From discrete to continuous time evolutionary finance models

ARTICLE IN PRESS Journal of Economic Dynamics & Control 34 (2010) 913–931

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

From discrete to continuous time evolutionary finance models Jan Palczewski a,c,, Klaus Reiner Schenk-Hoppe´ a,b, a

School of Mathematics, University of Leeds, UK Leeds University Business School, University of Leeds, UK c Faculty of Mathematics, University of Warsaw, Poland b

a r t i c l e in fo

abstract

Article history: Received 9 October 2008 Accepted 11 December 2009 Available online 6 January 2010

This paper aims to open a new avenue for research in continuous-time financial market models with endogenous prices and heterogenous investors. To this end we introduce a discrete-time evolutionary stock market model that accommodates time periods of arbitrary length. The dynamics is time-consistent and allows the comparison of paths with different frequency of trade. The main result in this paper is the derivation of the limit model as the length of the time period tends to zero. The resulting model in continuous time generalizes the workhorse model of mathematical finance by introducing asset prices that are driven by the market interaction of investors following self-financing trading strategies. Our approach also offers a numerical scheme for the simulation of the continuous-time model that satisfies constraints such as market clearing at every time step. An illustration is provided. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.

Keywords: Evolutionary finance Market interaction Wealth dynamics Self-financing strategies Endogenous prices Continuous-time limit

1. Introduction Research in evolutionary finance models, which are employed to study the wealth dynamics driven by the market interaction of investment strategies, has seen tremendous progress in the last few years. This approach has shed light on optimal long-term investment strategies and the dynamics of asset prices in incomplete markets; see Evstigneev et al. (2009) for a survey of the current state of the art in evolutionary finance. All of the existing models in this theory are formulated in discrete time with a fixed length of the time period, cf. Evstigneev et al. (2006, 2008). This approach is suitable, for instance, if investors trade only once every day, quarter or year because of institutional or market constraints. Models with myopic investors, in which the planning horizon and frequency of trade coincide, share this property; see e.g. Brock et al. (2009), Chiarella et al. (2006) and the references in the surveys by Chiarella et al. (2009) and Hommes and Wagener (2009). The comparison of models with different frequency of trade and/or investors’ planning horizon is potentially problematic because the dynamics might lack consistency or scalability with respect to time. The evolutionary finance model of stock markets introduced in this paper allows for time periods of arbitrary length and is time-consistent, i.e. time series with different frequency of trade are related. This model generalizes the approach presented in Evstigneev et al. (2009, 2006, 2008) and Hens and Schenk-Hoppe´ (2005). Time-consistency is achieved by defining investment strategies and asset payoffs as continuous-time processes and then deriving quantities in a discrete-time setting which are meaningful from an economic point of view. Investors adjust portfolios to fit their investment strategies at the beginning of

 Corresponding author at: School of Mathematics, University of Leeds, UK.  Also corresponding author.

E-mail addresses: [email protected] (J. Palczewski), [email protected] (K.R. Schenk-Hoppe´). 0165-1889/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jedc.2009.12.005

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each time period only, akin to tracking a benchmark. Asset payoffs are aggregated over time to lump-sum payments which are paid to the asset holders at the end of each period. The main result in this paper is the derivation of a continuous-time model which describes the limit of the discrete-time dynamics when the length of the time period tends to zero. The resulting dynamics has an explicit representation as a system of random differential equations. It turns out that the limit model generalizes the classical continuous-time ¨ dynamics of self-financing portfolios in mathematical finance (see, for instance, the textbooks by Bjork, 2004 or Shreve, 2004) by introducing market interaction of investors. Our findings highlight the potential of dynamic economic theory, combined with evolutionary ideas, for mathematical finance. Our results on the existence of a continuous-time limit (and the explicit description of the limit model) are useful for practical finance because they provide a numerical scheme for the simulation of the continuous-time dynamics. We prove that the continuous-time dynamics is uniformly approximated (on finite time intervals) by the discrete-time model with small time steps. The approximation error decreases linearly with the length of the time step. This numerical method possesses an advantage over standard numerical techniques because it approximates the continuous-time wealth dynamics with a scheme that ensures clearing of the asset markets in each time period. An illustration of the convergence result is provided. In mathematical finance discrete-time approximations of continuous-time models have a long tradition, see Prigent (2003) for a comprehensive account of approximation theory for financial markets. Most continuous-time models however do not take into account the market interaction of investors. For instance, the market impact of trades (and its implications for optimal behavior of large investors) is usually studied for a single trader facing an exogenous price impact function (e.g. Bank and Baum, 2004). The existence of continuous-time limits for other discrete-time agent-based finance models with finitely many traders (see the surveys by Chiarella et al., 2009; Hommes and Wagener, 2009), by-and-large, is an open problem. One exception is the paper by Buchmann and Weber (2007) who derive the continuous-time limit of a betting market model a la Kelly (1956). This framework, however, rules out capital gains and therefore cannot appropriately model stock markets. There is a strand of literature studying continuous-time models inspired by locally interacting particle models in physics. These limit models are deterministic (though can display chaotic dynamics) and describe the market dynamics on an aggregate level only, see e.g. Lux (2009) and the references therein. The derivation of these models usually rests on quite restrictive assumptions on the types of traders in the market. Related stochastic models in continuous time with finitely many noise traders are given in Alfarano et al. (2008) and Lux (1997). A stochastic agent-based model with chartists and fundamentalists in continuous time is presented in Chiarella et al. (2007); but their approach relies on ad hoc assumptions on agent behavior and the sluggish adjustment of asset prices. Rheinlaender and Steinkamp (2004) consider a closely related model in continuous time which is a stochastic extension of Zeeman’s (1974) one-dimensional deterministic model. Section 2 presents a generalization of the discrete-time model (Evstigneev et al., 2006, 2008) with arbitrarily small time steps and time-dependent investment strategies. A heuristic derivation of its continuous-time limit is given in Section 3. Section 4 shows that this continuous-time model with market interaction of heterogenous investors leads to the classical continuous-time dynamics of self-financing portfolios with consumption. The explicit representation of both models, which are useful for numerical and analytical studies, are derived in Section 5. Section 6 contains the main result on the convergence of the dynamics of the discrete-time evolutionary stock market model to the continuous-time model. An example is discussed in Section 6.2. Section 7 shows how to extend the results to more general dividend processes. Section 8 concludes. All proofs and auxiliary results are collected in Appendixes A–E. 2. Discrete-time evolutionary model This section introduces a discrete-time evolutionary stock market model with time periods of arbitrary length. The model extends the approach presented in Evstigneev et al. (2006, 2008). Given the length n 4 0 of a time period, time is discrete and proceeds through the set tnn ¼ nn, n ¼ 0; 1; 2; . . . . There are K long-lived assets (stocks) with random dividend intensity dðtÞ ¼ ðd1 ðtÞ; . . . ; dK ðtÞÞ, t Z0, with dk ðtÞ Z0 for k ¼ 1; . . . ; K. Although the dependence on the random event o 2 O is suppressed in this notation, the process dðtÞ depends on two arguments and is assumed to be measurable with respect to the product s algebra. Each asset is in positive net supply of one. The total dividend paid by asset k at time tnn þ 1 (to the investors who hold the asset over the time period ½tnn ; tnn þ 1 Þ) is given by Z tn nþ1 Dn;k ðtnn þ 1 Þ ¼ dk ðsÞ ds: tnn

Dividends are paid in terms of a perishable consumption good, with price normalized to one. P There are I investors with initial wealth Vni ð0Þ Z0, i ¼ 1; . . . ; I, such that V n ð0Þ ¼ Ii ¼ 1 Vni ð0Þ 4 0. Each investor is P i n i n i i n represented by a (possibly random) investment strategy l ðtn Þ ¼ ðl1 ðtn Þ; . . . ; lK ðtn ÞÞ, n Z 0. It is assumed that Kk ¼ 1 lk ðtnn Þ ¼ 1 i n i n and lk ðtn Þ 4 0 for all k ¼ 1; . . . ; K. The component lk ðtn Þ describes the investor’s budget share invested in asset k at the point in time tnn .

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Every investor consumes a constant fraction, cn, of his wealth in every period with the remainder being invested in assets. The constant c 40 (with cn o 1) is the same for all investors and represents the intensity of consumption. The i market value of investor i’s investment in asset k held between time tnn and tnn þ 1 is given by ð1cnÞlk ðtnn ÞVn ðtnn Þ. The discrete-time dynamics is defined by an equation describing the evolution of the vector of investors’ wealth between two consecutive points in time. The wealth of investor i, i ¼ 1; . . . ; I, evolves as K X

Vni ðtnn þ 1 Þ ¼

yin;k ðtnn Þ½Sn;k ðtnn þ 1 Þ þ Dn;k ðtnn þ 1 Þ

ð1Þ

k¼1

with portfolio i

ð1cnÞlk ðtnn ÞVni ðtnn Þ ; Sn;k ðtnn Þ

yin;k ðtnn Þ ¼

k ¼ 1; . . . ; K:

ð2Þ

i

The quantity yn;k ðtnn Þ represents the number of shares of asset k owned by investor i at the beginning of the period ½tnn ; tnn þ 1 Þ. P i The market for asset k clears if the total number of shares owned by investors is equal to 1: Ii ¼ 1 yn;k ðtnn Þ ¼ 1. The asset price Sn;k ðtnn Þ is then given by Sn;k ðtnn Þ ¼ ð1cnÞ/lk ðtnn Þ; Vn ðtnn ÞS: P Here /x; yS ¼ Ii ¼ 1 xi yi denotes the scalar product. Inserting (2) and (3) into (1) yields Vni ðtnn þ 1 Þ ¼

ð3Þ

lik ðtnn ÞVni ðtnn Þ ½ð1cnÞ/lk ðtnn þ 1 Þ; Vn ðtnn þ 1 ÞS þDn;k ðtnn þ 1 Þ /lk ðtnn Þ; Vn ðtnn ÞS k¼1 K X

ð4Þ

with i ¼ 1; . . . ; I or, equivalently, in vector notation Vn ðtnn þ 1 Þ ¼ YðLðtnn Þ; Vn ðtnn ÞÞ½ð1cnÞLðtnn þ 1 ÞVn ðtnn þ 1 Þ þ Dn ðtnn þ 1 Þ; where, for each

t ¼ tn , n

the matrix LðtÞ 2 R

KI

is given by Lki ðtÞ ¼

ð5Þ

lik ðtÞ

and

i

Yik ðL; VÞ ¼

Lki V : ðLVÞk

ð6Þ

Eq. (5) is an implicit description of the wealth dynamics of heterogenous investors trading in a stock market. As we show in this paper, the dynamics can be represented in explicit form by solving the intertemporal problem in Vn ðtnn þ 1 Þ which stems from the short-term equilibrium of the asset market, see Section 5. The assets traded between the investors are ‘long-lived,’ they provide a random payoff stream over time and, at any period in time, have a market price. Trade between the investors takes place as an exchange of assets and consumption good (dividend). The behavior of investors is described by investment strategies which are ‘primitives’ (i.e. exogenous) while asset prices and portfolios are endogenous. This marks a departure from the conventional general equilibrium paradigm in which market dynamics is derived from the maximization of utility from consumption. Here the dynamics is driven by the interaction of investment strategies. These strategies are taken as fundamental characteristics of the investors, while the optimality of individual behavior and the coordination of beliefs are not reflected in formal terms but are rather left to the interpretation of the revealed behavior. The components of our model are observable and can be estimated empirically, which makes the theory closer to practical applications, see the discussion in Evstigneev et al. (2008, Chapter 1, 2010). The evolutionary view to this model, which places the emphasis on the processes of selection and mutation of investment strategies, is explained in detail in Evstigneev et al. (2009, 2006). An interpretation of the approach from the perspective of demand theory, which revives the Marshall concept of temporary equilibrium, is provided in Evstigneev et al. (2008). Indeed the models considered in these papers are obtained by setting, in (5), the length of the time period to n ¼ 1. Remark. The derivation of the corresponding model with an arbitrary, strictly increasing sequence of time points ðtn Þ is straightforward. The dynamics is given by (5) when replacing tnn by tn , tnn þ 1 by tn þ 1 , cn by cðtn þ 1 tn Þ and Dn ðtnn þ 1 Þ by R tn þ 1 dðsÞ ds. tn Any solution to (5) possesses the following property on the aggregate wealth of investors. Summation of (4) over i ¼ 1; . . . ; I yields V n ðtnn Þ ¼

I X i¼1

Vni ðtnn Þ ¼

K 1 X D ðt n Þ cn k ¼ 1 n;k n

ð7Þ

with n Z1. This relation says that the market for the consumption good clears, which is simply Walras’ law. The definition of prices ensures clearing of all K asset markets. Since investors exhaust their budgets, Walras’ law implies market clearing for the consumption good.

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Define the set of strictly positive budget shares ( ) K X KI L ¼ L 2 ð0; 1 : Lki ¼ 1 for all i : k¼1

Assumption. To simplify the presentation, we will assume from now on that the sum of the dividend intensities is P constant and, without further loss of generality, set this value equal to one, i.e. d ðtÞ ¼ Kk ¼ 1 dk ðtÞ ¼ 1. The case of a time-dependent process d ðtÞ is discussed in detail in Section 7. Under the above assumption, (7) implies for all n Z 1 V n ðtnn Þ ¼

1 : c

ð8Þ

Define the set ( D¼

) 1 V 2 ½0; 1Þ : V ¼ : c i¼1 I

I X

i

Theorem 1. Fix any n 40 with 0 ocn o1. Suppose Lðtnn Þ 2 L for all n Z 0. (i) For every Vn ðtnn Þ 2 ½0; 1ÞI with V n ðtnn Þ 4 0, there exists a unique Vn ðtnn þ 1 Þ that solves (5). This solution satisfies Vn ðtnn þ 1 Þ 2 D. (ii) For every initial value Vn ð0Þ 2 ½0; 1ÞI with V n ð0Þ 4 0 and a realization of the dividend process dðtÞ, the discrete-time dynamics (5) generates a sample path Vn ðtnn Þ 2 D, n ¼ 1; 2; . . . . The proof of Theorem 1, as well as that of any other result, is provided in the Appendix. This result ensures that the wealth dynamics (5) is well-defined for every length n of the time period. The solution Vn ðtnn Þ has the property (8) at each moment tnn when dividends are paid and investors rebalance their portfolios in a self-financing way. This does not apply at the time when investors enter the market. Therefore, initial values only have to satisfy Vn ð0Þ 2 ½0; 1ÞI with V n ð0Þ 4 0 while Vn ðtnn Þ 2 D, n Z1. The next step is the derivation of the limit of the discrete-time model’s sample paths as n-0. Our main finding is that this limit is the solution of a continuous-time model which is a random differential equation. The most surprising property of this limit model, which is derived in the next two sections, is that it possesses a natural interpretation as the wealth dynamics of self-financing strategies with market interaction. The limit model corresponds to the workhorse model of ¨ mathematical finance (e.g. Bjork, 2004 or Shreve, 2004) but extends the usual wealth dynamics by having endogenous asset prices and heterogenous investors. 3. Heuristic derivation of the limit model It is instructive to present first a short-cut leading to the limit of the discrete-time model (5) as n-0. While the continuous-time model obtained is correct, this approach is heuristic. The proper mathematical proof of the approximation property of sample paths has to proceed differently, see Section 6. The derivation presented here might also provide a valuable shortcut for classroom presentations. i We assume that investors describe their benchmark budget shares l ðtÞ, i ¼ 1; . . . ; I, at every time moment t Z 0, but n portfolios are rebalanced only at the discrete points in time tn , n ¼ 1; 2; . . . . The shorter the length of the time period n, the higher the frequency of rebalancing. A smaller n enables investors to better track their benchmark strategy. In the limit the benchmark is perfectly matched. Using that Vn ðtÞ ¼ YðLðtÞ; Vn ðtÞÞLðtÞVn ðtÞ for all t ¼ tnn , one can rewrite (5) as Vn ðtnn þ 1 ÞVn ðtnn Þ ¼ YðLðtnn Þ; Vn ðtnn ÞÞ½ð1cnÞ½Lðtnn þ 1 ÞVn ðtnn þ 1 ÞLðtnn ÞVn ðtnn Þ þ Dn ðtnn þ 1 ÞcnVn ðtnn Þ: tn

ð9Þ

tn

This representation allows to express the change in each investor’s wealth from time n to n þ 1 as the sum of (from left to right) changes in the asset prices, income from asset payoffs and consumption expenditure. Suppose LðtÞ is differentiable in t for every sample-path. Dividing both sides of (9) by n 4 0 and letting n-0 gives a differential equation dVðtÞ ¼ YðLðtÞ; VðtÞÞ½dðLðtÞVðtÞÞ þ dðtÞ dtcV ðtÞ dt:

ð10Þ

The dynamics described by (10), which is an implicit differential equation, is in continuous time. Analogously to the discrete-time case, marginal changes in the investors’ wealth stem from price changes, asset payoffs and consumption. Unfortunately the derivation is as incomplete as it is brief. One cannot conclude that the sample paths generated by the discrete-time system (9) converge, as n-0, to the solution of the differential equation (10) describing the continuous-time

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system. This would require the estimation of the distance between both solutions over finite time horizons which cannot be derived from the convergence of the difference quotients to the differentials. In this paper we follow a route akin to the convergence of numerical schemes to prove that the sample paths of (10) actually are the limit of the sample paths of the discrete-time model (5) as the length of the time periods, n, tends to zero. This approach relies on deriving a bound on the difference between the paths of continuous and discrete-time models and showing that this bound tends to zero on any compact time interval as n-0. The proof that the limit of the discrete-time model is actually described by the solution to (10) is provided in Section 6. The interpretation of the continuous-time system (10) is put on a solid foundation in the next section. This will be done by presenting an alternative derivation of the limit model. The derivation is based on the continuous-time wealth dynamics of self-financing strategies with consumption and the market interaction of investors. 4. Continuous-time evolutionary model This section shows that the evolutionary stock market model with continuous time (10) generalizes the wealth dynamics of an investor with a self-financing strategy as used in mathematical finance. The main innovation is the incorporation of the market interaction of traders by introducing endogenous prices in this standard framework. We will proceed in two steps. First it is proved that the dynamics (10) is well-defined. Then we show that this evolutionary model satisfies the standard definition of the wealth dynamics of a self-financing strategy in mathematical finance. The stock price process is defined through market clearing. When referring to the continuous-time dynamics we will use the term differential equation throughout this paper; but it is important to point out that these equations are actually integral equations (i.e. solution is always meant in the sense of Carathe´dory). All functions are assumed to be measurable in the sense defined at the beginning of Section 2. The dynamics (10) can be written as i

dV ðtÞ ¼

lik ðtÞV i ðtÞ ½d/lk ðtÞ; VðtÞSþ dk ðtÞ dtcV i ðtÞ dt l ðtÞ; VðtÞS / k k¼1 K X

for i ¼ 1; . . . ; I. Define the set ( L0 2 RKI :

0 ¼ LM

K X

ð11Þ

) 0 Lki ¼ 0 for all i and JL0 J r M

k¼1

with J  J denoting the Euclidean norm. 0 Theorem 2. Suppose LðtÞ 2 L for all t Z 0 and there is a constant M Z 0 such that ð@=@tÞLðtÞ 2 LM for all t Z 0. Then the system (10) has a unique solution VðtÞ 2 D for every initial value Vð0Þ 2 D. The solution is continuous and global, i.e. defined for all t Z 0. 0 means that LðtÞ is differentiable with uniformly bounded derivative. This rules out The assumption ð@=@tÞLðtÞ 2 LM arbitrarily fast changes in the investors’ strategies. It is essentially a technical condition needed to interpret the dynamics (10) as a random differential equation. i i i Suppose investment strategies l ðtÞ ¼ ðl1 ðtÞ; . . . ; lK ðtÞÞ, i ¼ 1; . . . ; I, are given and each asset is in unit supply. Define the price of asset k at time t by 1

I

Sk ðtÞ ¼ lk ðtÞV 1 ðtÞ þ    þ lk ðtÞV I ðtÞ ¼ /lk ðtÞ; VðtÞS:

ð12Þ

Under the assumptions of Theorem 2 the asset price is well-defined with Sk ðtÞ 40. Investor i’s portfolio is defined as

yik ðtÞ ¼

lik ðtÞV i ðtÞ

ð13Þ

Sk ðtÞ 1

I

for k ¼ 1; . . . ; K. The market for each asset k clears because yk ðtÞ þ    þ yk ðtÞ ¼ 1, i.e. demand is equal to supply. Finally denote the consumption process of investor i by i

dC ðtÞ ¼ cV i ðtÞ dt:

ð14Þ

Inserting (12)–(14) into (11), one obtains the dynamics i

dV ðtÞ ¼

K X

yik ðtÞ½dSk ðtÞ þ dk ðtÞ dtdC i ðtÞ:

ð15Þ

k¼1

This equation is the standard mathematical finance textbook definition of the wealth dynamics of a trader employing a ¨ (2004) or Shreve (2004). self-financing strategy with consumption in a market with K assets, see e.g. Bjork Summarizing the discussion, our continuous-time model generalizes the workhorse of mathematical finance (15). In contrast to the standard model, the evolutionary stock market model derives endogenous asset prices from the market interaction of heterogenous investors.

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5. Representation as random dynamical system The evolutionary stock market model, both in discrete and in continuous time, is defined by an implicit equation. The wealth of investors, resp. the differential of the wealth, appears on both sides of the equation. In this section an explicit formulation of the dynamics is derived. The availability of a description of the dynamics as a random dynamical system (Arnold, 1998) is indispensable for efficient numerical simulations and analytical studies of the dynamics. The explicit representation of the discrete-time system is presented in Section 5.1. This result requires invertibility of the ‘capital gains’ matrix. It turns out that a reduction of the dimension of the dynamics is possible by using the market clearing condition for the consumption good. While this dimension-reduction is optional for the discrete-time model, it is necessary to derive an explicit representation of the continuous-time dynamics, Section 5.2. The explicit formulation of both models as random dynamical systems is used in the proof of existence and uniqueness of solution as well as in the main result on the convergence of sample paths as n-0. 5.1. Discrete-time system Rewrite (5) as ½Idð1cnÞYðLðtnn Þ; Vn ðtnn ÞÞLðtnn þ 1 ÞVn ðtnn þ 1 Þ ¼ YðLðtnn Þ; Vn ðtnn ÞÞDn ðtnn þ 1 Þ: An explicit representation of the dynamics can be obtained only if the matrix on the far left of this equation is invertible. This follows from Lemma E.1(i). Therefore the dynamics has the explicit form Vn ðtnn þ 1 Þ ¼ ½Idð1cnÞYðLðtnn Þ; Vn ðtnn ÞÞLðtnn þ 1 Þ1 YðLðtnn Þ; Vn ðtnn ÞÞDn ðtnn þ 1 Þ:

ð16Þ

This representation is used to prove existence and uniqueness of solution which is asserted in Theorem 1 in Section 2. The dimension of the system (16) can be reduced by one. Eq. (8) implies that for n Z 1 VnI ðtnn Þ ¼

I1 1 X V i ðt n Þ:  c i¼1 n n

ð17Þ

Relation (17) might not be satisfied at the initial time, but can be assumed without loss of generality. The vector of initial wealth Vn ð0Þ can be scaled by any positive constant without changing the sample paths Vn ðtnn Þ, n ¼ 1; 2; . . . . This invariance property follows from Eq. (16) and the fact that the investors’ portfolios YðL; VÞ are unchanged if the vector V is multiplied by a positive constant, i.e.

YðL; aVÞ ¼ YðL; VÞ for all a 4 0: From now on we assume that Vn ð0Þ satisfies (17). Define ( ) I1 X i 1 I1 ^ ^ ^ : : V r D ¼ V 2 ½0; 1Þ c i¼1 ^ using (17). Given V 2 D, define V^ ¼ ðV 1 ; . . . ; V I1 Þ, There is a one-to-one correspondence between elements in D and D P 1 I1 ^i ^ Conversely, for V^ 2 D ^ define V ¼ ðV^ ; . . . ; V^ which is in D. ; ð1=cÞ I1 i ¼ 1 V Þ 2 D. Consider the discrete-time dynamics (5). The representation of the corresponding system in terms of V^ is given by   ^ ðLðt n Þ; V^ n ðt n ÞÞ ð1cnÞðL ^ ðt n ÞL ^ I ðt n ÞÞV^ n ðt n Þ þ 1cn lI ðt n Þ þDn ðt n Þ ð18Þ V^ n ðtnn þ 1 Þ ¼ Y n n nþ1 nþ1 nþ1 nþ1 nþ1 c ^ : L  D^ RðI1ÞK defined as with function Y ^ ðL; V^ Þ ¼ L Y ik ki P

i V^ I1 ^j j ¼ 1 ðLkj LkI ÞV þ LkI =c

ð19Þ

^ ðtÞ 2 RKðI1Þ is obtained from LðtÞ by omitting the last column: with i ¼ 1; . . . ; I1 and k ¼ 1; . . . ; K. The matrix L ^ ðtÞ ¼ li ðtÞ L ki k

ð20Þ

with k ¼ 1; . . . ; K and i ¼ 1; . . . ; I1. LI ðtÞ 2 RKðI1Þ is a matrix with I1 identical columns, each being equal to the vector lI ðtÞ:

LIki ðtÞ ¼ lIk ðtÞ

ð21Þ

with k ¼ 1; . . . ; K and i ¼ 1; . . . ; I1. As above one obtains the semi-explicit form   ^ ðLðt n Þ; V^ n ðt n ÞÞðL ^ ðt n ÞL ^ I ðt n ÞÞV^ n ðt n Þ ¼ Y ^ ðLðt n Þ; V^ n ðt n ÞÞ 1cn lI ðt n Þ þDn ðt n Þ : ½Idð1cnÞY n n nþ1 nþ1 nþ1 n n nþ1 nþ1 c

ð22Þ

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Invertibility of the matrix on the far left of the equation is ensured by Theorem E.1 under the assumption of fully diversified investment strategies, i.e. Lðtnn Þ 2 L. One arrives at the explicit form   ^ ðLðt n Þ; V^ n ðt n ÞÞðL ^ ðt n ÞL ^ I ðt n ÞÞ1 Y ^ ðLðt n Þ; V^ n ðt n ÞÞ 1cn lI ðt n Þ þDn ðt n Þ : V^ n ðtnn þ 1 Þ ¼ ½Idð1cnÞY ð23Þ n n nþ1 nþ1 n n nþ1 nþ1 c This representation is used in the proof of the convergence result for the wealth dynamics as n-0. An alternative proof of existence and uniqueness of solution for the discrete-time model can be given using the reduced version (23). The argument is as follows. The mapping of V^ n ðtnn Þ into V^ n ðtnn þ 1 Þ defined by (23) is ^ is more cumbersome in this case than in the one considered in uniquely determined. The direct proof that V^ n ðtnn þ 1 Þ 2 D Theorem 1. 5.2. Continuous-time system While the reduction of dimension is optional for the discrete-time system, it turns out to be a necessary step in deriving an explicit differential equation in the continuous-time case. To verify this claim, rewrite the continuous-time model (10) in semi-explicit form ½IdYðLðtÞ; VðtÞÞLðtÞ dVðtÞ ¼ YðLðtÞ; VðtÞÞdðtÞ dtcVðtÞ dt:

ð24Þ

The matrix on the far left is not invertible because all column sums are equal to zero. The system suffers from an ‘overspecification’ of solution through the existence of at least one linearly dependent row in the capital gains matrix. Therefore one cannot simply proceed as in the discrete-time case. We will show that reducing the dimension by one leads to an invertible matrix in the new system and therefore gives an explicit system. The relation (17) holds for any solution of the continuous-time model: simply take the sum of (11) over i ¼ 1; . . . ; I to P I i verify that VðtÞ 2 D. In addition, one has dV ðtÞ ¼  I1 i ¼ 1 dV ðtÞ. Using these two properties, (24) can be written (analogously to the discrete-time case) in semi-explicit form   ^ ðtÞLI ðtÞÞdV^ ðtÞ ¼ Y ^ ðLðtÞ; V^ ðtÞÞÞ ðd½L ^ ðtÞLI ðtÞÞV^ ðtÞ þ 1 dlI ðtÞ þ dðtÞ dt cV^ ðtÞ dt: ^ ðLðtÞ; V^ ðtÞÞðL ð25Þ ½IdY c As in Section 5.1, Theorem E.1 allows to take the inverse of the matrix on the far left of this equation. Therefore the dynamics (25) can be expressed in explicit form     ^ ðLðtÞ; V^ ðtÞÞðL ^ ðtÞL ^ I ðtÞÞ1 Y ^ ðtÞL ^ I ðtÞÞV^ ðtÞ þ 1 dlI ðtÞ þ dðtÞ dt cV^ ðtÞ dt: ^ ðLðtÞ; V^ ðtÞÞ ðd½L dV^ ðtÞ ¼ ½IdY ð26Þ c Analytical results on the long-term dynamics of the random dynamical system generated by (26) are obtained in Palczewski and Schenk-Hoppe´ (2010). 6. Convergence of sample paths This section presents the main result on the convergence of sample paths generated by the discrete-time model to that of the continuous-time model as the length n of the time period tends to zero. We also give a numerical illustration of the approximation method. 6.1. The result Fix an initial value Vð0Þ ¼ Vn ð0Þ 2 D for both models. Then Theorem 1 ensures, for any fixed n 40, existence and uniqueness of a sample path Vn ðtnn Þ, n ¼ 0; 1; 2; . . . ; while Theorem 2 gives the corresponding result for the sample path VðtÞ, t Z0. Both sample paths take values in the set D. Define the distance between the two sample paths at time tnn by

ann ¼ JVðtnn ÞVn ðtnn ÞJ

ð27Þ

an

with Euclidean norm J  J. Our aim is to derive an upper bound on n (independent of the dividend intensity process dðtÞ) that uniformly converges to zero on compact time intervals of the form ½0; T as n-0. The convergence result requires a slightly stronger assumption (discussed in detail after the theorem) on the investment strategies than is needed for the existence and uniqueness of solution. Define for every e 4 0 the set of fully e diversified strategies ( ) K X Lki ¼ 1 for all i : Le ¼ L 2 ½e; 1KI : k¼1

One has the following result on the convergence of sample paths of the discrete-time model as the length of the time period tends to zero.

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0 Theorem 3. Suppose there exist e; M 40 such that LðtÞ 2 Le and ð@=@tÞLðtÞ 2 LM for all t Z 0. Let Vn ð0Þ ¼ Vð0Þ 2 D. Then, for every T 4 0, there exists a constant C1 40, depending on T; e; M, but independent of Vð0Þ, n 4 0 and ðdðtÞÞt2½0;T , such that ann rC1 n for n ¼ 0; 1; . . . ; bT=nc.

This result implies that the continuous-time model (10) is approximated by the model (5). For small time steps the sample path generated by the discrete-time system is close to the continuous-time sample path. This approximation property shows that the random differential equation (10) is the correct limit model and that the heuristic derivation provides the right answer. The discrete-time model does not correspond to a standard numerical approximation scheme (such as Euler or Runge–Kutta). Its main advantage over standard methods is that asset markets clear at any point in time along all sample paths. No correction procedure (such as projection) is required to ensure that the numerical solution stays in the set D. The condition of fully e diversified investment strategies, which bounds the budget shares strictly away from zero, is needed to ensure a minimal degree of ‘niceness’ in the behavior of sample paths. Under this condition, prices are bounded away from zero and, thus, dividend-yields are bounded away from infinity. This restricts the local movement of the vector of investors’ wealth from being arbitrarily fast. Conditions with a similar spirit are standard in approximation results for numerical schemes of deterministic and stochastic differential equations. It is noteworthy that the convergence in Theorem 3 is uniform in dðtÞ. Viewed as a numerical approximation, the upper bound on the difference of the sample paths holds true for any potential realization of the dividend intensity. Let us give a short intuition for the proof of the convergence result. It is based on an estimate of the distance between sample paths of the discrete- and continuous-time system. The dynamics are compared at the times tnn , n ¼ 1; 2; . . . . The explicit representations for the system with reduced dimension is employed, see Section 5. Lemma B.2 states that the change of the wealth vector between two points in time tnn and tnn þ 1 can be expressed as follows: in the discrete-time case Z n Fðdðtnn þ hÞ; cn; V^ n ðtnn Þ; Lðtnn Þ; Lðtnn þ 1 Þ; L0 ðtnn þhÞÞ dh; ð28Þ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ 0

and in the continuous-time case Z n Fðdðtnn þ hÞ; 0; V^ ðtnn þhÞ; Lðtnn þ hÞ; Lðtnn þ hÞ; L0 ðtnn þhÞÞ dh; V^ ðtnn þ 1 ÞV^ ðtnn Þ ¼

ð29Þ

0

where L0 ðtÞ ¼ ð@=@tÞLðtÞ denotes the matrix of marginal changes of the components of the investment strategies. An upper bound on the absolute distance between both systems’ sample paths can be defined in terms of the difference of the function F over subsets of the domain. This difference can be bounded from above. Finally Gronwall’s lemma is applied to derive the upper bound whose existence is asserted in Theorem 3.

6.2. Example We consider an example to illustrate the continuous-time dynamics and the convergence property of the discrete-time model. Suppose there are three investors and three dividend-bearing long-lived assets (shares). The dividend intensities dðtÞ are driven by a continuous-time Markov process XðtÞ with two states f1; 2g. The Markov process has an intensity of transition from state 1 to 2 equal to 2.2 and an intensity of the opposite transition equal to 1.05. The dividend intensity process is determined by XðtÞ in the following way: ( ð0:7; 0:3; 0Þ; XðtÞ ¼ 1; dðtÞ ¼ ð0:3; 0:2; 0:5Þ; XðtÞ ¼ 2: Investors follow constant proportions investment strategies: 0 1 0:5 0:3 0:1 C LðtÞ ¼ B @ 0:3 0:4 0:2 A: 0:2 0:3 0:7 The strategy of investor i is described by column i. The consumption rate is set to c ¼ 0:05, and the initial wealth is evenly distributed among the investors. The time series of the wealth dynamics and stock prices for one realization of the dividend process are presented in Fig. 1. The solid lines represent the solution to the continuous-time system (computed using a time step of 0.1) and the bold lines are obtained for the discrete-time model with time step length n ¼ 10. This approximation consists of 20 points over the time horizon depicted in the figure. Fig. 1(a) collects the time series of all three investors’ wealth V i ðtÞ resp. Vni ðtnn Þ with i ¼ 1; 2; 3: investor 1 (top), investor 2 (middle) and investor 3 (bottom). Fig. 1(b) shows the corresponding asset prices Sk ðtÞ resp. Sn;k ðtnn Þ with k ¼ 1; 2; 3. Over the

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10

8

6

4 0

50

100

150

200

0

50

100

150

200

8.0

7.5

7.0

6.5

6.0

Fig. 1. Time series of investors’ wealth and stock prices over the horizon [0,200]. (a) Investors’ wealth: exact solution (solid line) and approximation with the step size n ¼ 10 (bold line). (b) Stock prices: exact solution (solid line) and approximation with the step size n ¼ 10 (bold line).

time horizon [0,200], the price of asset 1 increases from 6.0 to about 7.3, the price of asset 2 is nearly constant and the price of asset 3 falls from 8.0 to about 6.5. The results reported in Fig. 1 suggest that the discrete-time approximation of the continuous-time dynamics works well. This statement can be quantified by measuring the approximation error for different lengths of the time step. The approximation error measured by the norm of the difference between the exact (continuous-time) solution and the approximate (discrete-time) solution of the wealth dynamics is reported in Fig. 2. Fig. 2(a) depicts the error for different lengths of the time step. Each simulation run uses the same trajectory of the dividend process as in Fig. 1. Over the short time horizon [0, 200] the error ann seems to grow slowly and linearly. Fig. 2(b) looks at a time horizon 10 times longer than in the previous case. The error appears to be bounded rather than to grow with time. This observation confirms the common wisdom that the numerical approximation of continuous-time dynamical systems is often much better than theory predicts. Despite the fact that Theorem 3 only ensures convergence of sample paths on compact time intervals, the simulation results indicate that the discrete-time dynamics approximates the solution to the continuous-time system well even over very long time horizons. Indeed this feature turns out to be consistent across runs. The reason for the high precision of the numerical approximation over long time horizons could be the existence of a random attractor (consisting only of one point) of the continuous-time dynamical system (10). Some related results on asymptotic behavior of the continuous-time model can be found in Palczewski and Schenk-Hoppe´ (2010).

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0.006 0.005 0.004 0.003 0.002 0.001 0.000 0

50

100

150

200

0

500

1000

1500

2000

0.006

0.004

0.002

0.000

Fig. 2. Approximation error ann ¼ JV ðtnn Þ-Vn ðtnn ÞJ for runs of different lengths. (a) Approximation error for n ¼ 1 (dashed line), n ¼ 5 (bold line) and n ¼ 10 (solid line) over the time horizon [0,200]. (b) Approximation error for n ¼ 10 over the time horizon [0,2000].

7. Time-dependent aggregate dividend intensity This section discusses the validity of the above results under a weaker condition on the sum of the dividend intensities P

d ðtÞ ¼ Kk ¼ 1 dk ðtÞ. Assume that every sample path of d ðtÞ is strictly positive and differentiable. Denote its derivative by d 0 ðtÞ. Suppose also that every sample path of the investment strategy LðtÞ is differentiable—with its derivative denoted by L0 ðtÞ. 7.1. Discrete-time system The discrete-time dynamics (5) as well as its representation in explicit form (16) are both valid when the aggregate dividend intensity is time-dependent. Summation of (4) over i ¼ 1; . . . ; I gives V n ðtnn Þ ¼ D n ðtnn Þ=ðcnÞ for n Z1 (see (7)), where Z tnn K X D n ðtnn Þ ¼ Dn;k ðtnn Þ ¼ d ðtÞ dt: ð30Þ n tn1

k¼1

Define the set ( Dn

ðt n Þ ¼ n

I X

D n ðtnn Þ V 2 ½0; 1Þ : V ¼ cn i¼1 I

)

i

for each n ¼ 1; 2; . . . . Note that Dn ðtnn Þ ¼ D for n Z1 if d ðtÞ ¼ 1 for t 2 ½0; 1Þ. One has the following assertion which generalizes Theorem 1.

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Lemma 1. Fix any n 4 0 with 0 o cn o 1. Suppose Lðtnn Þ 2 L for all n Z 0. Assume that every sample path of d ðtÞ is strictly positive. (i) For every Vn ðtnn Þ 2 ½0; 1ÞI with V n ðtnn Þ 40, there exists a unique Vn ðtnn þ 1 Þ that solves (5). This solution satisfies Vn ðtnn þ 1 Þ 2 Dn ðtnn þ 1 Þ. (ii) For every initial value Vn ð0Þ 2 ½0; 1ÞI with V n ð0Þ 4 0 and a realization of the dividend process dðtÞ, the discrete-time dynamics (5) generates a sample path Vn ðtnn Þ 2 Dn ðtnn Þ, n ¼ 1; 2; . . . . The reduction of the dimension is similar to the procedure outlined in Section 5.1. Here we employ the relation VnI ðtnn Þ ¼ D n ðtnn Þ=ðcnÞ

I1 X

Vni ðtnn Þ

ð31Þ

i¼1

for all n Z 1 and the fact that the sample path is invariant to the scaling of the initial wealth vector Vn ð0Þ. We will assume without loss of generality that Vn ð0Þ 2 Dn ð0Þ, where Dn ð0Þ is given by (30) with D n ð0Þ ¼ nd ð0Þ (which is strictly positive by our assumption on the dividend intensity). Using (31) one has a one-to-one correspondence between ^ n ðt n Þ, where Vn 2 Dn ðtnn Þ and V^ n ¼ ðVn1 ; . . . ; VnI1 Þ 2 D n ( ) I1 n X ^ i r D n ðtn Þ : ^ n ðt n Þ ¼ V^ 2 ½0; 1ÞI1 : D V n cn i¼1 The system (5) can be written in terms of V^ as   ^ n ðLðt n Þ; V^ n ðt n Þ; D n ðt n ÞÞ ð1cnÞðL ^ ðt n ÞL ^ I ðt n ÞÞV^ n ðt n Þ þ 1cn lI ðt n ÞD n ðt n Þ þDn ðt n Þ ; V^ n ðtnn þ 1 Þ ¼ Y n n n nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 cn

ð32Þ

^ n : Hn -RðI1ÞK , with where the function Y ( ) I1 X i D Hn ¼ ðL; V^ ; DÞ 2 L  ½0; 1ÞI1  ð0; 1Þ : V^ r cn i¼1 is defined as ^ ^ Y n;ik ðL; V ; DÞ ¼ Lki P

i V^

I1 ^j j ¼ 1 ðLkj LkI ÞV þ LkI D=ðc

nÞ:

By virtue of Theorem E.1, the explicit representation of (32) is given by   ^ n ðLðt n Þ; V^ n ðt n Þ; D n ðt n ÞÞðL ^ ðt n ÞL ^ I ðt n ÞÞ1 Y ^ n ðLðt n Þ; V^ n ðt n Þ; D n ðt n ÞÞ 1cn lI ðt n Þ þDn ðt n Þ : V^ n ðtnn þ 1 Þ ¼ ½Idð1cnÞY n n n nþ1 nþ1 n n n nþ1 nþ1 c ð33Þ This equation generalizes (23) to the case of a time-dependent aggregate dividend intensity process. 7.2. Continuous-time system We now show how to extend the results in Section 5.2. First note that the dynamics (10) and (11) are both correct for a time-dependent aggregate dividend intensity. The aggregate wealth at time t in the continuous-time model is given by d ðtÞ=c (sum (11) over i ¼ 1; . . . ; I), i.e. VðtÞ 2 Dðd ðtÞÞ, where ( ) I X Dðd Þ ¼ V 2 ½0; 1ÞI : V i ¼ d =c : i¼1

The following lemma generalizes Theorem 2. 0 Lemma 2. Suppose LðtÞ 2 L for all t Z0 and there exists an M 4 0 such that L0 ðtÞ 2 LM and jd 0 ðtÞj rM for all t Z 0. Then the system (10) has a unique solution VðtÞ 2 Dðd ðtÞÞ, t Z 0, for every initial value Vð0Þ 2 Dðd ð0ÞÞ. The solution is continuous and global.

Lemma 2 implies that for all t Z 0 V I ðtÞ ¼ d ðtÞ=c

I1 X i¼1

V i ðtÞ:

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Using this relation, the dimension of the system (10) can be reduced by one. It is straightforward to show that   ^ ðLðtÞ; V^ ðtÞ; d ðtÞÞ ðL ^ ðtÞLI ðtÞÞdV^ ðtÞ þðd½L ^ ðtÞLI ðtÞÞV^ ðtÞ þ 1 d½d ðtÞlI ðtÞ þ dðtÞ dt cV^ ðtÞ dt; dV^ ðtÞ ¼ Y c

ð34Þ

^ : H-RðI1ÞK , with where the function Y ( ) I1 X i d H ¼ ðL; V^ ; d Þ 2 L  ½0; 1ÞI1  ð0; 1Þ : V^ r c i¼1 is given by ^ ðL; V^ ; d Þ ¼ L Y ik ki P

i V^ I1 ^j j ¼ 1 ðLkj LkI ÞV þ LkI d =c:

Eq. (34) can be written in explicit form (see Theorem E.1)   ^ ðLðtÞ; V^ ðtÞÞðL ^ ðtÞL ^ I ðtÞÞ1  Y ^ ðLðtÞ; V^ ðtÞÞððd½L ^ ðtÞL ^ I ðtÞÞV^ ðtÞ þ 1 d½d ðtÞlI ðtÞ þ dðtÞ dtÞcV^ ðtÞ dt : dV^ ðtÞ ¼ ½IdY c

ð35Þ

7.3. Convergence result The dynamics of the discrete-time model (33) and the continuous-time model (35) (both with time-dependent aggregate dividend intensity d ðtÞ) are close in the sense that the distance between both paths at times tnn ,

ann ¼ JVðtnn ÞVn ðtnn ÞJ; converges to zero as n-0. The convergence result, Theorem 3, is generalized by the following theorem. 0 Theorem 4. Suppose d ðtÞ is twice differentiable and there exist e; M 40 such that L0 ðtÞ 2 LM , jd 0 ðtÞj r M, jd 00 ðtÞj rM, d ðtÞ Z e and LðtÞ 2 Le . Let Vn ð0Þ ¼ Vð0Þ 2 Dðd ð0ÞÞ. For every T 40 there exists a constant C1 4 0 depending on T; e; M and d ð0Þ (but independent of Vð0Þ, n 40 and ðdðtÞÞt2½0;T ), such that ann r C1 n for n ¼ 0; 1; . . . ; bT=nc.

Note that one has Dn ð0Þ ¼ Dðd ð0ÞÞ for n 40. Therefore, the set of initial conditions for the discrete- and continuous-time models are identical. 8. Conclusion This paper marks the birth of continuous-time evolutionary finance. It opens an avenue for the study of the wealth dynamics of interacting investment strategies (and the endogenous asset price dynamics it entails) in continuous time. We derive the continuous-time limit of a generalization of the discrete-time evolutionary stock market model by Evstigneev et al. (2006, 2008). This limit model, which is obtained by letting the length of the time period tend to zero, has an explicit representation as a random dynamical system and possesses a meaningful interpretation from an economics and finance point of view. The continuous-time model extends the standard framework of mathematical finance by introducing (endogenous) stock prices which are driven by the market interaction of investors. An efficient numerical simulation of the limit model is possible because our approximation results provide an explicit scheme which converges uniformly on finite time intervals. Our numerical method has the advantage of delivering market clearing in every time step without the need of projection techniques. Future research will focus on analytical and numerical studies of the dynamics of the continuoustime model. First results are obtained in Palczewski and Schenk-Hoppe´ (2010).

Acknowledgments Financial support by the Finance Market Fund, Norway (project Stability of Financial Markets: An Evolutionary Approach) and the National Center of Competence in Research ‘Financial Valuation and Risk Management,’ Switzerland (project Behavioural and Evolutionary Finance) is gratefully acknowledged. We thank the referees and the editor (Carl Chiarella) for their comments. Appendix A

Proof of Theorem 1. Lemma E.1(i) implies that the discrete-time dynamics (5) is equivalent to the explicit representation (16). The assumption Lðtnn Þ 2 L and V n ðtnn Þ 40 imply that YðLðtnn Þ; Vn ðtnn ÞÞ is well-defined. Therefore, (16) (and thus (5)) has a unique solution.

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Lemma E.1(i) also states that the inverse matrix in (16) maps the non-negative orthant ½0; 1ÞI into itself. Since the vector

YðLðtnn Þ; Vn ðtnn ÞÞDn ðtnn þ 1 Þ has only non-negative coordinates, one finds that Vn ðtnn þ 1 Þ is non-negative. Eq. (8) further gives that the coordinates of Vn ðtnn þ 1 Þ sum up to 1=c. This implies Vn ðtnn þ 1 Þ 2 D.

&

Proof of Theorem 2. The proof of existence and uniqueness of the solution to the model in continuous time employs standard results from the theory of random differential equations. Lemma B.2(ii) states that the explicit system (26) is equivalent to dV^ ðtÞ ¼ FðdðtÞ; 0; V^ ðtÞ; LðtÞ; LðtÞ; L0 ðtÞÞ dt:

ð36Þ

Lemma B.1 implies local Lipschitz continuity of F in the argument V^ and local integrability in all arguments (due to their boundedness). Arnold (1998, Theorem 2.2.1) ensures existence and uniqueness of the solution. The ^ is forward solution is global because all components are non-negative and their sum is finite for all t. In fact, the set D invariant. & Appendix B Define the set of dividend intensities ( ) K X dk ¼ 1 : S ¼ d 2 ½0; 1ÞK : k¼1

^  L  L  RKI -RI1 as Define the function F : S  ½0; 1  D     ^ ðL1 ; V^ ÞðL ^ 2 L ^ I Þ1  Y ^ ðL1 ; V^ Þ d þ ð1aÞðL ^ 0 L ^ 0 I ÞV^ þ 1a0L;I cV^ : Fðd; a; V^ ; L1 ; L2 ; L0 Þ ¼ ½Idð1aÞY 2 c

ð37Þ

^ 2 RKðI1Þ denotes the matrix obtained from L by omitting the last column, and The vector 0L;I is the last column of L0 . L I KðI1Þ ^ has I1 identical columns, each being equal to the last column of L. the matrix L 2 R Lemma B.1. The function F is continuously differentiable on its domain. ^ ðL1 ; V^ ÞðL^2 LI Þ is invertible. Therefore the function Proof of Lemma B.1. Theorem E.1 implies that the matrix Idð1aÞY 2 F is well-defined. Direct computation shows that F is continuously differentiable. & Lemma B.2. Suppose LðtÞ 2 L and its derivative L0 ðtÞ exists for all t Z 0 along every sample path. (i) The solution to (23) fulfills Z n Fðdðtnn þ hÞ; cn; V^ n ðtnn Þ; Lðtnn Þ; Lðtnn þ 1 Þ; L0 ðtnn þ hÞÞ dh: V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ 0

(ii) The solution to (26) satisfies dV^ ðtÞ ¼ FðdðtÞ; 0; V^ ðtÞ; LðtÞ; LðtÞ; L0 ðtÞÞ dt: Proof of Lemma B.2. (i) The investor’s wealth V^ n ðtnn Þ, less his consumption, is fully invested in the available assets, i.e. V^ n ðtnn Þ ¼

1 ^ Y ðLðtnn Þ; V^ n ðtnn ÞÞSn ðtnn Þ; 1cn

see (2). Using the market clearing condition (3), this implies   ^ ðLðt n Þ; V^ n ðt n ÞÞ ðL ^ ðt n ÞL ^ I ðt n ÞÞV^ n ðt n Þ þ 1 lI ðt n Þ : V^ n ðtnn Þ ¼ Y n n n n n n c Inserting into (18) yields h ^ ðÞ ð1cnÞðL ^ ðt n ÞL ^ I ðt n ÞÞðV^ n ðt n ÞV^ n ðt n ÞÞ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ cnV^ n ðtnn Þ þ Y nþ1 nþ1 nþ1 n I

I

^ ðt n ÞL ^ ðt n ÞL ^ ðt n Þ þ L ^ ðt n ÞÞV^ n ðt n Þ þ þð1cnÞðL nþ1 n nþ1 n n

 1cn I n I ðl ðtn þ 1 Þl ðtnn ÞÞ þ Dn ðtnn þ 1 Þ ; c

^ ðLðt n Þ; V^ n ðt n ÞÞ. Lemma E.1(ii) implies that this equation can be written in ^ ðÞ is used as a shorthand notation for Y where Y n n explicit form h ^ ðÞðL ^ ðt n ÞL ^ I ðt n ÞÞ1  ð1cnÞY ^ ðÞðL ^ ðt n ÞL ^ ðt n ÞL ^ I ðt n Þ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ ½Idð1cnÞY nþ1 nþ1 nþ1 n nþ1  I n 1cn ^ I n I n n n n ^ ^ ^ ^ þ L ðtn ÞÞV n ðtn Þ þ Y ðÞðl ðtn þ 1 Þl ðtn ÞÞ þ Y ðÞDn ðtn þ 1 ÞcnV n ðtn Þ : c This proves (i).

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(ii) Differentiability of LðtÞ for every t implies that integration with respect to dLðtÞ can be substituted by the integration ^ ðtÞ and L ^ I ðtÞ. Therefore, the dynamics (26) is equivalent with respect to L0 ðtÞ dt, Rudin (1976, p. 325). This also applies to L to the formulation in (ii). &

Appendix C

Proof of Theorem 3. Denote the distance between the two sample paths of the models with reduced dimension derived in Section 5 by

a^ nn ¼ JV^ ðtnn ÞV^ n ðtnn ÞJ; where J  J denotes the Euclidean norm. n One has ðann Þ2 r Iða^ n Þ2 because JVðtnn ÞVn ðtnn ÞJ2

¼ JV^ ðtnn ÞV^ n ðtnn ÞJ2 þ

I1  X

 i i V^ ðtnn ÞV^ n ðtnn Þ

!2 r IJV^ ðtnn ÞV^ n ðtnn ÞJ2 ;

i¼1

which implies that it suffices to obtain the convergence result for the ‘hat’-system. According to the assumptions of the theorem there are e; M 40 such that LðtÞ 2 Le for all t Z0 and its derivative satisfies 0 L0 ðtÞ ¼ ð@=@tÞLðtÞ 2 LM for all t Z 0. The function F (which is defined in (37)) is continuously differentiable by Lemma B.1 0 ^  Le  Le  LM is compact. This implies existence of a constant C2 such that for any d 2 S, and the set S  ½0; 1  D ^ ^ ^ a; a 2 ½0; 1, V ; V  2 D, L1 ; L1; ; L2 ; L2; 2 Le and L0 2 LM0 JFðd; a; V^ ; L1 ; L2 ; L0 ÞJ r C2 ;

ð38Þ

JFðd; a; V^ ; L1 ; L2 ; L0 ÞFðd; a ; V^  ; L1; ; L2; ; L0 ÞJ rC2 ðJV^ V^  J þ jaa j þJL1 L1; J þ JL2 L2; JÞ:

ð39Þ

This result plays an important role in the derivation of estimates in the remainder of this proof. Subtracting Eq. (28) for the discrete-time system V^ n ðtnn þ 1 Þ from Eq. (29) for the continuous-time system V^ ðtnn þ 1 Þ and taking norms on both sides, yields Z  n

a^ nn þ 1 r a^ nn þ  

0

   ½Fðdðtnn þ hÞ; 0; V^ ðtnn þ hÞ; Lðtnn þ hÞ; Lðtnn þ hÞ; L0 ðtnn þhÞÞFðdðtnn þ hÞ; cn; V^ n ðtnn Þ; Lðtnn Þ; Lðtnn þ 1 Þ; L0 ðtnn þhÞÞ dh: 

This inequality remains valid if the norm is pulled inside the integral (due to Jensen’s inequality). The estimate (39) implies that Z   n  a^ nn þ 1 r a^ nn þ  C2 ðJV^ ðtnn þhÞV^ n ðtnn ÞJ þJLðtnn þ hÞLðtnn ÞJ þ JLðtnn þhÞLðtnn þ 1 ÞJ þcnÞ dh:  0  Boundedness of the derivative L0 ðtÞ gives JLðtnn þ hÞLðtnn ÞJ rMh; JLðtnn þ hÞLðtnn þ 1 ÞJ r MðnhÞ: The triangle inequality and the relation (38) yield n JV^ ðtnn þ hÞV^ n ðtnn ÞJ rJV^ ðtnn þ hÞV^ ðtnn ÞJ þ JV^ ðtnn ÞV^ n ðtnn ÞJ rC2 h þ a^ n :

Inserting these estimates provides the following upper bound on the approximation error at time tnn þ 1 :

a^ nn þ 1 r a^ nn þ

Z n 0

n

n

C2 ðC2 h þ a^ n þMh þ MðnhÞ þ cnÞ dh r a^ n þ

n

¼ ð1 þ nC2 Þa^ n þ

1 2 2 1 n C n þ nC2 a^ n þ C2 M n2 þ C2 cn2 2 2 2

C2 2 n ðC2 þ 2M þcÞ: 2

^ This implies a^ n ¼ 0. Gronwall’s lemma gives At the initial time, V^ ð0Þ ¼ V^ n ð0Þ 2 D. 0

a^ nn r

ð1þ nC2 Þn C2 2 1 n ðC2 þ 2M þcÞ r ennC2 ðC2 þ2M þ cÞn; nC2 2 2

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where the second estimate uses the inequality ð1 þ aÞn rena for a Z0. If nn r T, the expression ennC2 is bounded by eTC 2 . Thus, the constant C1 in the assertion of the theorem is given by pffiffi I TC 2 e ðC2 þ 2M þ cÞ: C1 ¼ 2 This completes the proof.

&

Appendix D

Proof of Lemma 1. The proof is analogous to that of Theorem 1.

&

Proof of Lemma 2. The proof is analogous to that of Theorem 2.

&

Proof of Theorem 4. We derive integral representations of the wealth increments of the continuous- and discrete-time dynamics. These formulations generalize those obtained in Lemma B.2. Define G : Ae -RI1 by I

^ ðL1 ; V^ ; wÞðL ^ 2 L ^ Þ1 Gðd; a; V^ ; L1 ; L2 ; L0 ; d; d 0 ; wÞ ¼ ½Idð1aÞY 2   0 ^ ^ ^ 0 I ÞV^ þ 1a ðd0L ^ ;I þd 0 L ^ ;I ÞÞcV^ ; ^  Y ðL1 ; V ; wÞðd þ ð1aÞðL L c

ð40Þ

where ^ ðL; V^ ; wÞ ¼ L Y ik ki P

i V^

I1 ^j j ¼ 1 ðLkj LkI ÞV þ LkI w=c

and ( Ae ¼

ðd; a; V^ ; L1 ; L2 ; L0 ; d; d 0 ; wÞ 2 ½0; 1ÞK  ½0; 1  ½0; 1ÞI1  Le  Le

0 LM

 ½e; 1Þ  ½M; M  ½e; 1Þ :

I1 X

i V^ r w=c and d; w;

i¼1

K X

)

dk r d ð0Þ þTM :

k¼1

Using the function G, the continuous-time model (35) can be written as dV^ ðtÞ ¼ GðdðtÞ; 0; V^ ðtÞ; LðtÞ; LðtÞ; L0 ðtÞ; d ðtÞ; d 0 ðtÞ; d ðtÞÞ dt: Therefore, V^ ðtnn þ 1 ÞV^ ðtnn Þ ¼

Z

tnn þ 1 tnn

GðdðtÞ; 0; V^ ðtÞ; LðtÞ; LðtÞ; L0 ðtÞ; d ðtÞ; d 0 ðtÞ; d ðtÞÞ dt:

ð41Þ

We now consider the discrete-time system. The identity   ^ n ðLðt n Þ; V^ n ðt n Þ; D n ðt n ÞÞ ðL ^ ðt n ÞL ^ I ðt n ÞÞV^ n ðt n Þ þ 1 lI ðt n ÞD n ðt n Þ V^ n ðtnn Þ ¼ Y n n n n n n n n cn inserted into (32) gives h ^ n ðLðt n Þ; V^ n ðt n Þ; D n ðt n ÞÞ  ð1cnÞðL ^ ðt n ÞL ^ I ðt n ÞÞðV^ n ðt n ÞV^ n ðt n ÞÞ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ cnV^ n ðtnn Þ þ Y n n n nþ1 nþ1 nþ1 n I

I

^ ðt n ÞL ^ ðt n ÞL ^ ðt n Þ þ L ^ ðt n ÞÞV^ n ðt n Þ þ þ ð1cnÞðL nþ1 n nþ1 n n

1cn I n I ðl ðtn þ 1 ÞD n ðtnn þ 1 Þl ðtnn ÞD n ðtnn ÞÞ þ Dn ðtnn þ 1 Þ: cn

This equation can be written in explicit form (see Theorem E.1) h ^ n ðÞðL ^ ðt n ÞL ^ I ðt n ÞÞ1  ð1cnÞY ^ n ðÞðL ^ ðt n ÞL ^ ðt n ÞL ^ I ðt n Þ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ ½Idð1cnÞY nþ1 nþ1 nþ1 n nþ1 1cn ^ ^ n ðÞDn ðt n ÞcnV^ n ðt n Þ; Y n ðÞðlI ðtnn þ 1 ÞD n ðtnn þ 1 ÞlI ðtnn ÞD n ðtnn ÞÞ þ Y nþ1 n cn n n n ^ n ðLðt Þ; V^ n ðt Þ; D n ðt ÞÞ. Using the function G, this dynamics has the integral representation for n Z1: ^ n ðÞ denotes Y where Y n n n Z n D n ðtnn þhÞ d ðtnn þhÞd ðtnn þhnÞ D n ðtnn Þ V^ n ðtnn þ 1 ÞV^ n ðtnn Þ ¼ Gðdðtnn þ hÞ; cn; V^ n ðtnn Þ; Lðtnn Þ; Lðtnn þ 1 Þ; L0 ðtnn þhÞ; ; ; Þ dh; ð42Þ I

^ ðt n ÞÞV^ n ðt n Þ þ þL n n

0

n

n

n

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where the definition (30) of D n is extended to arbitrary values t Z n by setting Z t D n ðtÞ ¼ d ðsÞ ds: tn

The relation (42) does not apply for n ¼ 0 because d ðtÞ is defined for positive arguments only. Subtracting (42) from (41), taking norms on both sides and applying Jensen’s inequality, one obtains for n Z 1 Z tn

nþ1

n n a^ n þ 1 r a^ n þ

GðdðtÞ; 0; V^ ðtÞ; LðtÞ; LðtÞ; L0 ðtÞ; d ðtÞ; d 0 ðtÞ; d ðtÞÞ

n tn !

D n ðtÞ d ðtÞd ðtnÞ D n ðtnn Þ

G dðtÞ; cn; V^ n ðtnn Þ; Lðtnn Þ; Lðtnn þ 1 Þ; L0 ðtÞ; ; ;

dt;

n n n n where a^ n ¼ JV^ ðtnn ÞV^ n ðtnn ÞJ. The same arguments as in Lemma B.1 can be applied to show that the function G is continuously differentiable on its domain Ae . Compactness of Ae implies existence of a constant C3 such that JGðaÞGða0 ÞJ r C3 Jaa0 J and JGðaÞJ r C3 for any a; a0 2 Ae . Therefore

a^ nn þ 1 r a^ nn þ

Z

tnn þ 1 tnn

C3 cn þ JV^ ðtÞV^ n ðtnn ÞJ þ JLðtÞLðtnn ÞJ þ JLðtÞLðtnn þ 1 ÞJ þ jd ðtÞD n ðtÞ=nj

  !  d ðtÞd ðtnÞ  þ d 0 ðtÞ  þjd ðtÞD n ðtnn Þ=nj dt:   n Boundedness of the derivative L0 ðtÞ yields JLðtÞLðtnn ÞJ r Mðttnn Þ and JLðtÞLðtnn þ 1 ÞJ r Mðtnn þ 1 tÞ for all t 2 ½tnn ; tnn þ 1 . The triangle inequality and boundedness of G on Ae imply n

JV^ ðtÞV^ n ðtnn ÞJ r a^ n þC3 ðttnn Þ: By Jensen’s inequality jd ðtÞD n

   

1  ðt n Þ=nj ¼ d ðtÞ n

n

Z

tnn n tn1

  1 Z tnn  d ðsÞ ds r jd ðtÞd ðsÞj ds:  n tn n1

Since the function d ðtÞ has a bounded derivative jd 0 ðtÞj r M for every t Z 0, one finds jd ðtÞd ðsÞj rMjtsj and, for all t Z tnn , Z n M tn M jd ðtÞD n ðtnn Þ=nj r ðtsÞ ds ¼ n þMðttnn Þ: n n tn1 2 A similar argument gives jd ðtÞD n ðtÞ=nj rM n=2: Boundedness of the second derivative jd 00 ðtÞj r M implies that for t Z n    d ðtÞd ðtnÞ  0  r M n=2: d ðtÞ   n n Inserting these estimates and changing variables by setting h ¼ ttnn , provides the following estimate for a^ n þ 1 :  Z n M M M n a^ nn þ 1 r a^ nn þ C3 cn þ a^ n þ C3 hþ Mh þ MðnhÞ þ n þ Mh þ n þ n dh 2 2 2 0 Z n n n n ¼ a^ n þ C3 ða^ n þð5M=2 þcÞn þ ðC3 þ MÞhÞdh ¼ ð1 þ C3 nÞa^ n þC3 n2 ðc þ C3 =2þ 3MÞ: 0

Recursive application of this inequality gives, for all n Z1, the upper bound

a^ nn r a^ n1 ð1þ C3 nÞn1 þ

ð1 þ C3 nÞn1 n C3 n2 ðc þ C3 =2 þ 3MÞ r a^ 1 eðn1ÞC3 n þ eðn1ÞC3 n ðc þ C3 =2þ 3MÞ n; C3 n

ð43Þ

where one has to use the inequality ð1 þ aÞn r ena for a Z0 to obtain the last estimate. n The estimation of a^ 1 requires a different approach as Eq. (42) is not defined for n ¼ 0 (d ðtÞ is given only for t Z 0). The increment of the discrete-time system between t0n ¼ 0 and t1n ¼ n is given by !   Z n Z t 1 d ðtÞd ð0Þ n n 0 ^ ^ ^ G dðtÞ; cn; V n ð0Þ; Lð0Þ; Lðt Þ; L ðtÞ; ðntÞd ð0Þ þ d ðsÞ ds ; ; d ð0Þ dt ð44Þ V n ðt ÞV n ð0Þ ¼ 1

0

1

n

0

n

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because D n ð0Þ ¼ nd ð0Þ and

lI ðt1n ÞD n ðt1n ÞlI ð0ÞD n ð0Þ ¼

Z n(

lI ðhÞðd ðhÞd ð0ÞÞ þ

0



)  Z h d I l ðhÞ ððnhÞd ð0Þ þ d ðsÞ dsÞ dh: dh 0

Subtracting (44) from (41) and proceeding as above, one obtains     Z n Z t  1 a^ n1 r C3 ðcn þ JV^ ðtÞV^ n ð0ÞJ þJLðtÞLð0ÞJ þ JLðtÞLðnÞJ þ  ðntÞd ð0Þ þ d ðsÞ ds d ðtÞ n 0 0    d ðtÞd ð0Þ  0 þ d ðtÞ  þ jd ðtÞd ð0ÞjÞ dt:   n

ð45Þ

We now estimate the terms on the right-hand side of (45). The first three terms that contain norms are bounded as above (using that an0 ¼ 0). The remaining three terms with absolute values can be bounded as follows:     Z t  1 Mt 2  ðntÞd ð0Þ þ d ðsÞ ds d ðtÞ rMt ; n 2n 0    d ðtÞd ð0Þ Mt 2  0 d ðtÞ rMþ    n 2n and jd ðtÞd ð0Þj r Mt: Inserting these upper bounds into (45), one obtains the estimate

a^ n1 rC3 ðc þ C3 =2 þ2MÞn2 þ C3 Mn: Therefore, using the inequality C3 n reC3 n , estimate (43) gives for n Z0

a^ nn renC 3 n ðc þ C3 =2 þ2MÞn þ eðn1ÞC3 n ðC3 M þ c þ C3 =2 þ3MÞn: Thus the constant C1 in the assertion of the theorem is given by pffiffi C1 ¼ IeTC 3 ½2c þC3 ð1 þMÞ þ 5M; because nn r T for n r bT=nc.

&

Appendix E This appendix collects auxiliary results on the invertibility of certain matrixes. They might be of independent interest. Lemma E.1. Suppose A 2 RNK and B; C 2 RKN are matrices with non-negative entries, N Z 1, K Z 1. Assume

(a) all column sums of A are strictly less than 1: N X

Aij o 1

for all j ¼ 1; . . . ; K;

i¼1

(b) all column sums of B are equal to 1; and (c) C has identical columns and the column sum is equal to 1. Then (i) the matrix IdAB is invertible and its inverse maps the non-negative orthant into itself; and (ii) the matrix IdAðBCÞ is invertible. Proof of Lemma E.1. The matrix D ¼ IdAB has a column-dominant diagonal. Each diagonal entry strictly dominates the sum of absolute values of the remaining entries in the corresponding column: Dii 4

N X j ¼ 1;jai

jDji j;

i ¼ 1; . . . ; N:

ð46Þ

ARTICLE IN PRESS J. Palczewski, K.R. Schenk-Hoppe´ / Journal of Economic Dynamics & Control 34 (2010) 913–931

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Indeed, the ði; jÞ entry of the matrix D is given by 1i ¼ j 

K X

Aik Bkj :

k¼1

All off-diagonal entries are non-positive and the entries on the diagonal are non-negative. The condition (46) is equivalent to N X K X

Aik Bkj o1;

j ¼ 1; . . . ; N:

i¼1k¼1

The following computation proves this inequality: ! N X K K N K X X X X Aik Bkj ¼ Aik Bkj o Bkj ¼ 1; i¼1k¼1

k¼1

i¼1

k¼1

where the strict inequality follows from assumption (a). Property (46) implies that the matrix D is invertible and D1 maps the non-negative orthant into itself (see Corollary, p. 22, and Theorem 23, p. 24, in Murata, 1977). Invertibility of ½IdAðBCÞ is equivalent to the invertibility of ½IdABþ ACD1 ¼ Id þACD1 : It suffices to prove that x ¼ 0 is the only solution to the linear equation x ¼ ACD1 x:

ð47Þ

N

For any y 2 R , the particular form of the matrix C implies ACy ¼ by, where " #T K K N X X X A1k Ck1 ; . . . ; ANk Ck1 and y ¼ yi : b¼ k¼1

i¼1

k¼1

The linear equation (47) therefore can only have solutions of the form x ¼ bb with b 2 R. All coordinates of b are nonnegative because the matrices A and C have non-negative entries. Assume that x ¼ bb is the solution to (47), with ba0 and P 1 bÞi . This further bi 40 for at least one i ¼ 1; . . . ; N. The condition ba0 implies that b ¼ bD1 b, where D1 b ¼ N i ¼ 1 ðD 1 1 1 yields D b ¼ 1. Since the matrix D maps the non-negative orthant into itself, all coordinates of D b are non-negative and D1 b Z0F a contradiction. This implies that the only solution to (47) is x ¼ 0, which proves the invertibility of the matrix ½IdAðBCÞ. & Theorem E.1. Assume that all investment strategies are fully diversified, i.e. L 2 L. Then the matrix ^ ðL; V^ ÞðL ^ LI Þ Idð1aÞY ^,L ^ and LI are defined in (19)–(21).) ^ and a 2 ½0; 1. (The matrices Y is invertible for every V^ 2 D ^ ðL; V^ Þ. This matrix has only non-negative entries. The sum of all entries in the k-th Proof of Theorem E.1. Let A ¼ ð1aÞY column is given by I1 X i¼1

XI1 Aik ¼ ð1aÞ i ¼ 1 Lki P

i V^ I1 ^i i ¼ 1 ðLki LkI ÞV þ LkI =c

¼ ð1aÞ

i V^

XI1

Lki P i¼1

I1 i¼1

i

Lki V^ þ LkI ð1=c

PI1 ^ i i ¼ 1 V Þ:

P ^i If a 4 0, the above sum is strictly less than 1. For a ¼ 0 and I1 i ¼ 1 V o 1=c, the full diversification of the I-th investor’s strategy (i.e. LkI 4 0 for all k) implies that the sum of entries in every column of A is strictly less than 1. In both cases, ^ LI Þ is invertible. Lemma E.1(ii) asserts that the matrix IdAðL PI1 ^ i The proof in the case of a ¼ 0 and i ¼ 1 V ¼ 1=c requires a different argument. Invertibility of the matrix ^ LI Þ is equivalent to D having full rank. By reordering rows, which does not change the rank, we can assume D ¼ IdAðL that the last row of the matrix A contains only strictly positive entries (i.e. by placing an investor with non-zero wealth in i the last row which is possible because V^ 4 0 for at least one i ¼ 1; . . . ; I1). We now use that adding rows does not change the rank of a matrix. First, adding the sum of rows 1 to I2 to row I1 gives 1 0 . . . D1;I2 D1;I1 D11 B D . . . D2;I2 D2;I1 C C B 21 C B C: B ^ ^ ^ C B C B @ DI2;1 . . . DI2;I2 DI2;I1 A 1 ... 1 1

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Second, we subtract from each row j, j ¼ 1; . . . ; I2, the last row multiplied by Dj;I1 . This leads to a matrix that has zeros in the last entry in each row j ¼ 1; . . . ; I2. The resulting matrix is given by 1 0 ... D1;I2 D1;I1 0 D11 D1;I1 B D D ... D2;I2 D2;I1 0C C B 21 2;I1 C B C: B ^ ^ ^ C B C B @ DI2;1 DI2;I1 . . . DI2;I2 DI2;I1 0 A 1 ... 1 1 ~ where The rank of the above matrix is equal to 1 plus the rank of the matrix D 1 0 D11 D1;I1 ... D1;I2 D1;I1 C B D D . . . D2;I2 D2;I1 C B 21 2;I1 ~ ¼B C: D C B ^ ^ A @ DI2;1 DI2;I1 . . . DI2;I2 DI2;I1 ~ is given by The ði; jÞ entry in D ~ ij ¼ 1i ¼ j  D

K X

Aik ðLkj LkI Þ þ

k¼1

K X k¼1

AI1;k ðLk;I1 LkI Þ ¼ 1i ¼ j 

K X

Aik ðLkj Lk;I1 Þ:

k¼1

In matrix notation, ~ B ~ ¼ IdAð ~ C~ Þ; D ^ omitting the last column, and where A~ 2 RðI2ÞK is given by the matrix A omitting the last row, B~ 2 RKðI2Þ is the matrix L C~ 2 RKðI2Þ has all columns equal to ðL1;I1 ; . . . ; LK;I1 Þ. Each column sum of A~ is strictly less than 1 because the last row of the matrix A contains only strictly positive entries and the sum of entries in each column of A equals one. Lemma E.1(ii) ~ is invertible. Therefore D has full rank, i.e. it is invertible. & implies that the matrix D References Alfarano, S., Lux, T., Wagner, F., 2008. Time variation of higher moments in a financial market with heterogeneous agents: an analytical approach. Journal of Economic Dynamics and Control 32, 101–136. Arnold, L., 1998. Random Dynamical Systems. Springer, Berlin. Bank, P., Baum, D., 2004. Hedging and portfolio optimization in financial markets with a large trader. Mathematical Finance 14, 1–18. ¨ Bjork, T., 2004. Arbitrage Theory in Continuous Time, second ed. Oxford University Press, Oxford. Brock, W.A., Hommes, C.H., Wagener, F.O.O., 2009. More hedging instruments may destabilize markets. Journal of Economic Dynamics and Control 33, 1912–1928. Buchmann, B., Weber, S., 2007. A continuous time limit of an evolutionary stock market model. International Journal of Theoretical and Applied Finance 10, 1229–1253. Chiarella, C., Dieci, R., Gardini, L., 2006. Asset price and wealth dynamics in a financial market with heterogeneous agents. Journal of Economic Dynamics and Control 30, 1755–1786. Chiarella, C., Dieci, R., X.-Z., 2009. He, Heterogeneity, Market Mechanisms, and Asset Price Dynamics. In: Hens, T., Schenk-Hoppe´, K.R. (Eds.), Handbook of Financial Markets: Dynamics and Evolution. North-Holland, Amsterdam, pp. 277–344 (Chapter 5). Chiarella, C., He, X.-Z., Zheng, M., December 2007. The stochastic dynamics of speculative prices, Quantitative Finance Research Centre Working Paper No. 208, University of Technology, Sydney. Evstigneev, I.V., Hens, T., Schenk-Hoppe´, K.R., 2009. Evolutionary Finance. In: Hens, T., Schenk-Hoppe´, K.R. (Eds.), Handbook of Financial Markets: Dynamics and Evolution. North-Holland, Amsterdam pp. 507–566 (Chapter 9). Evstigneev, I.V., Hens, T., Schenk-Hoppe´, K.R., 2006. Evolutionary stable stock markets. Economic Theory 27, 449–468. Evstigneev, I.V., Hens, T., Schenk-Hoppe´, K.R., 2008. Globally evolutionarily stable portfolio rules. Journal of Economic Theory 140, 197–228. Evstigneev, I.V., Hens, T., Schenk-Hoppe´, K.R., 2010. The Kelly capital growth investment criterion: theory and practice. In: MacLean, L.C., Thorp, E.O., Ziemba, W.T. (Eds.), Survival and Evolutionary Stability of the Kelly Rule. World Scientific, Singapore. Hens, T., Schenk-Hoppe´, K.R., 2005. Evolutionary stability of portfolio rules in incomplete markets. Journal of Mathematical Economics 41, 43–66. Hommes, C.H., Wagener, F.O.O., 2009. Complex Evolutionary Systems in Behavioral Finance. In: Hens, T. Schenk-Hoppe´, K.R. (Eds.), Handbook of Financial Markets: Dynamics and Evolution. North-Holland, Amsterdam, pp. 217–276 (Chapter 4). Kelly, J.L., 1956. A new interpretation of information rate. Bell System Technical Journal 35, 917–926. Lux, T., 2009. Stochastic Behavioral Asset Pricing Models and the Stylized Facts. In: Hens, T., Schenk-Hoppe´, K.R. (Eds.), Handbook of Financial Markets: Dynamics and Evolution. North-Holland, Amsterdam, pp. 161–215 (Chapter 3). Lux, T., 1997. Time variation of second moments from a noise trader/infection model. Journal of Economic Dynamics and Control 22, 1–38. Murata, Y., 1977. Mathematics for Stability and Optimization of Economic Systems. Academic Press, New York. Palczewski, J., Schenk-Hoppe´, K.R., 2010. Market selection of constant proportions investment strategies in continuous time, Journal of Mathematical Economics, in press, doi:10.1016/j.jmateco.2009.11.011. Rheinlaender, T., Steinkamp, M., 2004. A stochastic version of Zeeman’s market model. Studies in Nonlinear Dynamics & Econometrics 8 (4) Article 4. Rudin, W., 1976. Principles of Mathematical Analysis. McGraw-Hill, New York. Prigent, J.-L., 2003. Weak Convergence of Financial Markets. Springer, Berlin. Shreve, S.E., 2004. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, Berlin. Zeeman, E.C., 1974. On the unstable behaviour of stock exchanges. Journal of Mathematical Economics 1, 39–49.