arXiv subjects

Vassili N. Kolokoltsov

Publications and source records attributed to Vassili N. Kolokoltsov.

24 records · Page 2Linked to original sources

arXiv2012

Nonlinear diffusions and stable-like processes with coefficients depending on the median or VaR

In June 2012 on a conference in Bielefeld, after the author made the presentation of his theory of nonlinear Markov processes, Tom Kurtz asked him whether his methods would allow to get well-posedness for nonlinear McKean-Vlasov type diffusions with coefficients depending on the distribution via the median. The present paper is the answer to this question. The aim here is not to find the weakest possible conditions for the affirmative answer, but a sensible general result with a quick and transparent proof.

math.PR↗

arXiv2012

Mean Field Games and Nonlinear Markov Processes

In this paper, we investigate the mean field games with $K$ classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of Lévy-Khintchine type (with variable coefficients). We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N-Nash equilibria for approximating systems of N agents.

math.PR↗

arXiv2012

On some models of many agent systems with competition and cooperation

We review and develop a selection of models of systems with competition and cooperation, with origins in economics, where deep insights can be obtained by the mathematical methods of game theory. Some of these models were touched upon in authors' book 'Understanding Game Theory', World Scientific 2010, where also the necessary background on games can be found.

math.OC↗

arXiv2011

Nonlinear Lévy and nonlinear Feller processes: an analytic introduction

The program of studying general nonlinear Markov processes was put forward in V. N. Kolokoltsov "Nonlinear Markov Semigroups and Interacting Lévy Type Processes" (Journ. Stat. Physics 126:3 (2007), 585-642), and was developed by the author in monograph "Nonlinear Markov processes and kinetic equations". Cambridge University Press, 2010, where, in particular, nonlinear Lévy processes were introduced. The present paper is an invitation to the rapidly developing topic of noninear Markov processes. We provide a quick (and at the same time more abstract) introduction to the basic analytical aspects of the theory developed in Part II of the above mentioned book.

math.PR↗

arXiv2009

The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Ito's construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

math.PR↗

arXiv2007

Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics

Functional limit theorem for continuous-time random walks (CTRW) are found in general case of dependent waiting times and jump sizes that are also position dependent. The limiting anomalous diffusion is described in terms of fractional dynamics. Probabilistic interpretation of generalized fractional evolution is given in terms of the random time change (subordination) by means of hitting times processes.

math.PR↗