Markov Processes, Semigroups and Generators

Abstract. We consider the motion of a particle along the geodesic lines of the Poincaré half-plane. The particle is specularly reflected when it hits randomlydistributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.

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“…which is the Duhamel expansion giving (see [11], page 52) the solution to ∂ ∂t f = (A + L)f and this concludes the proof. 3.4.…”

Section: 3supporting

confidence: 58%

Motion Among Random Obstacles on a Hyperbolic Space

2016

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“…The convergence of the chains in distribution follows from the convergence of transition operators by a well known general result, see e.g. (Kallenberg, 2002), or (Kolokoltsov, 2011). Moreover, weak convergence of random variables to a constant implies their convergence in probability.…”

Section: Auxiliary Lemmasmentioning

confidence: 91%

“…These class of models is often referred to as mean-field games. Related equilibrium concept was called the Nash certainty equivalence principle, see (Lasry & Lions, 2006;Achdou & Capuzzo-Dolcetta, 2010;Gomes, Mohr, & Souza, 2010;Huang, Caines, & Malhamé, 2003;Huang, Malhamé, & Caines, 2006;Huang, 2010;Kolokoltsov, Li, & Yang, 2011) and references therein. Unlike our centralized setting above, analysis of these models does not lead to the control of distributions in the limit, but rather to certain consistency condition on homogeneous individual controls (Nash certainty equivalence).…”

Section: Conclusion and Bibliographical Commentsmentioning

confidence: 99%

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Kolokoltsov¹

2012

IJSP

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Managing large complex stochastic systems, including competitive interests, when one or several players can control the behavior of a large number of particles (agents, mechanisms, vehicles, subsidiaries, species, police units, etc), say N k for a player k, the complexity of the game-theoretical (or Markov decision) analysis can become immense as N k → ∞. However, under rather general assumptions, the limiting problem as all N k → ∞ can be described by a well manageable deterministic evolution. In this paper we analyze some simple situations of this kind proving the convergence of Nashequilibria for finite games to equilibria of a limiting deterministic differential game.

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“…Equation (27) defines an integral operator with a positive stochastic kernel, which, taking into account assumption (26), is in fact a probability kernel. Hence this operator extends naturally to a positivity preserving conservative contraction in B(R) (68)), so that…”

Section: Duality For F Depending On the Difference Of Its Argumentsmentioning

confidence: 99%

Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

Kolokoltsov

2015

Journal of Applied Probability

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We introduce a notion of kth order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put -call symmetries in option pricing. Our general kth order duality can be financially interpreted as put -call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of kth order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries.

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Markov Processes, Semigroups and Generators

Cited by 169 publications

References 0 publications

Motion Among Random Obstacles on a Hyperbolic Space

Motion Among Random Obstacles on a Hyperbolic Space

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

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