Multi-Time Systems of Conservation Laws

Motivated by the work of P.L. Lions and J-C. Rochet [12], concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multi-time systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.Comment: 2

Procyclicality of financial systems: is there a need to modify current accounting and regulatory rules?

Financial systems have an intrinsic tendency to exacerbate business cycle fluctuations rather than smoothing them out. The current crisis is a perfect illustration of this. Some commentators have argued that the recent reforms to international bank regulation (Basel II) and accounting rules (IAS 39) are likely to increase this intrinsic procyclicality in the future. This article examines whether this accusation is founded and what policy decisions could be envisaged to alleviate this undesirable feature of financial systems.

The Biais-Martimort-Rochet equilibrium with direct mechanisms

In this note we show that the equilibrium characterized by Biais, Martimort and Rochet (Econometrica, 78, 2000) could have been characterized by using direct mechanisms.Common Agency, Revelation Principle, Direct Mechanisms, Nonlinear Prices

Three-types models of multidimensional screening

This paper analyzes the variety of optimal screening contracts in a relatively simple multidimensional framework a` la Armstrong and Rochet (1999), when only three types of agents are present. It is shown, among other things, that the well known principle in optimal contract theory of `no distortion at the top' does not carry over to the multidimensional caseAsymmmetric information; multidimensional screening; optimal contract

On the strategic use of risk and undesirable goods in multidimensional screening

A monopolist sells goods with possibly a characteristic consumers dislike (for instance, he sells random goods to risk averse agents), which does not affect the production costs. We investigate the question whether using undesirable goods is profitable to the seller. We prove that in general this may be the case, depending on the correlation between agents types and aversion. This is due to screening effects that outperform this aversion. We analyze, in a continuous framework, both 1D and multidimensional cases

On the Existence of Linear Equilibria in the Rochet-Vila Model of Market Making

This paper derives necessary and sucient conditions for the existence of linear equilibria in the Rochet-Vila model of market making. In contrast to most previous work on the existence of linear equilibria in models of market making, we do not impose independence of the underlying random variables. For distributions that are determined by their moments we show that a linear equilibrium exists if and only if the joint distribution of noise trade and asset payoff is elliptical.Market Microstructure, Market Making, Linear Equilibria

Liquidity and Capital Requirements and the Probability of Bank Failure

Using the model of Rochet and Vives (2004), this note shows that a prudential regulator can in general not mitigate a bank’s failure risk solely by means of liquidity requirements. However, their effectiveness can be restored if, in addition, minimum capital requirements are met. This provides a rationale for capital requirements beyond the commonly envoked reasoning that they are to be used to control the riskiness of banks’ asset portfolios.prudential regulation, liquidity requirements, minimum capital requirements, global games

Fluctuations for analytic test functions in the Single Ring Theorem

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.Comment: 29 pages, 1 figure. In Version v2, we slightly modified the assumptions, in order to fix a problem un the control of the tails (see Assumption 2.3). In v3, some minors typos were corrected. In v4, some explanations were added in the introduction and some typos were corrected. To appear in Indiana Univ. Math.

Outliers in the Single Ring Theorem

This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type $A+P$, where the rank of $P$ stays bounded as the dimension goes to infinity and where the matrix $A$ is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if $P$ has some eigenvalues out of the maximal circle of the single ring, then $A+P$ has some eigenvalues (called outliers) in the neighborhood of those of $P$, which is not the case for the eigenvalues of $P$ in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of $A$ around the eigenvalues of $P$ and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of $P$) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.Comment: Version v2 contains a major improvement with respect to the first one: we now consider the general case for fluctuations of the outliers. In version v4, we slightly weakened the hypotheses. In v5, we simplified notation and added a remark about the real case. 42 pages, 4 figures, to appear in Probab. Theory Related Field