111,914 research outputs found
Stochastic integrals and SDE driven by nonlinear Lévy noise
No full textWe develop the theory of SDE driven by nonlinear Lévy noise, aiming at applications to Markov processes. 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 generates a Markov semigroup, where the measures are metricized by the Wasserstein–Kantorovich metrics W p . The analysis of SDE driven by nonlinear Lévy noise was initiated by the author in (“Kolokoltsov, Probability Theory Related Fields, DOI: 10.1007/s00440-010-0293-8, 2009”) (inspired partially by “Carmona and Nualart, Nonlinear Stochastic Integrators, Equations and Flows, Stochatic Monographs, v. 6, Gordon and Breach, 1990”), see also (“Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Monograph. To appear in CUP, 2010”). Here, we suggest an alternative (seemingly more straightforward) approach based on the path-wise interpretation of these integrals as nonhomogeneous Lévy processes. Moreover, we are working with more general W p -distances rather than with W 2
8. J. M. Binner, L. R. Fletcher, V. Kolokoltsov, F. Ciardiello (2013). External Pressure on Alliances: What does Prisoners Dilemma Reveal?, Games, 4(4), 754-775.ISSN 2073-4336
No full textExistence of solutions to path-dependent kinetic equations and related forward-backward systems
No full textThis paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward - forward systems of equations of mean field games. We present local well-posedness, global existence and some regularity results for these equations
Nonlinear Levy and nonlinear Feller processes : an analytic introduction
No full textThe program of studying general nonlinear Markov processes was put forward in V. N. Kolokoltsov "Nonlinear Markov Semigroups and Interacting L\'evy 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 L\'evy 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
Idempotent structures in optimization
No full textConsider the set A = R ∪ {+∞} with the binary operations o1 = max
and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries
in A. Let the generalised sum u o1 v of two vectors denote the vector with
entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector
v ∈ An denote the vector with the entries a o2 vj . With these operations,
the set An provides the simplest example of an idempotent semimodule.
The study of idempotent semimodules and their morphisms is the subject
of idempotent linear algebra, which has been developing for about
40 years already as a useful tool in a number of problems of discrete optimisation.
Idempotent analysis studies infinite dimensional idempotent
semimodules and is aimed at the applications to the optimisations problems
with general (not necessarily finite) state spaces. We review here
the main facts of idempotent analysis and its major areas of applications
in optimisation theory, namely in multicriteria optimisation, in turnpike
theory and mathematical economics, in the theory of generalised solutions
of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games
and controlled Marcov processes, in financial mathematics
Time dependent ecological model
No full textKolokoltsov V, Kondratiev Y. Time dependent ecological model . Interdisciplinary Studies of Complex Systems. 2022;(20):11-15.We study the asymptotic behaviour of an ecological model in the presence of a time dependent birth rate
The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
No full textIto'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 coeffcients (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
Nonlinear Markov semigroups and interacting Lévy type processes
No full textSemigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R-d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models
Evolutionary game of coalition building under external pressure
No full textWe study the fragmentation-coagulation, or merging and splitting, model as introduced in [16], where N small players can form coalitions to resist to the pressure exerted by the principal. It is a Markov chain in continuous time and the players have a common reward to optimize. We study the behavior as N grows and show that the problem converges to a (one player) deterministic optimization problem in continuous time, in the infinite dimensional state space `1. We apply the method developed in [8], adapting it to our different framework. We use tools involving dynamics in `1, generators of Markov processes, martingale problems and coupling of Markov chains
Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics
No full textFunctional limit theorems for continuous-time random walks (CTRW) are found in the 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
- …
