A single-item continuous double auction game

A double auction game with an infinite number of buyers and sellers is introduced. All sellers posses one unit of a good, all buyers desire to buy one unit. Each seller and each buyer has a private valuation of the good. The distribution of the valuations define supply and demand functions. One unit of the good is auctioned. At successive, discrete time instances, a player is randomly selected to make a bid (buyer) or an ask (seller). When the maximum of the bids becomes larger than the minimum of the asks, a transaction occurs and the auction is closed. The players have to choose the value of their bid or ask before the auction starts and use this value when they are selected. Assuming that the supply and demand functions are known, expected profits as functions of the strategies are derived, as well as expected transaction prices. It is shown that for linear supply and demand functions, there exists at most one Bayesian Nash equilibrium. Competitive behaviour is not an equilibrium of the game. For linear supply and demand functions, the sum of the expected profit of the sellers and the buyers is the same for the Bayesian Nash equilibrium and the market where players behave competitively. Connections are made with the ZI-C traders model and the $k$-double auction.Comment: 37 pages, 15 figure

Recognizing sparse perfect elimination bipartite graphs

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For $n \times n$ matrices with m non-zero elements, the currently best known algorithm has a time complexity of $O(n^3/\log n)$. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new algorithms for the recognition of sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time $O(n m)$

Dimensional Reduction for Conformal Blocks

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.Comment: 12 pages, 1 figur

GMM estimation with noncausal instruments under rational expectations

There is hope for the generalized method of moments (GMM). Lanne and Saikkonen (2011) show that the GMM estimator is inconsistent, when the instruments are lags of noncausal variables. This paper argues that this inconsistency depends on distributional assumptions, that do not always hold. In particular under rational expectations, the GMM estimator is found to be consistent. This result is derived in a linear context and illustrated by simulation of a nonlinear asset pricing model.generalized method of moments, noncausal autoregression, rational expectations

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift $A$ which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of nonadapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.Comment: Minor revision. Accepted for publication in Journal of Differential Equation