Abstract: This survey examines recent results on evolutionary stability in $n$-player games and argues that (i) while evolutionary ideas are indeed important for Economics, the relevant evolutionary stability concepts are not those directly imported from biology through ``classical" evolutionary game theory, but those associated to playing-the-field, finite-population settings; and (ii) once this step is made, evolutionary stability is strongly related to results commonly associated with perfect competition. This can be formalized in a large class of games which we call ``aggregative." In these games, the payoffs to each player can be expressed as a function of the own strategy and some aggregate of the opponents' (or all) strategies.

Abstract: (jointly with Giulio Bottazzi) We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return, and one risky, paying a stochastic dividend, and we assume trading to take place in discrete time inside an endogenous price formation setting. Traders demand for the risky asset is expressed as a fraction of their individual wealth and is based on future prices forecast obtained on the basis of past market history.
The general case is studied in which an arbitrary large number of heterogeneous traders operates in the market and any smooth function which maps the infinite information set to the present investment choice is allowed as agent's trading behavior. A complete characterization of equilibria is given and their stability conditions are derived. We find that this economy can only possess isolated generic equilibria where a single agent dominates the market and continuous manifolds of non-generic equilibria where many agents hold finite wealth shares. We show that irrespectively of agents number and of their behavior, all possible equilibria belong to a one dimensional "Equilibrium Market Line". Finally we discuss the relative performances of different strategies and the selection principle governing market dynamics.

Abstract: If an asset X is represented by an sde, financial quantities like prices of options, hedging portfolio, are obtained by stochastic calculus as functionals of X. If we suppose that the sde is solved using the Euler scheme, the asymptotic error, discovered by Kurtz and Protter in the spirit of a central limit theorem, takes the form of a process solution to an other sde.
In order to propagate this asymptotic error through financial calculus, we have, in some sense, to take the derivative of non differentiable functionals like stochastic integrals. We show that this may be performed thanks to the theory of Dirichlet forms. The main step is the construction of a Dirichlet form on the Wiener space which is an Ornstein-Uhlenbeck form with weight, the key result being that this weight is random, even non-necessarily adapted.

Abstract: On the high end of incomes distribution, the Pareto law with the exponent value of 2 is a universal distribution for the class of hierarchy models with multiplicative-invariant interaction.

Abstract: This paper compares models of fractional processes and associated weak convergence re- sults based on moving average representations in the time domain with spectral representa- tions. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between "two-sided" processes depending on both leads and lags from one-sided or "causal" processes, since in the case of fractional processes these models yield different limiting properties. We derive new representations of fractional Brownian motion, and show how different results are obtained for, in particular, the distribution of stochastic integrals in the multivariate con- text. Our results have implications for valid statistical inference in fractional integration and cointegration models.

Abstract: The topic of the talk will be SPDEs of parabolic type driven by Poisson Random measures. I will present present some Existence and Uniqueness results for Lipschitz and non Lipschitz coefficients.

Abstract: We establish a central limit theorem for queuing models where the arrival and departure rates depend on stationary semi-Markov processes with heavy tailed sojourn time distributions. We show that after suitable rescaling the fluctuations around the deterministic first order approximation can be described by a fractional Ornstein-Uhlenbeck process. We apply our limit theorem to model price fluctuations in financial markets with many inert investors that trade only occasionally. With our choice of scaling long range dependence emerges in financial time series. We also link investor inertia to the observed fall is the Hurst coefficient of the S&P 500 index over the late 1990s.
The talk is based on joint work with E. Bayraktar (Michigan) and R. Sircar (Princeton)

Abstract: Authors: Emilio De Santis, Carlo Marinelli MSC-class: 82C20; 91A18; 91A25
We introduce and study a class of infinite-horizon non-zero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove ``fixation'', i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zero-temperature Glauber dynamics on random graphs of possibly infinite volume.

Abstract: This paper proposes a method to analyze endogenous fluctuations of aggregate investment when firm-level investment follows an (S,s) policy and has a spillover effect on other firms' investments. First, we derive the distribution function of aggregate fluctuations in a partial equilibrium of differentiated product markets, under the assumption that a firm's position in its (S,s) band follows a uniform distribution. Second, the variance of the growth rate of average capital is shown to converge to a non-zero value when the number of firms tends to infinity, if the technology exhibits constant returns to scale. Third, we numerically compute the equilibrium paths, in a partial and a general equilibrium, in which the firms' positions evolve deterministically. The simulations uphold our analytical results as well as exhibit echo effects in the output series.

Abstract: In [1] convergence for time-space discretization was obtained with measurable and bounded drift and constant drift. In [2] convergence for time discretization was obtained with measurable and bounded drift and state-dependent drift. Here convergence for space-time dicretization is obtained with measurable and bounded drift and state-dependent drift. Ideas from [1] and [2] are used. Fine properties of the kernel related to the numerical approximation is needed. Joint work with M Signahl.

[1] Gyöngy, I. and Nualart, D. (1997) Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Pot. Anal. 7, 725-757.

[2] Gyöngy I. (1999) Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II, Pot. Anal. 11, 1-37.