of stopping times τ i,j → τ i +1 , as j → ∞ , such that on each interval [ τ i,j , τ i,j +1 ) the wealth of all the investors, whohave non-zero wealth at τ i,j , can be bounded away from zero by an F τ i,j -measurablevariable. The wealth of some of those investors may become zero at τ i,j +1 , but only “bya jump”. If τ i, ( ω ) < ∞ , it will also hold that τ i,j ( ω ) < ∞ for all j .The reason why we need to treat differently the moments when the wealth reacheszero in a continuous way and by a jump is that we do not assume that the function C ( a ) is bounded in a neighborhood of zero (this is necessary, for example, to apply thetheorem to the strategy b L – see the proof of Proposition 2).Now we will proceed to the construction of τ i,j and Y i,j . Let τ , = 0, and for all t ≥ Y , t = y , where y ∈ R M ++ is the given initial wealth. Suppose τ i,j and Y i,j are constructed. We will now show how to construct τ i,j +1 , Y i,j +1 . For brevity, i will beassumed fixed and omitted in the notation, so we will simply write τ j , Y j , while Y j,m will denote the m -th coordinate of Y j .Let A ( ω ) = { m : Y j,mτ j ( ω ) > } denote the set of the investors who are still active(i.e. have positive wealth) at τ j ; for ω such that τ j ( ω ) = ∞ we put A ( ω ) = ∅ . Observethat A is an F τ j -measurable random set (since it is finite, the measurability means thatI( m ∈ A ) are F τ j -measurable functions for each m ).On the set { ω : A ( ω ) = ∅} , define τ j +1 = τ j + 1 (with τ j +1 ( ω ) = ∞ if τ j ( ω ) = ∞ ),and on the set Ω ′ = { ω : A ( ω ) = ∅} define γ = ( δ τ j + 1) C ( Y jτ j )and τ j +1 = inf (cid:26) t > τ j : | X t − X τ j | ≥ M γ ∧ min m ∈ A Y j,mτ j , (26)or G t − G τ j ≥ γ (cid:18) M ∧ min m ∈ A Y j,mτ j (cid:19) , (27)or δ t ≥ δ τ j + 1 , or t ≥ τ j + 1 (cid:27) . (28)Observe that we have the strict inequality τ j +1 > τ j on Ω ′ since the processes X, G, δ are c`adl`ag. Also, τ j +1 ≤ τ j + 1 by the condition in (28).For each ω define the complete metric space E ( ω ) consisting of c`adl`ag functions16 : R + → R M + satisfying the conditions f t = Y jt ( ω ) for t ≤ τ j ( ω ) , (29) f mt ∈ (cid:20) Y j,mτ j ( ω ) , Y j,mτ j ( ω ) (cid:21) for t > τ j ( ω ) , m = 1 , . . . , M, (30)and the metric d ( f, e f ) = sup t ≥ | f t − e f t | (note that if A ( ω ) = ∅ , then E ( ω ) consists of one element).From now on, we will assume that ω is fixed and omit it in the notation. Consider theoperator U on E , which maps a function f ∈ E to the c`adl`ag function g := U ( f ) : R + → R M + defined by the formula g mt = Y j,mt ∧ τ j − Z t | v ms ( f s − ) | I( τ j < s < τ j +1 , m ∈ A ) dG s + Z t F m ( l s ( f s − )) I( τ j < s < τ j +1 ) dX s , (31)where F : R MN → R MN is the function defined in (7), and l m,ns ( ω, z ) = v m,ns ( ω, z ) I( m ∈ A ( ω )) . (32)Let us show that U is a contraction mapping of E to itself. If A ( ω ) = ∅ , this isobvious, so consider ω such that A ( ω ) = ∅ . Suppose f ∈ E , g = U ( f ). First we willshow that g ∈ E . It is clear that g satisfies (29), and, if m / ∈ A , then g m satisfies(30). To show that the lower bound in (30) is satisfied for m ∈ A , consider the firstintegral in (31). Since f ∈ E , by condition (13) we have v m,ns ( f s − ) ≤ C ( Y jτ j ) δ s ≤ γ ,using that δ s < δ τ j + 1 for s < τ j +1 . Hence the integral can be bounded from aboveby γ ( G τ j +1 − − G τ j ), and this quantity does not exceed Y j,mτ j by the choice of τ j +1 (see(27)). Therefore, g mt ≥ Y j,mτ j for t ≥ τ j .The upper bound from (30) for m ∈ A follows from that the second integral in (31)is bounded from above by | X τ j +1 − − X τ j | since F m,n ( l ) ≤ τ j +1 (see (26)) we have | X τ j +1 − − X τ j | ≤ Y j,mτ j . Thus, g satisfies conditions (29)–(30), so g ∈ E .Now we will show that U is contracting. Consider f, e f ∈ E and m ∈ A . Then | U ( f ) mt − U ( e f ) mt | ≤ Z ( τ j ,τ j +1 ) | v ms ( f s − ) − v ms ( e f s − ) | dG s + N X n =1 Z ( τ j ,τ j +1 ) | F m,n ( l s ( f s − )) − F m,n ( l s ( e f s − )) | dX ns := I m + I m . By conditions (14) and (28), we have | v m,ns ( f s − ) − v m,ns ( e f s − ) | ≤ γd ( f, e f ) for s ∈ ( τ j , τ j +1 ).Hence I m ≤ γd ( f, e f )( G τ j +1 − − G τ j ) ≤ M d ( f, e f ) , where the last inequality is due to (27). 17o bound I m , observe that for each n and ( ω, t ) ∈ ( τ j , τ j +1 ) \ Π m,n we have (by (12)) | l · ,ns ( f s − ) | ≥ min m ∈ A v m,ns ( f s − ) ≥ γ , (33)and a similar inequality is true for | l · ,ns ( e f s − ) | . It is straightforward to check that F satisfies the property (cid:12)(cid:12)(cid:12)(cid:12) ∂F m,n ∂l p,q ( l ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | l · ,n | for any m, n, p, q. Hence, for any l, e l ∈ R MN + such that | l · ,n | ≥ α and | e l · ,n | ≥ α for all n with some α > | F m,n ( l ) − F m,n ( e l ) | ≤ α − | l − e l | . From this and (33), we find that on the set( τ j , τ j +1 ) \ Π m,n | F m,n ( l s ( f s − )) − F m,n ( l s ( e f s − )) | ≤ γ | l s ( f s − ) − l s ( e f s − ) | ≤ γ d ( f, e f ) . On Π m,n we have F m,n ( l s ( f s − )) − F m,n ( l s ( e f s − )) = 0 , and, consequently, obtain the bound I m ≤ γ d ( f, e f ) | X τ j +1 − − X τ j | ≤ M d ( f, e f ) . Now we see that U is a contraction mapping: d ( U ( f ) , U ( e f )) ≤ d ( f, e f ).As a result, U ( ω ) has a fixed point f ∗ ( ω ) for any ω . Observe that the operator U preserves adaptedness, i.e. if f t ( ω ) is a c`adl`ag adapted process with values in R M + andsatisfies conditions (29)–(30), then U ( ω, f ( ω )) is such a process as well. Hence f ∗ is ac`adl`ag adapted process since it can be obtained, for example, as the limit U ( n ) ( Y jt ∧ τ j )as n → ∞ where n stands for the n -times application of U .Now we can define the process Y j +1 as follows: for each m put Y j +1 ,mt = f ∗ ,mt for t < τ j +1 ,Y j +1 ,mt = f ∗ ,mτ j +1 − − | v mτ j +1 ( f ∗ ,mτ j +1 − ) | I( m ∈ A )∆ G τ j +1 + N X n =1 F m,n ( v τ j +1 ( f ∗ ,mτ j +1 − )) I( m ∈ A )∆ X nτ j +1 for t ≥ τ j +1 (note that Y j +1 t = Y j +1 τ j +1 for all t ≥ τ j +1 ). Inserting Y j +1 ,m in (31), we obtain theequation for t ∈ [ τ j , τ j +1 ] Y j +1 ,mt = Y j,mτ j − Z ( τ j ,t ] | v ms ( Y j +1 s − ) | I( Y j,mτ j > dG s + Z ( τ j ,t ] F m ( l s ( Y j +1 s − )) dX s . (34)The indicator here can be equivalently replaced by I(inf u 1) for any c > 0, whichcan be obtained from (21), (23) using that ζ t ( c ) ∈ [0 , c ].To prove (13), on the set Γ we can use the estimate b λ nt ( c ) = b nt c + Z R N + x n c + | x | K t ( dx ) ≤ h nt c ∧ ≤ c ∧ . (35)The last inequality here holds since | h t | ≤ | b t | + R R N + (1 ∧ | x | ) K t ( dx ) = 1 (see (20)). Onthe set Γ ∪ Γ , we can use the estimate b λ nt ( c ) = Z R N + x n ζ t ( c ) + | x | K t ( dx ) ≤ K t ( R N + ) = ¯ ν t ∆ G t ≤ G t (36)(note that if ( ω, t, c ) ∈ Γ ∪ Γ , then ∆ G t > b t = 0). Therefore, we obtain v m,nt ( z ) ≤ z m max (cid:18) | z | ∧ , I(∆ G t > G t (cid:19) ≤ (2 | a | ∨ δ t ≤ C ( a ) δ t , (37)so (13) holds.To prove (14), suppose z, e z, a ∈ R N + and z k , e z k ∈ [ a k / , a k ] for all k . If z m = e z m = 0,then v m,nt ( z ) = v m,nt ( e z ) = 0, so (14) holds. If e z m = 0, but z m > 0, then, using (37), weobtain | v m,nt ( z ) − v m,nt ( e z ) | = v m,nt ( z ) ≤ (2 | a | ∨ δ t ≤ (2 | a | ∨ δ t a m / | z − e z | ≤ C ( a ) δ t | z − e z | , where we used the inequality | z − e z | ≥ z m ≥ a m / 2. In a similar way, (14) is satisfied if z m = 0, but e z m > z m > e z m > 0. Denote c = | z | , e c = | e z | . Then | v m,nt ( z ) − v m,nt ( e z ) | ≤ b λ nt ( e c ) | z m − e z m | + z m | b λ nt ( c ) − b λ nt ( e c ) | . (38)Using (35)–(36), the first term in the right-hand side can be bounded as follows: b λ nt ( e c ) | z m − e z m | ≤ δ t | a | ∧ | z − e z | . For the second term in the right-hand side of (38) we have z m | b λ nt ( c ) − b λ nt ( e c ) | ≤ a m (cid:26) | c − e c | c e c b nt + | c − e c | (cid:18)Z R N + x n ( c + | x | )( e c + | x | ) K t ( dx ) (cid:19) I(∆ G t = 0)+ | ζ t ( c ) − ζ t ( e c ) | (cid:18)Z R N + x n ( ζ t ( c ) + | x | )( ζ t ( e c ) + | x | ) K t ( dx ) (cid:19) I(∆ G t > (cid:27) := 2 a m { A + A + A } , A i denote the three terms in the braces. Using that | c − e c | ≤ | z − e z | and | b t | ≤ | h t | ≤ 1, we obtain A ≤ | c − e c || a | b nt ≤ | z − e z || a | , and A ≤ | c − e c | c ( e c ∧ Z R N + x n | x | K t ( dx ) ≤ | z − e z || a | ( | a | ∧ h nt ≤ | z − e z || a | ∧ . Let us bound A . Assume c ≥ e c (hence also ζ t ( c ) ≥ ζ t ( e c )) and ζ t ( c ) > 0. Then we have A ≤ ( ζ t ( c ) − ζ t ( e c )) (cid:18)Z R N + ζ t ( c ) + | x | K t ( dx ) (cid:19) I(∆ G t > ≤ ζ t ( c ) − ζ t ( e c ) c ∆ G t I(∆ G t > ≤ ( ζ t ( c ) − ζ t ( e c )) 2 I(∆ G t > | a | ∆ G t , (39)where in the second inequality we used the bound Z R N + ζ t ( c ) + | x | K t ( dx ) = 1 c ∆ G t Z R N + cζ t ( c ) + | x | ν { t } ( dx )= 1 c ∆ G t (cid:18) − cζ t ( c ) (1 − ¯ ν t ) (cid:19) ≤ c ∆ G t . Here the second equality follows from (17) – notice that ( ω, t, c ) ∈ Γ because we assume ζ t ( c ) > ζ t ( c ) − ζ t ( e c ) in (39). Let Q t be the random measure on R N + defined by Q t ( A ) = ν { t } ( A ) + (1 − ¯ ν t ) I(0 ∈ A ). Observe that Q t ( R N + ) = 1. Since( ω, t, c ) ∈ Γ and ( ω, t, e c ) ∈ Γ ∪ Γ , from (17) and (18) we find that Z R N + ζ t ( c ) + | x | Q t ( dx ) = 1 c , Z R N + ζ t ( e c ) + | x | Q t ( dx ) ≤ e c . From this, we obtain1 e c − c ≥ ( ζ t ( c ) − ζ t ( e c )) Z R N + Q t ( dx )( ζ t ( c ) + | x | )( ζ t ( e c ) + | x | ) ≥ ( ζ t ( c ) − ζ t ( e c )) Z R N + Q t ( dx )( c + | x | ) ≥ ( ζ t ( c ) − ζ t ( e c )) Z R N + ν { t } ( dx )( c + | x | ) ≥ ζ t ( c ) − ζ t ( e c ) c ∨ p t . Hence, we conclude that ζ t ( c ) − ζ t ( e c ) ≤ ( c − e c )( c ∨ c e cp t ≤ | z − e z | ( | a | ∧ p t . (40)From (39) and (40), we find A ≤ | z − e z | I(∆ G t > | a | ∧ p t ∆ G t . z m > e z m > | v ( z ) m,nt − v ( e z ) m,nt | ≤ (cid:18) | a | ∧ | a | (cid:18) | a | + 4 | a | ∧ | a | ∧ (cid:19)(cid:19) δ t | z − e z |≤ | a || a | ∧ δ t | z − e z | ≤ C ( a ) δ t | z − e z | . Thus, condition (C2) holds, which finishes the proof. The key idea of the proof of the first claim of the theorem is to show that ln r t is a σ -submartingale by showing that its drift rate is non-negative. Since ln r t is a non-positiveprocess, it will be then a usual submartingale [15, Proposition 3.1]. For the reader’sconvenience, let us briefly recall the related notions and known results; details can befound in, e.g., [15].A scalar semimartingale Z with Z = 0 is called a σ -submartingale if there existsa non-decreasing sequence of predictable sets Π n ∈ P such that Z Π n t := R t I s (Π n ) dZ s is a submartingale for each n and S n Π n = Ω × R + . Suppose the triplet ( B h , C, ν ) ofpredictable characteristics of Z with respect to a truncation function h ( z ) admits therepresentation B h = b h · G , C = c · G , ν = K ⊗ G with predictable processes b ht , c t , atransition kernel K t ( dz ), and a non-decreasing predictable c`adl`ag process G t . Then Z is a σ -submartingale if and only if P ⊗ G -a.e. on Ω × R + Z | z | > | z | K t ( dz ) < ∞ and d t := b ht + Z R ( z − h ( z )) K t ( dz ) ≥ d is called thedrift rate of Z with respect to G . One can see that it does not depend on the choice ofthe truncation function h (see [14, Proposition II.2.24]).Observe that if Z R | z | K ( dz ) < ∞ , then d t = b t + R R zK t ( dz ), where b t = b ht − R R h ( z ) K t ( dz ) is a well-defined predictableprocess, which does not depend on the choice of h . From this we obtain the corollarythat will be used further in the proof: if Z is a non-positive semimartingale, then it willbe a submartingale if P ⊗ G -a.s. Z z< zK t ( dz ) > −∞ and d t = b t + Z R zK t ( dz ) ≥ . (41)In particular, observe that for a non-positive semimartingale it holds that R z> zK ( dz ) < ∞ since K t ( { z : z > − Z t − } ) = 0. As a consequence, if (41) is satisfied, then the process d is G -integrable and the compensator of Z t is A t = d · G t . (42)Let us also state one auxiliary inequality, which generalizes well-known Gibbs’ in-equality, and will play an important role in the proof. Suppose α, β ∈ R N + are twovectors such that | α | , | β | ≤ n it holds that if β n = 0, then also α n = 0.Then α (ln α − ln β ) ≥ k α − β k | α | − | β | , (43)22here α n (ln α n − ln β n ) = 0 if α n = 0. A short direct proof can be found in [9, Lemma 2].Now we can proceed to the proof of the first claim of the theorem. Assume that thestrategy b L is used by investor m = 1, and the wealth equation has a unique solution Y t .We will use the notation of Section 5 and introduce the predictable R N + -valued processes λ t , Λ t for investor 1 and e λ t , e Λ t , e Λ ( s ) t for the other investors. To keep the notation concise,from now on the superscript “1” for investor 1 will be omitted, so we will simply write λ t , Λ t . It will be also convenient to assume that the particular version of λ t is selected: λ t ( ω ) = b λ t ( ω, W t − ( ω )) for all ( ω, t ) with the function b λ t ( c ) defined in (21).Let W t = Y t + e Y t denote the total market wealth, and r t = Y t /W t the relative wealthof investor 1. Define the predictable process F with values in R N + by F nt = λ nt r t − λ nt + (1 − r t − ) e λ nt , where 0 / Y and W can be written as stochastic exponents Y = Y E (cid:18) −| Λ | + FW − · X (cid:19) ,W = W E (cid:18) − r − · | Λ | − (1 − r − ) · | e Λ | + 1 W − · | X | (cid:19) . (44)Recall that the stochastic exponent of a semimartingale S is the process E ( S ) whichsolves the equation d E ( S ) t = E ( S ) t dS t with E ( S ) = 1. It is known that E ( S ) > E ( S ) − > S = − 1, see [14, § II.8a]. From the definition of b λ , one cancheck that ∆( −| Λ | + ( F/W − ) · X ) > − Y > Y − > ζ t ( ω ) denote the predictable process ζ t ( ω, W t − ( ω )). As follows from the definitionof b L and ζ , we have ζ t = (1 −| ∆Λ t | ) W t − . Let e ζ t = (1 −| ∆ e Λ t | ) W t − . Define the predictablefunction f ( ω, t, x ) by f t ( x ) = ln ζ t + F t xr t − ζ t + (1 − r t − ) e ζ t + | x | ! . Using the Dol´eans–Dade formula, which for a process of bounded variation S takes theform E ( S ) t = exp( S ct + P u ≤ t ln(1 + ∆ S u )), we obtainln r t = ln r + (1 − r − ) · ( | e Λ ct | − | Λ ct | ) + F − W − · X ct + X s ≤ t f s (∆ X s ) . For the further analysis, it will be convenient to split the process ln r t into several parts.Let f t ( x ) = f t ( x ) + f t ( x ) + f t ( x ), where f t ( x ) = f t ( x ) I(∆ G t = 0 , ∆ e Λ t = 0) ,f t ( x ) = f t ( x ) I(∆ G t > ,f t ( x ) = f t ( x ) I(∆ G t = 0 , ∆ e Λ t > . Then ln r t = ln r + Z t + e Z t (45)23ith the processes Z t = (1 − r − ) · ( | e Λ ct | − | e Λ ( s ) ct | − | Λ ct | ) + F − W − · X ct + X s ≤ t ( f s + f s )(∆ X s ) , (46) e Z t = (1 − r − ) · | e Λ ( s ) ct | + X s ≤ t f s (∆ X s ) , (47)where e Λ ( s ) ct = e Λ ( s ) t − P u ≤ t ∆ e Λ ( s ) u is the continuous part of the singular part of theLebesgue decomposition of e Λ with respect to G .Observe that I(∆ X = 0 , ∆ G = 0 , ∆ e Λ = 0) = 0 since the set { ∆ X = 0 , ∆ G = 0 } istotally inaccessible and the process e Λ is predictable. Therefore, X s ≤ t f s (∆ X s ) = X s ≤ t f s (0) = − X s ≤ t ln(1 − (1 − r s − ) | ∆ e Λ ss | ) . (48)From this formula and (47), it follows that e Z t is a non-decreasing predictable c`adl`agprocess, so in order to show that ln r t is a σ -submartingale, it is enough to show that Z t is a σ -submartingale.We will make use of condition (41). Since the process Z is of bounded variation, it isnot difficult to see (from, e.g., the canonical representation of a semimartingale) that itscontinuous part can be represented as Z ct = b · G t , where b is the predictable processfrom (41). From (46), we find b t = (1 − r t − )( | e λ t | − | λ t | ) I(∆ G t = 0) + ( F t − b t W t − . The measure of jumps µ Z of Z is such that for a function g ( ω, t, z ) with g ( ω, t, 0) = 0we have g ∗ µ Zt = g ( f + f ) ∗ µ t + X s ≤ t g ( f s (0)) I(∆ X s = 0) , so its compensator can be represented in the form ν Z = K Z dG with the kernel K Z suchthat Z R g t ( z ) K Zt ( dz ) = Z R N + g t ( f t ( x ) + f t ( x )) K t ( dx ) + 1 − ¯ ν t ∆ G t g t ( f t (0))(when ∆ G t ( ω ) = 0, we have f t ( ω, x ) = 0, so we treat the last term in the right-hand sideas zero). Consequently, the drift rate of Z with respect to G is d t = b t + R R zK Zt ( dz ) = h t + h t with the predictable processes h t = (1 − r t − )( | e λ t | − | λ t | ) I(∆ G t = 0) + ( F t − b t W t − + Z R N + f t ( x ) K t ( dx ) ,h t = Z R n + f t ( x ) K t ( dx ) + 1 − ¯ ν t ∆ G t f t (0) . (49)We need to show that h , h ≥ 0. For h , using the inequality x − ≥ ln x for x > F t − b t ≥ b t ln( F t ) , (50)where we put b nt ln( F nt ) = 0 if F nt = 0 (notice that if F nt = 0, then λ nt = 0, so also b nt = 0). 24ntroduce the set X t ( ω ) = { x ∈ R N + : x n = 0 if F nt ( ω ) = 0 , n = 1 , . . . , N } . Onthe set { ( ω, t, x ) : ∆ G t ( ω ) = 0 , x ∈ X t ( ω ) } , using the concavity of the logarithm, theequality ∆Λ t = 0 if ∆ G t = 0, and the inequality e ζ t ≤ W t − we obtain f t ( x ) ≥ ln (cid:18) W t − + F t xW t − + | x | (cid:19) ≥ x ln F t W t − + | x | , (51)where we put x n ln( F nt ) = 0 if F nt = x n = 0. Denote a t = Z R N + xW t − W t − + | x | K t ( dx ) . (52)As follows from (21), we have K t ( ω, R N + \ X t ( ω )) = 0. Then from (51)–(52) we obtain Z R N + f t ( x ) K t ( dx ) = Z X t f t ( x ) K t ( dx ) ≥ a t ln F t W t − I(∆ G t = 0) . Together with (50), this implies h t ≥ (cid:18) (1 − r t − )( | e λ t | − | λ t | ) + ( a t + b t ) ln F t W t − (cid:19) I(∆ G t = 0) . From (21), it follows that we have λ t = ( a t + b t ) /W t − when ∆ G t = 0, so on the set { ∆ G = 0 } h t ≥ (1 − r t − )( | e λ t | − | λ t | ) + λ t ln F t = (1 − r t − )( | e λ t | − | λ t | ) + λ t (ln λ t − ln( r t − λ t + (1 − r t − ) e λ t )) . Applying inequality (43), we obtain h t ≥ 14 (1 − r t − ) k λ t − e λ t k I(∆ G t = 0) ≥ . (53)Let us prove that h ≥ 0. Consider the set { ∆ G > } , on which we have f t ( x ) = ln ζ t + F t xr t − ζ t + (1 − r t − ) e ζ t + | x | ! = ln (cid:18) ζ t + F t xζ t + | x | (cid:19) + ln ζ t + | x | r t − ζ t + (1 − r t − ) e ζ t + | x | ! , and, using the concavity of the logarithm, we find that for x ∈ X t ( ω ) f t ( x ) ≥ x ln F t ζ t + | x | + ln ζ t + | x | r t − ζ t + (1 − r t − ) e ζ t + | x | ! := A t ( x ) + B t ( x ) . (54)For the term A t ( x ), applying inequality (43), we get Z R N + A t ( x ) K t ( dx ) = λ t ln( F t ) = λ t (ln λ t − ln( r t − λ t + (1 − r t − ) e λ t )) ≥ 14 (1 − r t − ) k λ t − e λ t k + (1 − r t − )( | λ t | − | e λ t | ) . B t ( x ), using the inequality ln x ≥ − x − , we obtain B t ( x ) ≥ (1 − r t − )( ζ t − e ζ t ) ζ t + | x | . From the definition of ζ t (see (17)), it follows that Z R N + ζ t + | x | K t ( dx ) ≥ W t − ∆ G t − − ¯ ν t ζ t ∆ G t , so we have Z R N + B t ( x ) K t ( dx ) ≥ (1 − r t − )( | e λ t | − | λ t | ) − (1 − r t − )(1 − ¯ ν t )( ζ t − e ζ t ) ζ t ∆ G t , (55)where for the first term in the right-hand side we used that ζ t − e ζ t = ( | e λ t |− | λ t | ) W t − ∆ G t .Thus, using (54)–(55) and that K t ( ω, R N + \ X t ( ω )) = 0, we find Z R N + f t ( x ) K t ( dx ) ≥ 14 (1 − r t − ) k λ t − e λ t k − (1 − r t − )(1 − ¯ ν t )( ζ t − e ζ t ) ζ t ∆ G t . (56)Also, using again the inequality ln x ≥ − x − , we obtain f t (0) = ln (cid:18) ζ t r t − ζ t + (1 − r t − ) e ζ t (cid:19) ≥ (1 − r t − )( ζ t − e ζ t ) ζ t . (57)Hence, from (49), (56), and (57), we find that h t ≥ 14 (1 − r t − ) k λ t − e λ t k I(∆ G t > ≥ . (58)Thus, we have proved that h , h ≥ 0, so ln r t is a submartingale, which finishes theproof of the first claim of the theorem.To prove the second claim, suppose investors m = 1 , . . . , M − b L and investor M use some strategy L . If r Mt is a submartingale, then ln r t is asupermartingale by Jensen’s inequality, and hence a martingale by the first claim of thetheorem. Consequently, we find from (45) (with the same notation as above) e Z t = 0 a.s. for all t ≥ , h + h = 0 P ⊗ G -a.s.The first equality implies that L ( s ) ,M = 0, so L M ≪ G . The second equality, togetherwith (53) and (58), implies that e λ t = b λ t ( W t − ) P ⊗ G -a.s., and therefore λ Mt = b λ t ( W t − )P ⊗ G -a.s. Then from (22) we obtain L M = b L M ( Y ), which finishes the proof. Remark. As can be seen from the proof, the wealth of an investor who uses the strategy b L does not vanish ( Y m > Y m − > 0) on any solution of the wealth equation (if itexists). This fact is needed in the proof of Proposition 2.26 .4. Proof of Theorem 3 We will use the same notation as in the proof of Theorem 2. Since ln r t is a non-positive submartingale, there exists the limit r ∞ = lim t →∞ r t . As we have shown,ln r t = ln r + Z t + e Z t , where Z t is a submartingale with drift rate d t = h t + h t ≥ 14 (1 − r t − ) k λ t − e λ t k = 14 k λ t − ¯ λ t k . Hence the compensator A t = d · G t of Z t (see (42)) satisfies the inequality A t ≥ k λ − ¯ λ k · G t . Since Z t is bounded from above ( Z t ≤ − ln r ), A t converges to a finite limit A ∞ , so k λ − ¯ λ k · G ∞ < ∞ . Moreover, on the set {k λ − e λ k · G ∞ = ∞} we necessarily have r ∞ = 1, because otherwise we would have A ∞ = ∞ on this set.From the inequality ln(1 − (1 − r s − ) | ∆ e Λ ( s ) s | ) ≤ − (1 − r s − ) | ∆ e Λ ( s ) s | and (47), (48), weobtain e Z t ≥ (1 − r − ) · | e Λ ( s ) t | = | ¯Λ ( s ) t | . Since e Z converges, we have | ¯Λ ( s ) ∞ | < ∞ , and on the set {| e Λ ( s ) ∞ | = ∞} we have r ∞ = 1. Suppose all the investors use the strategy b L . By virtue of (44), W t = W E ( S ) t with theprocess S t = −| b λ ( W − ) | · G t + 1 W − · | X t | = − | x | ζ + | x | ∗ ν t + X s ≤ t | ∆ X s | W s − , where ζ denotes the predictable process ζ t ( W t − ). In particular, the continuous part S ct and the jumps ∆ S t are given by S ct = − | x | I(¯ ν = 0) W − + | x | ∗ ν t , ∆ S t = − Z R N + | x | ζ t + | x | ν { t } ( dx ) + | ∆ X t | W t − = ζ t + | ∆ X t | W t − − . From the formula E ( S ) t = exp( S ct + P u ≤ t ln(1 + ∆ S u )), we find V t = V E ( U ) t with theprocess U t = − S ct − X s ≤ t ∆ S s S s = − S ct + X s ≤ t (cid:18) W s − ζ s + | ∆ X s | − (cid:19) . (59)The continuous part of U t is U ct = − S ct = b U · G t with the predictable process b Ut = Z R N + | x | W t − + | x | K t ( dx ) I(∆ G t = 0) , and the measure of jumps µ U acts on functions f ( ω, t, u ) with f ( ω, t, 0) = 0 as f ∗ µ Ut = f (cid:18) W − ζ + | x | − (cid:19) ∗ µ t + X s ≤ t f s (cid:18) W s − ζ s − (cid:19) I(∆ X s = 0 , ¯ ν s > , 27o its compensator ν U is such that f ∗ ν Ut = f (cid:18) W − ζ + | x | − (cid:19) ∗ ν t + X s ≤ t f s (cid:18) W s − ζ s − (cid:19) (1 − ¯ ν s ) I(¯ ν s > . In particular, ν U = K U ⊗ G with the transition kernel K U such that Z R f t ( u ) K Ut ( du ) = Z R N + f t (cid:18) W t − ζ t + | x | − (cid:19) K t ( dx ) + f (cid:18) W t − ζ t − (cid:19) (1 − ¯ ν t )∆ G t I(∆ G t > . From the definition of ζ in Lemma 2, it follows that R R | u | K Ut ( du ) < ∞ , and hence thedrift rate of U with respect to G t is given by d Ut = b Ut + Z R uK Ut ( du ) ≤ , where the inequality follows from that on the set { ∆ G = 0 } we have ζ t = W t − , and onthe set { ∆ G > } we have Z R N + (cid:18) W t − ζ t + | x | − (cid:19) K t ( dx ) ≤ (cid:18) − W t − ζ t (cid:19) (1 − ¯ ν t )∆ G t in view of that K t ( dx ) = (∆ G t ) − ν { t } ( dx ) and the definition of ζ .Consequently, U t is a σ -supermartingale. This implies that V t is also a σ -super-martingale, and, hence, a usual supermartingale because it is non-negative. In particu-lar, it has an a.s.-limit V ∞ = lim t →∞ V t ∈ [0 , ∞ ), and therefore W ∞ = 1 /V ∞ ∈ (0 , ∞ ],which proves the first claim of the theorem.If ¯ ν ≡ 0, we have ζ t = W t − for all t , so equation (59) becomes U t = − | x | W − + | x | ∗ ( µ t − ν t ) , and, hence, U t is a purely discontinuous local martingale with bounded jumps, ∆ U t ∈ ( − , { V ∞ = 0 } = {| u | ∗ ν U ∞ = ∞} a.s., or equivalently { W ∞ = ∞} = { ( | x | W − + | x | ) ∗ ν ∞ = ∞} a.s. From this andthe existence of the limit W ∞ follows the second claim of the theorem. 7. Appendix: Lebesgue derivatives In this appendix we assemble several known facts about the Lebesgue decompositionand Lebesgue derivatives of σ -finite measures, and prove auxiliary results for randommeasures generated by predictable non-decreasing c`adl`ag processes. The Lebesgue decomposition of σ -finite measures. Let (Ω , F ) be a measurablespace. First recall the following known result, which can be found (in a slightly differentform), e.g., in Chapter 3.2 of [6]. Proposition 3. Let P , e P be σ -finite measures on (Ω , F ) . Then there exists a measurablefunction Z ≥ ( P -a.s. and e P -a.s.) and a set Γ ∈ F such that e P( A ) = Z A Zd P + e P( A ∩ Γ) for any A ∈ F , (60)28 nd P(Γ) = 0 . (61) Such Z is P -a.s. unique and Γ is e P -a.s. unique, i.e. if Z ′ and Γ ′ also satisfy the aboveproperties, then Z = Z ′ P -a.s., and e P(Γ △ Γ ′ ) = 0 (where Γ △ Γ ′ = Γ \ Γ ′ ∪ Γ ′ \ Γ denotesthe symmetric difference). The function Z – the Lebesgue derivative of e P with respect to P – is denoted in thispaper by d e P /d P. If e P ≪ P, the Lebesgue derivative coincides with the Radon–Nikodymderivative and one can take Γ = ∅ . When it is necessary to emphasize that the set Γ isrelated to e P and P, we use the notation Γ e P / P .In an explicit form, Z and Γ can be constructed as follows. Let Q be any σ -finitemeasure on (Ω , F ) such that P ≪ Q, e P ≪ Q (for example, Q = P + e P). Then Z = d e P d Q (cid:18) d P d Q (cid:19) − I (cid:18) d P d Q > (cid:19) , Γ = (cid:26) ω : d P d Q ( ω ) = 0 (cid:27) , where the derivatives are in the Radon–Nikodym sense.By approximating a measurable function with simple functions, from (60), it followsthat for any F -measurable function f ≥ Z Ω f d e P = Z Ω f d e P d P d P + Z Ω f I(Γ) d e P (62)(where the integrals may assume the value + ∞ ).The following proposition contains facts about Lebesgue derivatives that are used inthe paper. Proposition 4. Let P , e P , Q be σ -finite measures on (Ω , F ) . Then the following state-ments are true.(a) Suppose Q is representable in the form Q( A ) = R A f d P + R A e f d e P , where f, e f ≥ are measurable functions, and e f = 0 P -a.s. Then d P d Q = 1 f I( f > , e f = 0) , Γ P / Q = { f = 0 , e f = 0 } . (b) If R is a σ -finite measure such that R ≪ P and R ≪ Q , then d e P d P = d e P d Q d Q d P R -a.s. (63) (c) If R is as in (b), then d Q /d P > and d P /d Q > -a.s.Proof. (a) is obtained by straightforward verification of (60)–(61).(b) Observe that for any A ∈ F we have e P( A ) = Z A d e P d Q d Q + e P( A ∩ Γ e P / Q )= Z A d e P d Q d Q d P d P + Z Ω I( A ∩ Γ Q / P ) d e P d Q d Q + e P( A ∩ Γ e P / Q )= Z A d e P d Q d Q d P d P + e P( A ∩ (Γ Q / P ∪ Γ e P / Q )) , (64)29here to obtain the second equality we applied (62), and to obtain the third one weexpressed the second integral in the second line from the equality e P( A ∩ Γ Q / P ) = Z Ω I( A ∩ Γ Q / P ) d e P d Q d Q + e P( A ∩ Γ Q / P ∩ Γ e P / Q ) . Suppose for A = { d e P d P > d e P d Q d Q d P } we have R( A ) > 0. Then also R( A ′ ) > A ′ = A ∩ (Γ Q / P ∪ Γ e P / Q ∪ Γ e P / P ) c because R(Γ Q / P ) = R(Γ e P / Q ) = R(Γ e P / P ) = 0. Consequently,P( A ′ ) > 0. But this leads to a contradiction between decomposition (60) and equality(64) for e P( A ′ ), since according to them we would have Z A ′ d e P d P d P = Z A ′ d e P d Q d Q d P d P , which is impossible due to the choice of A . Hence R( d e P d P > d e P d Q d Q d P ) = 0. In the same waywe show that R( d e P d P < d e P d Q d Q d P ) = 0.(c) follows from (63) if one takes e P = P. The Lebesgue decomposition of non-decreasing predictable processes. Let(Ω , F , ( F t ) t ≥ , P) be a filtered probability space satisfying the usual assumptions, and P be the predictable σ -algebra on Ω × R + . For a scalar non-decreasing c`adl`ag predictableprocess G , denote by P ⊗ G the measure on P defined asP ⊗ G ( A ) = E( I ( A ) · G ∞ ) , A ∈ P . (65)Observe that P ⊗ G is σ -finite on P . Indeed, this can be shown by considering thepredictable stopping times τ n = inf { t ≥ G t ≥ n } . The stochastic intervals A n =[0 , τ n ) := { ( ω, t ) : t < τ n ( ω ) } are predictable, i.e. A n ∈ P , while P ⊗ G ( A n ) ≤ n and S n A n = Ω × R + . Proposition 5. (a) For any scalar non-decreasing c`adl`ag predictable processes G, e G there exists a predictable process ξ ≥ and a set Γ ∈ P such that up to P -indistingui-shability e G = e G + ξ · G + I(Γ) · e G and I(Γ) · G = 0 . (66) (b) A predictable process ξ ≥ and a set Γ ∈ P satisfy (66) if and only if ξ is a versionof the Lebesgue derivative d (P ⊗ e G ) /d (P ⊗ G ) and Γ is the corresponding set from theLebesgue decomposition. We denote any P ⊗ G -version of such a process ξ by d e G/dG or d e G t /dG t , and callit a predictable Lebesgue derivative of e G with respect to G . When it is necessary toemphasize that the set Γ is related to e G and G , we use the notation Γ e G/G . Proof. Without loss of generality assume e G = 0.(a) Let ξ = d (P ⊗ e G ) /d (P ⊗ G ) and Γ be the corresponding set from the Lebesguedecomposition. Define the process e G ′ = ξ · G + I(Γ) · e G. We have to show that e G ′ = e G . Since e G ′ and e G are c`adl`ag, it is enough to show that e G ′ t = e G t a.s. for any t ≥ 0, and this is equivalent to thatE( e G ′ t I( B )) = E( e G t I( B )) for any B ∈ F t . (67)30et M be the bounded c`adl`ag martingale such that M u = E(I( B ) | F u ). We haveE( e G ′ t I( B )) = E( e G ′ t M t ) = E( M − · e G ′ t ) , and, similarly, E( e G t I( B )) = E( M − · e G t ) , (68)where we used the following fact: if A t is a non-decreasing c`adl`ag predictable process and M t is a bounded c`adl`ag martingale, then for any stopping time τ we have E( M τ A τ ) =E( M − · A τ ). This result is proved in [14, Lemma I.3.12] in the case E A ∞ < ∞ , fromwhich our case follows by a localization procedure.Finally, from the definition of e G ′ and the Lebesgue decomposition of the measureP ⊗ e G , it follows that the measures P ⊗ e G and P ⊗ e G ′ coincide. Hence, for any non-negative P -measurable function f we have E( f · e G t ) = E( f · e G ′ t ), which finishes the proofby (67)–(68).(b) In view of the construction in (a), it only remains to show that if ξ, Γ satisfy(66), then ξ is the Lebesgue derivative and Γ is the corresponding predictable set. Thisfollows from straightforward verification of properties (60)–(61). References [1] P. H. Algoet and T. M. Cover. Asymptotic optimality and asymptotic equipartitionproperties of log-optimum investment. The Annals of Probability , 16(2):876–898,1988.[2] C. D. Aliprantis and K. C. Border. Infinite Dimensional Analysis: A Hitchhiker’sGuide . Springer, 3rd edition, 2006.[3] R. Amir, I. V. Evstigneev, T. Hens, and L. Xu. Evolutionary finance and dynamicgames. Mathematics and Financial Economics , 5(3):161–184, 2011.[4] R. Amir, I. V. Evstigneev, and K. R. Schenk-Hopp´e. Asset market games of survival:a synthesis of evolutionary and dynamic games. Annals of Finance , 9(2):121–144,2013.[5] L. Blume and D. Easley. Evolution and market behavior. Journal of EconomicTheory , 58(1):9–40, 1992.[6] V. I. Bogachev. Measure theory , volume 1. Springer Science & Business Media,2007.[7] L. Breiman. Optimal gambling systems for favorable games. In Proceedings ofthe 4th Berkeley Symposium on Mathematical Statistics and Probability , volume 1,pages 63–68, 1961.[8] C. Dellacherie and P.-A. Meyer. Probabilities and Potential B. Theory of Martin-gales . North-Holland, 1982.[9] Ya. Drokin and M. Zhitlukhin. Relative growth optimal strategies in an assetmarket game. Annals of Finance , published online, 2020.3110] I. Evstigneev, T. Hens, and K. R. Schenk-Hopp´e. Evolutionary behavioral finance.In E. Haven et al., editors, The Handbook of Post Crisis Financial Modelling , pages214–234. Palgrave Macmillan UK, 2016.[11] I. V. Evstigneev, T. Hens, and K. R. Schenk-Hopp´e. Evolutionary stable stockmarkets. Economic Theory , 27(2):449–468, 2006.[12] T. Hens and K. R. Schenk-Hopp´e. Evolutionary stability of portfolio rules in in-complete markets. Journal of mathematical economics , 41(1-2):43–66, 2005.[13] T. Holtfort. From standard to evolutionary finance: a literature survey. Manage-ment Review Quarterly , 69(2):207–232, 2019.[14] J. Jacod and A. Shiryaev. Limit Theorems for Stochastic Processes . Springer,Berlin, 2nd edition, 2002.[15] J. Kallsen. σ -localization and σ -martingales. Theory of Probability & Its Applica-tions , 48(1):152–163, 2004.[16] I. Karatzas and C. Kardaras. The num´eraire portfolio in semimartingale financialmodels. Finance and Stochastics , 11(4):447–493, 2007.[17] I. Karatzas and S. E. Shreve. Methods of Mathematical Finance . Springer, 1998.[18] J. L. Kelly, Jr. A new interpretation of information rate. Bell System TechnicalJournal , 35(4):917–926, 1956.[19] R. S. Liptser and A. N. Shiryaev. Theory of Martingales . Kluwer Academic Pub-lishers, 1989.[20] J. Palczewski and K. R. Schenk-Hopp´e. From discrete to continuous time evolution-ary finance models. Journal of Economic Dynamics and Control , 34(5):913–931,2010.[21] J. Palczewski and K. R. Schenk-Hopp´e. Market selection of constant proportionsinvestment strategies in continuous time. Journal of Mathematical Economics , 46(2):248–266, 2010.[22] E. Platen and D. Heath. A Benchmark Approach to Quantitative Finance . Springer-Verlag, Berlin, 2006.[23] L. Shapley and M. Shubik. Trade using one commodity as a means of payment. Journal of political economy , 85(5):937–968, 1977.[24] M. Zhitlukhin. Survival investment strategies in a continuous-time market modelwith competition. arXiv:1811.12491arXiv:1811.12491 dG s ( ω ) (9) (for s = 0 , we put y − = y ), and for all ω ∈ Ω , z ∈ R M + , t ∈ R + | v mt ( ω, z ) | ∆ G t ( ω ) ≤ z m . (10) (C2) There exist sets Π m,n ∈ P , n = 1 , . . . , N , a non-random function C m : R M + → (0 , ∞ ) , and a predictable c`adl`ag process δ m > such that v m,nt ( ω, z ) = 0 for all ( ω, t ) ∈ Π m,n and z ∈ R N + , (11) and for all ( ω, t ) / ∈ Π m,n and z, e z, a ∈ R M + such that z k , e z k ∈ [ a k / , a k ] for all k ,it holds that v m,nt ( ω, z ) ≥ ( C m ( a ) δ mt ( ω )) − if z m > , (12) v m,nt ( ω, z ) ≤ C m ( a ) δ mt ( ω ) , (13) | v m,nt ( ω, z ) − v m,nt ( ω, e z ) | ≤ C m ( a ) δ mt ( ω ) | z − e z | . (14) Then for any vector of initial wealth y ∈ R M ++ and a predictable non-decreasing c`adl`agprocess H such that G ≪ H , equation (4) has a unique solution (up to P -indistingui-shability). The proof is provided in Section 6. Let us comment on the conditions imposed inthe theorem.In condition (C1), equation (9) restricts the class of strategies under considerationto strategies that from the whole information of investors’ past wealth use only theknowledge of the current wealth y s − , on which depend the “instantaneous” investmentrates v m,nt . The indicator in the integrand appears for the purpose of ensuring thatthe process Y m is non-negative: if Y mu or Y mu − become zero for some u , such a strategystops investing afterwards. For the same reason we require (10) to hold, which meansthat an investor cannot spend more money than available. Note that (C1) implies thatthe realization of the strategy is absolutely continuous with respect to G , i.e. L m ≪ G ,which is a reasonable requirement since if a strategy does not have this property, thenit “wastes” money (invests in assets when the expected payoff is zero).Condition (C2) is needed because the proof is based on a contraction mapping ar-gument. Inequalities (12)–(13) are analogous to similar upper and lower bounds onequation coefficients in such proofs, while (14) is a Lipschitz continuity condition. Notethat it would be too restrictive to require v m,nt to be bounded away from zero globallyin inequality (12). Indeed, if asset n does not yield a payoff “predictably” at time t ,it would be natural to take v m,nt = 0. Therefore, we relax the lower bound on v m byintroducing the sets Π m,n where v m,n may vanish.The conditions of the theorem may look cumbersome, but it is possible to verify thatcertain strategies satisfy them. In particular, in Section 4.2 we do that for a candidateoptimal strategy under mild additional assumptions.9 . Optimal strategies If a strategy profile and a vector of initial wealth are feasible, we define the relativewealth of investor m as the process r mt = Y mt | Y t | , where r mt ( ω ) = 0 if | Y t ( ω ) | = 0.We will be interested in finding strategies for which the relative wealth of an investorgrows on average in the following sense. Definition 3. | L mt ( Y j +1 ) | − | L mτ j ( Y j +1 ) | by (9). In the second integral, on( τ j , τ j +1 ] we have (as follows from (9)) l m,nt ( Y j +1 t ) = d L m,nt ( Y j +1 ) dG t . Consequently, (34) implies that the process Y j +1 satisfies equation (6) for t ≤ τ j +1 .18roceeding by induction, for fixed i we obtain the non-decreasing sequence of stop-ping times τ i,j and the processes Y i,j . Let τ i +1 , = lim j τ i,j ∈ [0 , ∞ ]. On [0 , τ i +1 , ) definethe process Y i +1 , by joining Y i,j , i.e. for ( ω, t ) such that t < τ i +1 , ( ω ) put Y i +1 , t = Y i, t I( t < τ i, ) + ∞ X j =1 Y i,jt I( τ i,j − ≤ t < τ i,j ) . Observe that on the set { τ i +1 , < ∞} , the limit Y i +1 , τ i +1 , − exists, since for t < τ i +1 , theprocess Y i +1 , t satisfies equation (6), in which the integral processes are non-decreasingand bounded by X nτ i +1 , , and the term | L mt | is non-decreasing and bounded by Y m + | X τ i +1 , | . For t ≥ τ i +1 , put Y i +1 , t = Y i +1 , τ i +1 , − − | l m | + N X n =1 F m,n ( l )∆ X nτ i +1 , with l m,n = v m,nτ i +1 , ( Y i +1 , τ i +1 , − ) I( inf s<τ i +1 , Y i +1 , ,ms > G τ i +1 , (the process Y i +1 , stays constant after τ i +1 , ). One can see that now Y i + i, satisfiesequation (6) for t ≤ τ i +1 , . Then the proof of the existence of a solution is finished byinduction. The uniqueness follows from the uniqueness of the fixed point of the operator U on each step of induction. As follows from Theorem 2 (see also the remark after its proof on p. 26), if a solutionof the wealth equation exists and investor m uses the strategy b L , then the wealth ofthis investor does not vanish ( Y m > Y m − > L in which every investor uses either astrategy satisfying the conditions of Theorem 1, or the strategy b L ′ such that, when usedby investor m , its cumulative investment process is b L ′ t ( m ; y ) = Z t y ms − b λ s ( | y s − | ) I( inf u dG s (it differs from the strategy b L only by the presence of the indicator). In order to showthat such a profile is feasible, we will verify conditions (C1), (C2) of Theorem 1 for b L ′ ( m ).Let v m,nt ( ω, z ) = z m b λ nt ( ω, | z | ), so that b L ′ ( m ) can be represented in the form (9).Inequality (10) is satisfied because if ∆ G t ( ω ) > ν t ( ω ) > | b λ t ( ω, c ) | = R R N + | x | ( ζ t ( ω, c ) + | x | ) − ν { t } ( ω, dx ) ≤ b λ .Hence condition (C1) holds.In order to verify condition (C2), consider the setsΠ m,n = { ( ω, t ) : h nt ( ω ) = 0 } and define the function C ( a ) by C ( a ) = max (cid:18) | a | ∨ a m / , | a | ∨ , | a || a | ∧ (cid:19) if a m > , C ( a ) = 1 if a m = 0 , δ t by δ t = sup s ≤ t (cid:18) max n I( h ns > h ns ∨ I(∆ G s > p s ∆ G s (cid:19) ∨ . The local boundedness assumptions imply that δ t is finite-valued.Equality (11) clearly holds. To prove inequalities (12)–(13), consider z, a ∈ R N + suchthat z k ∈ [ a k / , a k ] for all k . Suppose z m > a m > m,n v m,nt ( z ) = z m b λ nt ( | z | ) ≥ z m | z | ∨ h nt ≥ a m / | a | ∨ δ t ≥ C ( a ) δ t , where in the first inequality we used the bound b λ t ( c ) ≥ h t / ( c ∨