Asymptotic minimization of expected time to reach a large wealth level in an asset market game
aa r X i v : . [ q -f i n . M F ] J u l Asymptotic minimization of expected time to reacha large wealth level in an asset market game
Mikhail Zhitlukhin ∗ Abstract
We consider a stochastic game-theoretic model of a discrete-time asset marketwith short-lived assets and endogenous asset prices. We prove that the strategywhich invests in the assets proportionally to their expected relative payoffs asymp-totically minimizes the expected time needed to reach a large wealth level. The re-sult is obtained under the assumption that the relative asset payoffs and the growthrate of the total payoff during each time period are independent and identicallydistributed.
Keywords: asset market game, crossing time, survival strategy, martingales.
MSC 2010:
JEL Classification:
C73, G11.
1. Introduction
One of classical problems in mathematical finance consists in finding an investmentstrategy which reaches a given wealth level as quick as possible. It is generally knownthat in market models with exogenously specified asset prices log-optimal strategiesasymptotically minimize the expected time of reaching a large wealth level, at leastwhen asset returns are specified by i.i.d. random variables; see e.g. the seminal paper [7]for a result in discrete time, or a more recent work [16] for a continuous-time model withL´evy processes. In the present paper we obtain an analogous result for a game-theoreticmodel of a market with endogenous prices – an asset market game .We consider a discrete-time model in which assets yield random payoffs that aredivided between agents (investors) proportionally to the number of shares of each assetheld by an investor. Asset prices are determined endogenously by an equilibrium ofsupply and demand and depend on investors’ strategies. As a result, the evolution ofinvestors’ wealth depends not only on their own strategies and realized asset payoffsbut also on strategies of the other investors in the market. Our goal is to identify aninvestment strategy that allows an investor to reach a large wealth level asymptoticallynot slower, on average, than any other investor in the market. We show that there existsa strategy with this property, and, moreover, it does not depend on the strategies usedby the other investors.This research was motivated by results in evolutionary finance – the field whichstudies financial markets from a point of view of evolutionary dynamics and investigates ∗ 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 proposedby Amir et al. [3] (among earlier models of a similar structure one can mention, e.g.,[6, 11, 13]). Short-lived assets can be purchased by investors at time t , yield payoffs at t + 1, and then the cycle repeats. They have no liquidation value, so investors can getprofit or loss only by receiving asset payoffs and paying for buying new assets. Certainly,such a model is a simplification of a real stock market, however models with short-livedassets have been widely studied in the literature because they are more amenable tomathematical analysis and ideas developed for them may be transferred to more realisticmodels.The main results of the present paper are related to the strategy λ ∗ of [3], which splitsan investment budget between assets proportionally to their expected payoffs. In thatpaper, it was shown that λ ∗ is a survival strategy in the sense that it allows investorsusing it to keep their relative wealth (the share in the total market wealth) boundedaway from zero on the whole infinite time interval with probability 1. As observed in[3], in view of such a structure, this strategy is analogous to the Kelly rule of “bettingone’s beliefs” in markets with exogenous asset prices (see [17]; a collection of paperson the Kelly rule can be found in [20]). Moreover, the key step to show that λ ∗ is asurvival strategy was to prove that it makes the logarithm of the relative wealth of aninvestor who uses it a submartingale, which is analogous to the log-optimality propertyin markets with exogenous prices (see, e.g., [1], or later literature where such strategiesare often called growth-optimal, benchmark, or num´eraire portfolios, [15, 18, 21]).Our first main result shows that the expected time needed for an investor using λ ∗ to reach a wealth level l is asymptotically, as l → ∞ , not greater than the same timefor any other investor in the market, i.e. if we denote these times by τ ∗ l and τ l , then ξ := lim sup l →∞ E τ ∗ l / E τ l ≤
1. Compared to [3], where no conditions on the distributionof asset payoffs are imposed, we require that the payoffs are generated by sequences ofi.i.d. random variables in a certain way, which is a usual assumption in various settingsof time minimization problems in asset market models (cf., e.g., [7, 16]), as well as inearlier works in evolutionary finance.The second main result states that, under some additional conditions, the strictinequality ξ < λ ∗ in some sense, and we find an upper bound for ξ which is strictly less than 1. Itis interesting to note that, among its assumptions, this results requires the payoffs to bestrictly random, and we provide a counterexample with non-random asset payoffs where ξ = 1. In other words, volatility, which we associate here with randomness of payoffs,helps λ ∗ to beat other strategies (see further discussion in Section 3).The paper is organized as follows. Section 2 describes the model, Section 3 statesthe main results, and Section 4 contains their proofs.
2. The model
The model we use is essentially equivalent to that of [3], but it will be more convenientto formulate it using the notation which is more common in stochastic analysis.For ease of exposition, let us first briefly describe the structure of the model in plainlanguage. The market consists of M ≥ N ≥ t = 1 , , . . . , the assets yield random payoffs, which are dividedbetween the investors proportionally to the number of shares of each asset purchased2y an investor at time t −
1. The supply of each asset is given exogenously (and,without loss of generality, is normalized to 1), while the demand depends on actionsof the investors, i.e. their investment strategies. An investment strategy consists ofan investor’s decisions, made at every moment of time simultaneously with the otherinvestors and independently of them, on what proportion of wealth to spend on buyingeach asset. Asset prices are determined by means of the market clearing mechanism,i.e. they are set in such a way that the demand becomes equal to the supply. Then,at the next moment of time, the assets purchased by the investors yield payoffs andthe cycle repeats. The important simplifying modeling assumption consists in that theassets bought at time t − t , i.e. they disappear after yielding payoffswithout any liquidation value and are replaced by their “copies”. Hence, we can saythat they live for just one period and call them short-lived .To define the model formally, introduce a probability space (Ω , F , P) with a filtration F = ( F t ) ∞ t =0 . Payoffs of asset n = 1 , . . . , N are specified by a sequence of randomvariables X nt ≥ t ≥
1, which is F -adapted (i.e. X nt are F t -measurable). It is assumedthat X nt are given exogenously, i.e. do not depend on actions of the investors, and that P n X nt > t ≥ m = 1 , . . . , M is specified by an adapted random sequence Y mt ≥
0. The initial wealth Y m of each investor is non-random and strictly positive.Further evolution of wealth depends on the investors’ actions and the asset payoffs.Actions of investor m are represented by a sequence of vectors of investment proportions λ mt = ( λ m, t , . . . , λ m,Nt ), t ≥
1, according to which this investor allocates the availablebudget Y mt − for buying assets at time t −
1. Short sales are not possible and the wholewealth is reinvested, so the vectors λ mt belong to the standard N -simplex ∆ = { λ ∈ R N + : P n λ n = 1 } .To allow dependence on a random outcome and the history of the market, we definea strategy of an investor as a sequence of functionsΛ t ( ω, y , λ , . . . , λ t − ) : Ω × R M + × ∆ M ( t − → ∆ , t ≥ , which are F t − ⊗ B ( R M + × ∆ M ( t − )–measurable ( B stands for the Borel σ -algebra). Theargument y ∈ R M + corresponds to the vector of initial wealth Y = ( Y , . . . , Y M ). Thearguments λ s = ( λ m,ns ), m = 1 , . . . , M , n = 1 , . . . , N , s = 1 , . . . , t −
1, are investmentproportions selected by the investors at the past moments of time (for t = 1, the functionΛ ( ω, y ) does not depend on λ s ). If this strategy is used by investor m , then thevalue of the function Λ t corresponds to the vector of investment proportions λ mt . Themeasurability of Λ t in ω with respect to F t − means that future payoffs are not knownto the investors at a moment when they decide upon their actions.After the investors have chosen their investment proportions at time t −
1, the equi-librium asset prices p nt − are determined from the market clearing condition that theaggregate demand of each asset is equal to the aggregate supply, which is normalizedto 1. Since investor m can buy x m,nt = λ m,nt Y mt − /p nt − units of asset n , we must have p nt − = M X m =1 λ m,nt Y mt − . If P m λ m,nt = 0, i.e. no one invests in asset n , we put x m,nt = 0 for all m ; in this case theprice p nt − = 0 can be defined in an arbitrary way with no effect on the investors’ wealth,3o we will put p nt − = 0 for convenience. At the next moment of time t , the total payoffreceived by investor m from the assets in the portfolio will be equal to P n x m,nt X nt .Consequently, the wealth sequence Y mt is defined by the recursive relation Y mt ( ω ) = N X n =1 λ m,nt ( ω ) Y mt − ( ω ) P k λ k,nt ( ω ) Y kt − ( ω ) X nt ( ω ) , (1)where λ m,nt ( ω ) denotes the realization of investor m ’s strategy in this market, which isdefined recursively as the sequence λ m,nt ( ω ) = Λ m,nt ( ω, Y , λ ( ω ) , . . . , λ t − ( ω )) . (2)Equation (1) expresses the wealth dynamics of an individual investor in the market.Observe that Y mt implicitly depends on the strategies of the other investors. At thesame time, if some investor m uses a fully diversified strategy, i.e. λ m,nt > t, n, (3)then Y mt > t and the total market wealth, which we will denote by W t = P m Y mt , does not depend on the investors’ strategies and is equal to P n X nt . Remark 1 (On extensions of the model) . The main features of the model which consid-erably simplify its mathematical analysis are that (a) the assets are short-lived, (b) thewhole wealth is reinvested and there is no risk-free asset, (c) there are no short-sales, and(d) the time runs discretely. There is a number of papers where these assumptions arerelaxed. Among them, one can mention, for example, [2], [5, 9], [4], [23] which address,respectively, the limitations (a), (b), (c), (d). However, to my knowledge, there is nogeneral model which would combine all these extensions together.
3. Main results
For a number l >
0, let τ ml denote the stopping time when the wealth of investor m reaches or exceeds the level l for the first time, i.e. τ ml = min { t ≥ Y mt ≥ l } , where min ∅ = ∞ . We are interested in finding a strategy which makes E τ ml smallcompared to other strategies asymptotically as l → ∞ .Our first result, Theorem 1 below, provides such a strategy in an explicit form. Wewill prove it under the following assumption on the payoff sequences. Assumption (A).
The sequences X nt can be represented in the form X n = ρ R n M X m =1 Y m , X nt = ρ t R nt N X i =1 X it − , t ≥ , where(A.1) ρ t > ρ t ) < ∞ , E ln ρ t >
0, and ρ t are independent of F t − for all t ;(A.2) R t = ( R t , . . . , R Nt ) is an adapted sequence of random vectors with values in ∆and there exists ε > R nt | F t − ) ≥ ε for all n and t .4he sequences ρ t and R t have a rather clear interpretation. Indeed, ρ t expresses thegrowth rate of the total payoff ( P n X nt = ρ t P n X nt − ), and R nt = X nt / P i X it are therelative payoffs of the assets. Observe that if (3) holds, then W t = P n X nt = ρ t W t − ,and (A.1) implies that lim t →∞ W t = ∞ by the strong law of large numbers.Introduce the following strategy Λ ∗ , which depends only on t and ω , and has thecomponents Λ ∗ ,nt ≡ λ ∗ ,nt = E( R nt | F t − ) . Note that this is the same strategy as λ ∗ in [3]. Theorem 1.
Let Assumption (A) hold and suppose investor uses the strategy Λ ∗ .Then E τ l < ∞ for any l > , and for any other investor m ∈ { , . . . , M } lim sup l →∞ E τ l E τ ml ≤ . (4)This theorem shows that no investor can reach a wealth level l faster asymptotically(as l → ∞ ) than an investor who uses the strategy Λ ∗ . The next theorem strengthensinequality (4) if the other investor uses an essentially different strategy. We will estab-lish it for the case of two investors in the market and when the following additionalassumption holds. Assumption (B).
The sequence of vectors R t from Assumption (A) is such that(B.1) R t are identically distributed and independent of F t − for all t ;(B.2) R t have linearly independent components, i.e. if P n c n R nt = 0 a.s. for c ∈ R N ,then c = 0.Observe that if this assumption holds, then the strategy Λ ∗ is constant and λ ∗ ,nt = λ ∗ ,n = E( R n ) > t and n , where the inequality follows from (A.2). For a > f ( a ) = sup ( E ln N X n =1 λ n R n λ ∗ ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ∈ ∆ and k λ − λ ∗ k ≥ a ) , where we put f ( a ) = − a is small enough), then it is compact, so the supremum is attained since the aboveexpectation is upper semicontinuous in λ as follows from the Fatou lemma. Moreover,by Jensen’s inequality, for any λ ∈ ∆, we have E ln P n λ n R n /λ ∗ ,n <
0. The inequalityis strict because the logarithm is strictly concave and P n λ n R n /λ ∗ ,n is non-constant asfollows from (B.2). Consequently, f ( a ) < a > Theorem 2.
Let M = 2 and Assumptions (A), (B) hold. Suppose investor 1 uses thestrategy Λ ∗ and investor 2 uses a strategy e Λ t such that its realization e λ t = e λ t ( ω ) satisfiesthe inequality k e λ t − λ ∗ k ≥ a a.s. for all t ≥ with some a > , and e λ nt are uniformlybounded away from zero (i.e. e λ nt > e ε for all t, n and some e ε > ).Then, with θ = E ln ρ > , we have lim sup l →∞ E τ l E τ l ≤ − | f ( a ) | ∧ θθ . (5)Note that the assumption about only two investors, M = 2, is not too restrictive. Inthe case M ≥
3, the above theorem can be used if one replaces investors m = 2 , . . . , M representative investor and let e λ be the realization of the strategy of this newinvestor (see (8)–(10) below). Since the time needed for an individual investor m ≥ τ l is replaced by τ ml .Inequality (4) generally cannot be improved if Assumption (B.2) does not hold. InExample 1 below, we demonstrate this for the case when R t are non-random. This factcan be compared with the known phenomenon of volatility-induced growth in modelswith exogenous asset prices, which consists in that a constant proportions strategy canachieve a growth rate strictly greater than the growth rate of any asset, if the relativeprices are non-constant. If the relative prices are constant this effect disappears, whichmay seem counter-intuitive since usually randomness (or volatility) is regarded as animpediment to financial growth. A popular intuitive explanation of this phenomenonconsists in that a constant proportions strategy “buys low and sells high” (see, e.g., [12]or Chapter 15 in [19]), but such an explanation have known flaws [8]. Example 1.
Suppose W = 1 and the asset payoffs are non-random and given by X nt = R n ρ t , (6)where ρ > R ∈ ∆ with R n ≥ ε for all n and some ε >
0. Clearly, this model satisfiesAssumptions (A) and (B.1), and the strategy Λ ∗ is of the form Λ ∗ t = R for all t . Proposition 1.
Suppose in model (6) investor 1 uses the strategy Λ ∗ and investor 2uses some constant strategy e Λ t = e λ . Then lim l →∞ τ l τ l = 1 .
4. Proofs
We begin with two simple lemmas, which will be needed in the proofs. Throughout thissection, for a vector x ∈ R N we will denote its L and L norms by | x | = P n | x n | and k x k = ( P n ( x n ) ) / . Lemma 1.
Suppose x, y ∈ R N have strictly positive coordinates and | x | = 1 . Then N X n =1 x n (ln x n − ln y n ) ≥ (cid:13)(cid:13)(cid:13)(cid:13) x − y | y | (cid:13)(cid:13)(cid:13)(cid:13) − ln | y | . (7)One can see that this lemma follows from a known inequality for the Kullback-Leibler divergence if x and y/ | y | are considered as probability distributions on a set of N elements. A short direct proof of (7) can be found in [3] (see there Lemma 2, whichis proved for | y | = 1, but easily implies our case as well). Lemma 2.
Let (Ω , F , ( F t ) ∞ t =0 , P) be a filtered probability space. Suppose X t , t ≥ , is anadapted sequence of identically distributed random variables such that X t is independentof F t − and E X t < ∞ for all t ≥ . Denote µ = E X t , σ = Var X t . Then for anystopping time τ ≥ X τ ≤ µ + 2 σ √ E τ . roof. Without loss of generality we may assume µ = 0 and E τ < ∞ . Introducing themartingale M t = P s ≤ t X s , we obtainE X τ ≤ E( M τ − min s ≤ τ M s I( τ > ≤ k min s ≤ τ M s k L ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16)X s ≤ τ X s (cid:17) / (cid:13)(cid:13)(cid:13)(cid:13) L = 2 σ √ E τ , where in the second inequality we applied Wald’s identity E M τ = 0, in the next one theBurkholder–Davis–Gundy inequality, and then Wald’s identity again (see, e.g., Chap-ter 7 of [22] for these results). Proof of Theorem 1 . Let us first show that the proof can be reduced to the case when M = 2 by replacing investors 2 , . . . , M with a representative investor. Let r mt = Y mt /W t denote the relative wealth of the investors and define e Y t = M X m =2 Y mt , e λ nt = M X m =2 r mt − − r t − λ m,nt , (8)where λ m,nt = λ m,nt ( ω ) are the realizations of the strategies defined in (2), and weput e λ nt = 0 when r t − = 1. Since λ ∗ ,nt > n by Assumption (A.2), we have W t = P n X nt >
0, so r mt are well-defined. Denote e τ l = min { t ≥ e Y t ≥ l } (9)and observe that e τ l ≤ τ ml for any m ≥
2. Also, it is straightforward to check that thewealth sequence of investor 1 satisfies the relation Y t = N X n =1 λ ∗ ,nt Y t − λ ∗ ,nt Y t − + e λ nt e Y t − X nt , (10)which is precisely relation (1) in the case of two investors who have the wealth Y t , e Y t and use the strategies Λ ∗ t , e Λ t = e λ t ( ω ), while e τ l is the first moment when the wealth ofthe second investor reaches or exceeds l . Consequently, to prove the theorem, it wouldbe enough to show that lim sup t →∞ E τ l E e τ l ≤ . So, from now on we will deal with the case M = 2. For brevity of notation, we willdenote the realizations of strategies and the wealth of the first and the second investors,respectively, by λ t (= λ ∗ t ), Y t and e λ t , e Y t ; their relative wealth will be denoted by r t and e r t = 1 − r t , and the moments of reaching or exceeding a wealth level l by τ l and e τ l .From (10), we find r t r t − = N X n =1 λ nt r t − λ nt + e r t − e λ nt R nt . (11)Denoting β nt = r t − λ nt + e r t − e λ nt , we obtain the relationE(ln r t | F t − ) − ln r t − ≥ E N X n =1 R nt ln λ nt β nt ! = N X n =1 λ nt (ln λ nt − ln β nt ) ≥ , (12)where in the first inequality we used the concavity of the logarithm, in the second onethat λ nt = E( R nt | F t − ), and in the last inequality applied Lemma 1 to the vectors λ t β t . Inequality (12) implies that ln r t is a submartingale (the integrability of ln r t follows from that r t ≥ r t − min n λ nt ≥ εr t − , which can be seen from (11)). In passing,observe that since this submartingale is non-positive, with probability 1 there existsthe finite limit lim t →∞ ln r t , so inf t ≥ r t >
0. This property allows to call Λ ∗ a survivalstrategy , i.e. an investor “survives” in the market by keeping a share of wealth boundedaway from zero. This result was proved in [3] for general payoff sequences (note that inearlier papers, e.g. [6], the term “survival” has a somewhat different meaning).If E e τ l < ∞ , we findln l ≤ E ln e Y e τ l = E(ln e r e τ l + ln W e τ l ) ≤ E ln W e τ l = θ E e τ l + ln W , (13)where in the last equality we applied Wald’s identity to the sequence of i.i.d. randomvariables ln( W t /W t − ) = ln ρ t . Here, as in Section 3, θ = E ln ρ .Inequality (13) gives us the lower bound E e τ l ≥ θ − ln( l/W ). Then we would liketo obtain an upper bound for E τ l of the same order. To do that, we will work with aslightly altered sequence e λ t , which we will define now.Let ε > δ = ε / r ′ t , t ≥
0, and e λ ′ t , t ≥
1, by the relations r ′ = r , e λ ′ t = e λ t + δλ t I(min n e λ nt ≤ ε/ , r ′ t − ≤ / , t ≥ ,r ′ t = N X n =1 λ nt r ′ t − λ nt r ′ t − + e λ ′ nt (1 − r ′ t − ) R nt , t ≥ . (14)By induction, one can check that r ′ t ≤ r t . Put Y ′ t = r ′ t W t and τ ′ = min { t ≥ Y ′ t ≥ l } .Then we have τ l ≤ τ ′ , so we will look for an upper bound for E τ ′ .Similarly to (12), we can show that ln r ′ t is a submartingale. Indeed, let β ′ t = r ′ t − λ t +(1 − r ′ t − ) e λ ′ t . ThenE(ln r ′ t | F t − ) − ln r ′ t − ≥ N X n =1 λ nt (ln λ nt − ln β ′ tn ) ≥ (cid:13)(cid:13)(cid:13)(cid:13) λ t − β ′ t | β ′ t | (cid:13)(cid:13)(cid:13)(cid:13) − ln | β ′ t | . (15)On the event { e λ ′ t = e λ t } we have | β ′ t | = 1, so the right-hand side of (15) is non-negative.On the event { e λ ′ t = e λ t + δλ t } , there exists a coordinate n = n ( ω ) such that e λ nt ≤ ε/ | β ′ t | = 1 + δ (1 − r ′ t − ) ≤ δ , we can estimate ln | β ′ t | ≤ δ and (cid:13)(cid:13)(cid:13)(cid:13) λ t − β ′ t | β ′ t | (cid:13)(cid:13)(cid:13)(cid:13) ≥ − r ′ t − | β ′ t | ( λ nt − e λ nt ) ≥ ε . Then the choice of δ implies that the right-hand side of (15) is non-negative on the event { e λ ′ t = e λ t + δλ t } as well. Thus, ln r ′ t is a non-positive submartingale, and, in particular,inf t ≥ r ′ t >
0. Since W t → ∞ , we also have τ ′ < ∞ .From now on, assume that l > Y (since we take l → ∞ ). Applying Fatou’s lemma,we obtainln l ≥ lim sup t →∞ E ln Y ′ τ ′ ∧ t − = lim sup t →∞ E (cid:18) ln W τ ′ ∧ t − ln ρ τ ′ ∧ t + ln r ′ τ ′ ∧ t − ln r ′ τ ′ ∧ t r ′ τ ′ ∧ t − (cid:19) . By Wald’s identity, E ln W τ ′ ∧ t = θ E( τ ′ ∧ t ) + ln W . From Lemma 2, E ln ρ τ ′ ∧ t ≤ θ +2 σ p E( τ ′ ∧ t ), where σ = Var(ln ρ ). Since ln r ′ t is a submartingale, E ln r ′ τ ′ ∧ t ≥ ln r .8inally, for all t ≥ r ′ t ≤ r ′ t − / ( δε ). Indeed, if r ′ t − > / r ′ t − ≤ / λ nt ≥ ε ≥ δε and e λ ′ nt ≥ min( ε/ , δλ nt ) ≥ δε to find r ′ t ≤ N X n =1 r ′ t − λ nt r ′ t − + e λ ′ nt (1 − r ′ t − ) R nt ≤ r ′ t − δε . Consequently, we obtain the inequalityln l > lim sup t →∞ (cid:16) θ E( τ ′ ∧ t ) − σ p E( τ ′ ∧ t ) (cid:17) − θ + ln W + ln r + ln( δε ) . Applying the monotone convergence theorem, we can see that E τ ′ should be finite, andhence ln l ≥ θ E τ ′ − σ √ E τ ′ − θ + ln( Y δε ) . (16)Now the claim of the theorem follows from (13), (16), and the relation E τ l ≤ E τ ′ .In the following proofs of Theorem 2 and Proposition 1, we will use the same no-tation for investors 1 and 2 as in the proof of Theorem 1, i.e. without and with tilde,respectively. Proof of Theorem 2 . We can assume that E e τ l < ∞ for all l >
0, as otherwise theproof becomes trivial. Let us begin with an auxiliary estimate. For c ∈ [0 ,
1) we define η c = P t ≥ I( r t < c ) and will now show that E η c < ∞ . As was shown in the proof ofTheorem 1, ln r t is a non-positive submartingale. If we denote by C t its compensator,i.e. the non-negative and non-decreasing sequence C t := t X s =1 E (cid:18) ln r s r s − (cid:12)(cid:12)(cid:12)(cid:12) F s − (cid:19) , then C t a.s.-converges to a limit C ∞ with E C ∞ < ∞ . This follows from the monotoneconvergence theorem since E C t = E ln( r t /r ) ≤ − ln r . Using Lemma 1, similarly to(12) and (15), we see that (with the same β t as in (12)) C ∞ ≥ ∞ X t =1 k λ − β t k = 14 ∞ X t =1 (1 − r t − ) k λ − e λ t k ≥ a ∞ X t =1 (1 − r t − ) . (17)Therefore, η c ≤ C ∞ / ( a (1 − c )) , so E η c < ∞ .From (11), we find e r t e r t − = N X n =1 e λ nt λ n r t − + e λ nt e r t − R nt , which impliesE (cid:18) ln e r t e r t − (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:19) ≤ E (cid:18) ln N X n =1 e λ nt R nt λ n r t − (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:19) ≤ f ( a ) − ln r t − . Since r t is a submartingale, we have E(ln( e r t / e r t − ) | F t − ) ≤ e r t is a supermartingale (its integrability follows from that e ε ≤ e r t / e r t − ≤ min( ε, e ε ) − ).Consequently, using that E e τ l < ∞ and applying Doob’s stopping theorem, we obtainE ln e r e τ l ≤ E e τ l X s =1 min( f ( a ) − ln r s − , . (18)9he possibility of applying Doob’s theorem can be justified by first applying it to thebounded stopping times e τ l ∧ t , then passing to the limit t → ∞ using Fatou’s lemmain the left-hand side of (18) (note that ln e r e τ l ∧ t is bounded from below by the integrablerandom variable e τ l ln e ε + ln e r ), and using the monotone convergence theorem in theright-hand side.Now, similarly to (13), for any c ∈ [ e f ( a ) ,
1) we findln lW ≤ E ln e r e τ l + θ E e τ l ≤ E e τ l X s =1 ( f ( a ) − ln( c )) I ( r s − ≥ c ) + θ E e τ l ≤ ( θ + f ( a ) − ln( c )) E e τ l + (ln c − f ( a )) E η c . Note that since we consider the case E e τ l < ∞ for all l >
0, we necessarily have θ + f ( a ) − ln( c ) >
0. Together with (16), this implieslim sup l →∞ E τ l E e τ l ≤ θ + f ( a ) − ln cθ . Taking c →
1, we obtain the claim of the theorem.
Proof of Proposition 1.
We will assume that e λ = λ , as otherwise the claim of theproposition is obvious. Since W t = ρ t , for investor 1 we have τ l ≥ θ − ln l , where θ = ln ρ .Therefore, it will be enough to show that for investor 2 we have e τ l ≤ ln lθ (1 + o (1)) . (19)Using the wealth equation (1) and that λ n = R n , we obtain e Y t e Y t − = ρ N X n =1 e λ n λ n λ n r t − + e λ n e r t − . (20)Inequality (17) implies that r t →
1, so the right-hand side of (20) is strictly greaterthan 1 for t large enough. Hence e Y t → ∞ , which implies that e τ l < ∞ for all l . Conse-quently, ln l > ln e Y e τ l − = ln W e τ l − + ln e r e τ l − = θ ( e τ l −
1) + ln e r e τ l − . (21)From (20), using the concavity of the logarithm and the inequality ln x ≥ − x − , weobtain the boundln e r t e r t − ≥ N X n =1 e λ n ln λ n λ n r t − + e λ n e r t − ≥ ln r − t − − N X n =1 ( e λ n ) e r t − λ n r t − ≥ e r t − (cid:18) − N X n =1 ( e λ n ) λ n r (cid:19) , where in the last inequality we estimated r t − ≥ r since r t is a non-decreasing sequence(in the proof of Theorem 1, we showed that it is a submartingale). Since e r t → e τ l → ∞ , we get e τ − l ln e r e τ l →
0. Then relation (21) implies (19), which is what is needed.
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