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The classical multidimensional version of Fatou’s lemma (Schmeidler in Proc Am Math Soc 24:300–306, 1970) originally obtained for unconditional expectations and the standard non-negative cone in a finite-dimensional linear space is extended to conditional expectations and general closed pointed cones.

This paper analyzes a dynamic stochastic equilibrium model of an asset market based on behavioral and evolutionary principles. The core of the model is a non-traditional game-theoretic framework combining elements of stochastic dynamic games and evolutionary game theory. Its key characteristic feature is that it relies only on objectively observable market data and does not use hidden individual agents’ characteristics (such as their utilities and beliefs). A central goal of the study is to identify an investment strategy that allows an investor to survive in the market selection process, i.e., to keep with probability one a strictly positive, bounded away from zero share of market wealth over an infinite time horizon, irrespective of the strategies used by the other players. The main results show that under very general assumptions, such a strategy exists, is asymptotically unique and easily computable. The paper resolves long-standing open problems that remained open for about a decade.

The purpose of this work is to develop an evolutionary finance model with a risk-free asset playing the role of a numeraire. The model describes a market where one risk-free and several “short-lived” risky assets (securities) are traded in discrete time. The risky securities live one period, yield random payoffs at the end of it, and then are re-born at the beginning of the next period. The main goal of the study is to identify investment strategies that make it possible for an investor to “survive” in the market selection process. It is shown that a strategy of this kind exists, is in a sense asymptotically unique and can be described by a simple explicit formula amenable for quantitative investment analysis.

The aim of this work is to extend the classical capital growth theory pertaining to frictionless nancial markets to models taking into account various kinds of frictions, including transaction costs and portfolio constraints. A natural generalization of the notion of a benchmark investment strategy (Platen, Heath and others) is proposed, and it is shown how such strategies can be used for the analysis of growth-optimal investments. The analysis is based on the classical von Neumann-Gale model of economic growth, a stochastic version of which is used in this study as a framework for the modeling of nancial markets with frictions. [The paper was received and accepted by the Editor on the same date: 15 Jan 2020.]

We consider a stochastic model of a financial market with one-period assets and endogenous asset prices. The model was initially developed and analyzed in the context of Evolutionary Finance with the main focus on questions of “survival and extinction” of investment strategies (portfolio rules). In this paper we view the model from a different, game-theoretic, perspective and analyze Nash equilibrium properties of survival portfolio rules.

International audience; The paper models evolution in pecunia—in the realm of finance. Financial markets are explored as evolving biological systems. Diverse investment strategies compete for the market capital invested in long-lived dividend-paying assets. Some strategies survive and some become extinct. The basis of our paper is that dividends are not exogenous but increase with the wealth invested in an asset, as is the case in a production economy. This might create a positive feedback loop in which more investment in some asset leads to higher dividends which in turn lead to higher investments. Nevertheless, we are able to identify a unique evolutionary stable investment strategy. The problem is studied in a framework combining stochastic dynamics and evolutionary game theory. The model proposed employs only objectively observable market data, in contrast with traditional settings relying upon unobservable investors’ characteristics (utilities and beliefs). Our method is analytical and based on mathematical reasoning. A numerical illustration of the main result is provided.

We consider symmetric oligopolies with positive network effects where each firm has its own proprietary network, which is incompatible with that of its rivals. We provide minimal conditions for the existence of (non-trivial) symmetric equilibrium in a general setting. We analyze the viability of industries with firm-specific networks and show that the prospects for successful launch decrease with more firms in the market. This is a major reversal from the case of single-network industries. A central part of the paper compares the viability and market performance of industries with compatible and incompatible networks and shows that viability, output, (endogenous) demand, and social welfare are higher for the former. However, the comparison of industry price, profit and consumer surplus requires respective qualifications, of a general nature for the former two but not for the latter. Overall, these results provide theoretical grounding in a general but not universal sense for the conventional view that compatibility leads to superior performance, which was hitherto based on case studies and stylized facts.

International audience; Evolutionary finance focuses on questions of “survival and extinction” of investment strategies (portfolio rules) in the market selection process. It analyzes stochastic dynamics of financial markets in which asset prices are determined endogenously by a short-run equilibrium between supply and demand. Equilibrium is formed in each time period in the course of interaction of portfolio rules of competing market participants. A comprehensive theory of evolutionary dynamics of this kind has been developed for models in which short selling is not allowed and asset supply is exogenous. The present paper extends the theory to a class of models with short selling and endogenous asset supply.

The paper models evolution in pecunia—in the realm of finance. Financial markets are explored as evolving biological systems. Investors pursuing diverse investment strategies compete for the market capital. Some `survive' and some `become extinct.' A central goal is to identify evolutionary stable, i.e. guaranteeing survival, investment strategies. The problem is studied in a framework combining stochastic dynamics and evolutionary game theory. The model proposed employs only objectively observable market data, in contrast with traditional settings relying upon unobservable investors characteristics (utilities and beliefs). The main result is a construction of an evolutionary stable strategy in the model at hand.

This paper analyzes a dynamic stochastic equilibrium model of an asset market based on behavioral and evolutionary principles. The core of the model is a non-traditional game-theoretic framework combining elements of stochastic dynamic games and evolutionary game theory. Its key characteristic feature is that it relies only on objectively observable market data and does not use hidden individual agents' characteristics (such as their utilities and beliefs). A central goal of the study is to identify an investment strategy that allows an investor to survive in the market selection process, i.e., to keep with probability one a strictly positive, bounded away from zero share of market wealth over an infinite time horizon, irrespective of the strategies used by the other players. The main results show that under very general assumptions, such a strategy exists, is asymptotically unique and easily computable.

The aim of this work is to extend the classical theory of growth-optimal investments (Shannon, Kelly, Breiman, Algoet, Cover and others) to models of asset markets with frictions -- transaction costs and portfolio constraints. As the modelling framework, we use discrete-time dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors|von Neumann-Gale dynamical systems. The main results are concerned with the construction and characterization of investment strategies possessing properties of asymptotic growth-optimality almost surely.This paper concludes a series of publications on growth-optimal investments under transaction costs, that resolved a number of long-standing open problems. It required developing a new modeling approach and new methods of mathematical analysis.

The paper provides necessary and sufficient conditions for the uniqueness of pure-strategy Nash equilibrium in the standard Bertrand duopoly with a homogeneous product. The main condition is elementary, easy to interpret, and nests all known sufficient conditions in the literature.

We analyze the concept of correlated equilibrium in the framework of two-player two-strategy games. This simple framework makes it possible to clearly demonstrate the characteristic features of this concept. We develop an intuitive and easily memorizable test for equilibrium conditions and provide a complete classification of symmetric correlated equilibria in symmetric games.

A famous result in game theory known as Zermelo's theorem says that ''in chess either White can force a win, or Black can force a win, or both sides can force at least a draw". The present paper extends this result to the class of all finite-stage two-player games of complete information with alternating moves. It is shown that in any such game either the first player has a winning strategy, or the second player has a winning strategy, or both have unbeatable strategies.

The paper provides new conditions for the existence, uniqueness, and symmetry of pure-strategy Nash equilibrium in the classical Cournot duopoly.

The aim of this work is to extend the classical capital growth theory pertaining to frictionless financial markets to models taking into account various kinds of frictions, including transaction costs and portfolio constraints. A natural generalization of the notion of a benchmark investment strategy (Platen, Heath and others) is proposed, and it is shown how such strategies can be used for the analysis of growth-optimal investments. The analysis is based on the classical von Neumann-Gale model of economic growth, a stochastic version of which is used in this study as a framework for the modeling of financial markets with frictions.

Von Neumann-Gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of such systems is played by a special class of paths (trajectories) called rapid: they grow over each time period t-1,t in a sense faster than others. The paper establishes existence and characterization theorems for such paths showing, in particular, that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid. The proof of this result is based on the methods of convex analysis in spaces of measurable functions. The study is motivated by the applications of the theory of von Neumann-Gale dynamical systems to the modeling of capital growth in financial markets with frictions -- transaction costs and portfolio constraints., Comment: 27 pages

Evolutionary Finance focuses on questions of "survival and extinction" of investment strategies (portfolio rules) in the market selection process. It analyzes stochastic dynamics of financial markets in which asset prices are determined endogenously by a short-run equilibrium between supply and demand. Equilibrium is formed in each time period in the course of interaction of portfolio rules of competing market participants. A comprehensive theory of evolutionary dynamics of this kind has been developed for models in which short selling is not allowed and asset supply is exogenous. The present paper extends the theory to a class of models with short selling and endogenous asset supply.

We consider a stochastic model of a financial market with one-period assets and endogenous asset prices. The model was initially developed and analyzed in the context of Evolutionary Finance with the main focus on questions of "survival and extinction" of investment strategies (portfolio rules). In this paper we view the model from a different, game-theoretic, perspective and analyze Nash equilibrium properties of survival portfolio rules.

The purpose of this work is to develop an evolutionary finance model with a risk-free asset playing the role of a numeraire. The model describes a market where one risk-free and several "short-lived" risky assets (securities) are traded in discrete time. The risky securities live one period, yield random payoffs at the end of it, and then are re-born at the beginning of the next period. The main goal of the study is to identify investment strategies that make it possible for an investor to "survive" in the market selection process. It is shown that a strategy of this kind exists, is in a sense asymptotically unique and can be described by a simple explicit formula amenable for quantitative investment analysis.

Michael I. Taksar was born on 9 September 1949, in Moscow, Russia. His mathematical abilities revealed themselves quite early, when he was a student in the special high school in Moscow organized i...

We develop a model of asset pricing and hedging for interconnected financial markets with frictions – transaction costs and portfolio constraints. The model is based on a control theory for random fields on a directed graph. Market dynamics are described by using von Neumann–Gale dynamical systems first considered in connection with the modelling of economic growth [13,24]. The main results are hedging criteria stated in terms of risk-acceptable portfolios and consistent price systems, extending the classical superreplication criteria formulated in terms of equivalent martingale measures.

Once upon a time there was a planet Π on which there was a financial market M. All the inhabitants of Π were participants in the market M. They traded stock, equity, derivative securities and other assets. They were heterogeneous and diverse. However, they all had a common feature. They all maximized expected utilities of their consumption subject to budget constraints. They were infinitely wise (or in the local language, fully rational), they had unlimited memory and computational possibilities as well as perfect foresight, and they had no other goals except establishing a general equilibrium in the market

The creation and protection of financial wealth is one of the most important roles of modern societies. People will commit to working hard and saving for future generations only if they can be sure that the efforts they exert every day will be rewarded by a better standard of living. This, however, can only be achieved with a well-functioning financial market. Unfortunately, a breakdown of the financial system as in the great financial crisis of 2007 and 2008 destroys the trust in this important social arrangement. To avoid such crises we need to improve our understanding of financial markets that, so far, has been built on totally unrealistic assumptions about the behavior of people acting in them. The most fundamental and at the same time the most questionable in modern economic theory is the hypothesis of full rationality of economic agents who are assumed to maximize their utility functions subject to their individual constraints, or in mathematical language, solve well-defined and precisely stated constrained optimization problems.

We provide conditions for the existence of measurable solutions to the equation ξ ( T ω ) = f ( ω , ξ ( ω ) ) \xi (T\omega )=f(\omega ,\xi (\omega )) , where T : Ω → Ω T:\Omega \rightarrow \Omega is an automorphism of the probability space Ω \Omega and f ( ω , ⋅ ) f(\omega ,\cdot ) is a strictly nonexpansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping D ( ω ) D(\omega ) of a random closed cone K ( ω ) K(\omega ) in a finite-dimensional linear space into the cone K ( T ω ) K(T\omega ) . Under the assumptions of monotonicity and homogeneity of D ( ω ) D(\omega ) , we prove the existence of scalar and vector measurable functions α ( ω ) > 0 \alpha (\omega )>0 and x ( ω ) ∈ K ( ω ) x(\omega )\in K(\omega ) satisfying the equation α ( ω ) x ( T ω ) = D ( ω ) x ( ω ) \alpha (\omega )x(T\omega )=D(\omega )x(\omega ) almost surely.

The paper examines a game-theoretic evolutionary model of an asset market with endogenous equilibrium asset prices. Assets pay dividends that are partially consumed and partially reinvested. The investors use general, adaptive strategies (portfolio rules), distributing their wealth between assets, depending on the exogenous states of the world and the observed history of the game. The main objective of the work is to identify strategies, allowing an investor to “survive”, i.e. to possess a positive, bounded away from zero, share of market wealth over the whole infinite time horizon. This work brings together recent studies on evolutionary finance with the classical topic of non-cooperative market games.

This paper introduces and analyzes an evolutionary model of a financial market with a risk-free asset. Focus is on the study of local stability of the wealth dynamics through the application of recent results on the linearization and stability of random dynamical systems (Evstigneev et al. Proc Am Math Soc 139:1061–1072, 2011). Conditions are derived for the linearization of the model at an equilibrium state which ensure local convergence of sample paths to this equilibrium. The paper also shows that the concept of local stability is closely related to the notion of evolutionary stability. A locally evolutionarily stable investment strategy in the evolutionary model with a risk-free asset is derived, extending previous research. The method illustrated here is applicable for the analysis of manifold economic and financial dynamic models involving randomness.

We consider a stochastic model of a financial market with long-lived dividend-paying assets and endogenous asset prices. The model was initially developed and analyzed in the context of evolutionary finance, with the main focus on questions of “survival and extinction” of investment strategies. In this paper we view the model from a different, game-theoretic, perspective and examine Nash equilibrium strategies, satisfying equilibrium conditions with probability one.

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems).

The paper develops a stochastic dynamic model of economic equilibrium with locally interacting agents. The main focus of the study is on the modeling of market interactions – those arising in connection with commodity exchange and regulated by price mechanisms. The mathematical framework is a control theory for random vector fields on directed graphs. The graphs involved serve to describe the spatio-temporal structure of commodity flows in the system. The main results are concerned with the existence, uniqueness and stability of stochastic dynamic equilibria.

The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of von Neumann–Gale dynamics is played by a special class of paths called rapid (they maximise properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades.

This paper examines a class of random dynamical systems related to the classical von Neumann and Gale models of economic dynamics. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. We establish a general existence theorem for equilibrium, which holds under conditions analogous to the standard deterministic ones. Our results answer questions that remained open for more than three decades.

The paper examines the problem of the existence of equilibrium for the stochastic analogue of the von Neumann–Gale model of economic growth. The mathematical framework of the model is a theory of set-valued random dynamical systems defined by positive stochastic operators with certain properties of convexity and homogeneity. Existence theorems for equilibria in such systems may be regarded as generalizations of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The known results of this kind are obtained under rather restrictive assumptions. We show that these assumptions can be substantially relaxed if one allows for randomization. The main result of the paper is an existence theorem for randomized equilibria. Some special cases (models defined by positive matrices) are considered in which the existence of pure equilibria can be established.

The goal of the chapter is to derive the Black-Scholes formula, one of the highlights of Mathematical finance. The proof is conducted by passing to the limit from the binomial model. The chapter begins with introducing some relevant notions: drift and volatility, continuous compounding, geometric random walk, etc. It then shows how to approximate the observed continuous-time price process with constant drift and volatility by price processes generated by suitable binomial models. The main theorem proved in the chapter establishes a general (probabilistic) version of the Black-Scholes formula for a European derivative security with a general payoff function. As a corollary to this theorem, an analytic version of the Black-Scholes formula for a European call option is obtained.

The chapter describes the Markowitz model, a cornerstone of portfolio theory, and examines the expected return and the variance of the return on a portfolio in the context of this model. The reader will learn how to apply the Markowitz approach to portfolio selection and about the trade-off between risk and return. The chapter considers the role of an investor’s risk tolerance and analyses the notion of an efficient portfolio. It provides a thorough discussion of the Markowitz optimization problem and of the underlying basic assumptions.

This chapter contains problems and exercises on Mean-Variance Portfolio Analysis. Some of these include additional theoretical material presented in the form of questions and answers. The others represent numerical examples on Mean-Variance Portfolio Analysis. Exercises of this kind can be used for tutorials, take-home tests and examinations. Detailed answers for all the questions are provided.

The chapter presents basic ideas and results related to capital growth theory. It introduces the dynamic securities market model with an infinite time horizon and self-financing strategies defined in terms of investment proportions. The highlight of the chapter is the study of the problem of growth optimal investments. It shows that the log-optimal portfolio rule is the solution to this problem in the class of simple (constant proportions) strategies. The chapter concludes with the classical example of the horse race model and the Kelly gambling rule of “betting your beliefs.”

The chapter introduces the reader to the celebrated Capital Asset Pricing Model (CAPM). It opens with the statement of a general version of the CAPM followed by a proof based on an explicit formula for the efficient portfolios. It then outlines an equilibrium model for the CAPM, defines the concepts of the market portfolio and capitalization weights and establishes the efficiency of the market portfolio. The highlight of the chapter is the Sharpe-Lintner-Mossin CAPM formula. Key notions related to it (the beta of an asset, risk premium, etc.) are defined and discussed.

The chapter introduces American derivative securities and develops the risk-neutral principle for their pricing. It explains the fundamental notion of an exercise strategy defined in terms of the property of non-anticipativity. The main result is a backward induction algorithm for computing the upper (seller’s) price of an American derivative security.

The chapter continues the study of the Markowitz model. The reader will learn how to compute an efficient portfolio with the given risk tolerance. The highlight is an explicit formula for efficient portfolios, rigorously derived and comprehensively discussed. The chapter analyses the minimum variance portfolio and the return-generating self-financing portfolio involved in the solution to the Markowitz optimization problem. It concludes with explaining the linear structure of the set of efficient portfolios.

The chapter continues the study of a financial market with a risk-free asset. It provides formulas for the expected return and the variance of the return on an efficient portfolio and shows how to represent the efficient frontier for the market with a risk-free asset through equations in the $∖sigma$-$m$ plane and in the $∖sigmaˆ{2}$-$m$ plane. The chapter introduces the notion of the tangency portfolio, examines conditions under which it exists and derives a formula for it. A discussion of the properties of the tangency portfolio is followed by a geometric illustration explaining the term “tangency.” The highlight of the chapter is the notion of the Sharpe ratio and the evaluation of the Sharpe ratio for the tangency portfolio. The chapter concludes with Tobin’s mutual fund theorem, which is formulated and proved.

The chapter focuses on a classical topic in Financial Economics: the general equilibrium analysis of financial markets. It introduces the simplest version of a two-period stochastic equilibrium model. An existence result (without proof) is given. An equilibrium pricing formula is derived in the case of von Neumann-Morgenstern utilities. The chapter concludes with an analysis of links between the no-arbitrage hypothesis and equilibrium.