The Steady-State Behaviour of a Stochastic Clearing System with Bounded Waiting Times

The stochastic clearing system considered in this paper is characterized by an uncontrollable Poisson input process and bounded customers' wating times. We assume that all the quantity currently present in the system is instantaneously removed whenever there are at least M items in the queue, or ev0ery t time units since the first arrival after the last clearing, whichever occurs first. The objective is to study the steady-state behaviour of this system. Knowledge of this steady-state behaviour can be used for the evaluation of the system performance as a function of the system's parameters. We present explicit expressions for the queue length and waiting time distribution, the average queue length, and the average waiting time under steady-state conditions. This work is related to dispatching in transportation systems with stationary Poisson arrivals

A new formula for some linear stochastic equations with applications

We give a representation of the solution for a stochastic linear equation of the form $X_t=Y_t+\int_{(0,t]}X_{s-} \mathrm {d}{Z}_s$ where $Z$ is a c\'adl\'ag semimartingale and $Y$ is a c\'adl\'ag adapted process with bounded variation on finite intervals. As an application we study the case where $Y$ and $-Z$ are nondecreasing, jointly have stationary increments and the jumps of $-Z$ are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When $Y$ and $Z$ are, in addition, independent L\'evy processes, the resulting $X$ is called a generalized Ornstein-Uhlenbeck process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP637 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evolutionary Model of Non-Durable Markets

Presented is an evolutionary model of consumer non-durable markets, which is an extension of a previously published paper on consumer durables. The model suggests that the repurchase process is governed by preferential growth. Applying statistical methods it can be shown that in a competitive market the mean price declines according to an exponential law towards a natural price, while the corresponding price distribution is approximately given by a Laplace distribution for independent price decisions of the manufacturers. The sales of individual brands are determined by a replicator dynamics. As a consequence the size distribution of business units is a lognormal distribution, while the growth rates are also given by a Laplace distribution. Moreover products with a higher fitness replace those with a lower fitness according to a logistic law. Most remarkable is the prediction that the price distribution becomes unstable at market clearing, which is in striking difference to the Walrasian picture in standard microeconomics. The reason for this statement is that competition between products exists only if there is an excess supply, causing a decreasing mean price. When, for example by significant events, demand increases or is equal to supply, competition breaks down and the price exhibits a jump. When this supply shortage is accompanied with an arbitrage for traders, it may even evolve into a speculative bubble. Neglecting the impact of speculation here, the evolutionary model can be linked to a stochastic jump-diffusion model.non-durables; evolutionary economics; economic growth; price distribution; Laplace distribution; replicator equation; firm growth; growth rate distribution; competition; jump-diffusion model

Heterogeneity, Market Mechanisms, and Asset Price Dynamics

This chapter surveys the boundedly rational heterogeneous agent (BRHA) models of financial markets, to the development of which the authors and several co-authors have contributed in various papers. We give particular emphasis to role of the market clearing mechanism used, the utility function of the investors, the interaction of price and wealth dynamics, portfolio implications, the impact of stochastic elements on the markets dynamics, and calibration of this class of models. Due to agents’ behavioural features and market noise, the BRHA models are both nonlinear and stochastic. We show that the BRHA models produce both a locally stable fundamental equilibrium corresponding to that of standard paradigm, as well as instability with a consequent rich range of possible complex behaviours characterised both indirectly by simulation and directly by stochastic bifurcations. A calibrated model is able to reproduce quite well the stylized facts of financial markets. The BRHA framework is thus able to accommodate market features that seem not easily reconcilable for the standard financial market paradigm, such as fat tail, volatility clustering, large excursions from the fundamental and bubbles.Bounded rationality; interacting heterogeneous agents; behavioural finance; nonlinear economic dynamics; complexity

"Hysteresis in Dynamic General Equilibrium Models with Cash-in-Advance Constraints"

In this paper, we investigate equilibrium cycles in dynamic general equilibrium models with cash-in-advance constraints. Our findings are two-fold. First, in such models, if an equilibrium cycle exists, then there also exists a continuum of equilibrium cycles in its neighborhood. Second, the limit cycle, to which a dynamic path converges, varies continuously according to the initial distribution of the money holdings. Thus, temporary shocks that affect the initial distribution have permanent effects in such models; that is, such models exhibit hysteresis. Furthermore, we also explore the logic behind the results.

A Market-Clearing Role for Inefficiency on a Limit Order Book

Using a stochastic sequential game in ergodic equilibrium, this paper models limit order book trading dynamics. It deduces investor surplus and some agents' strategies from depth's stationarity, while bypassing altogether agents' intricate forecasting problems. Market inefficiency adjusts to induce equal supply and demand for liquidity over time. Consequently, at a given bid-ask spread surplus per investor is invariant to faster, more regular or more sophisticated trading, or modified queuing rules: apparent improvements are offset as inefficiency adjusts back to market-clearing levels. Moreover, investor surplus decreases with the spread. In the model, price discreteness fixes the spread at the tick size. Narrowing the tick is beneficial, but may be resisted by sell-side traders.stochastic sequential game, ergodic equilibrium, market microstructure, limit order book, market depths, bid-ask spread