Competing particle systems evolving by interacting Lévy processes

Abstract

We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Lévy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Lévy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modeling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington-Kirkpatrick model of spin glasses.

Infinite systems of competing Brownian particles

“…A more precise definition is given in Definitions 6 and 7 later in this article. This system was studied in [35,18]. For g 1 = 1, g 2 = g 3 = .…”

“…Indeed, it is an easy exercise to show that a system of i.i.d standard Brownian motions starting from the same point is not rankable from bottom to top at any fixed time t > 0. Some sufficient conditions for weak existence and uniqueness in law are found in [35,18]. We restate them in Theorem 3.1 in a slightly different form: (5) lim i→∞ x i = ∞ and…”

“…There are other generalizations of competing Brownian particles: competing Lévy particles, [35]; a second-order stock market model, when the drift and diffusion coefficients depend on the name as well as the rank of the particle, [11,3]; competing Brownian particles with values in the positive orthant R N + , see [13]. Two-sided infinite systems (X i ) i∈Z of competing Brownian particles are studied in [33].…”

The Rank Effect for Commodities

“…Equation (3.1) together with these regularity conditions implies that the commodity price processes in this economy are continuous semimartingales, which represent a broad class of stochastic processes (Karatzas and Shreve, 1991). Furthermore, this analysis based on continuous semimartingales can also be extended to stochastic processes with occasional discrete jumps (Shkolnikov, 2011;Fernholz, 2016c).…”

“…1 Although we do not commit to a specific model of economic behavior in this paper, the generality of our methods implies that our econometric framework is applicable to both rational (Sharpe, 1964;Lucas, 1978;Cochrane, 2005) and behavioral (Shiller, 1981; De Bondt 1 A growing and extensive literature, including Banner, Fernholz, and Karatzas (2005), Pal and Pitman (2008), Ichiba, Papathanakos, Banner, Karatzas, andFernholz (2011), andShkolnikov (2011), among others, analyzes these rank-based methods.…”