arXiv subjects

Vassili Kolokoltsov

Publications and source records attributed to Vassili Kolokoltsov.

26 records · Page 2Linked to original sources

arXiv2013

Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift

We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators, where the nonlinear drift includes a mean field term of McKean type and the driving noise is Gaussian. Uniform convergence in L^2 sense is achieved by applying L^2-type estimates and the Gronwall Theorem. The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by stochastic nonlinear oscillators with differentiable trajectories

math.PR↗

arXiv2012

The turnpike theorems for Markov games

This paper has a two-folded purpose. First, we attempt to outline the development of the turnpike theorems in the the last several decades. Second, we study turnpike theorems in finite-horizon two-person zero-sum Markov games on a general Borel state space. Utilising the Bellman (or Shapley) operator defined for this game, we prove the stochastic versions of the early turnpike theorem on the set of optimal strategies and the middle turnpike theorem on the distribution of the state space.

math.PR↗

arXiv2011

Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black-Scholes equations

Expanding the ideas of the author's paper 'Nonexpansive maps and option pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black-Scholes type equations.

math.OC↗

arXiv2010

Stochastic monotonicity and duality for one-dimensional Markov processes

The theory of monotonicity and duality is developed for general one-dimensional Feller processes. Moreover it is shown that local monotonicity conditions (conditions on the Lévy kernel) are sufficient to prove the well-posedness of the corresponding Markov semigroup and process, including unbounded coefficients and processes on the half-line.

math.PR↗

arXiv2008

The optimal assignment problem for a countable state space

Given a square matrix B=(b_{ij}) with real entries, the optimal assignment problem is to find a bijection s between the rows and the columns maximising the sum of the b_{is(i)}. In discrete optimal control and in the theory of discrete event systems, one often encounters the problem of solving the equation Bf=g for a given vector g, where the same symbol B denotes the corresponding max-plus linear operator, (Bf)_i:=max_j (b_{ij}+f_j). The matrix B is said to be strongly regular when there exists a vector g such that the equation Bf=g has a unique solution f. A result of Butkovic and Hevery shows that B is strongly regular if and only if the associated optimal assignment problem has a unique solution. We establish here an extension of this result which applies to max-plus linear operators over a countable state space. The proofs use the theory developed in a previous work in which we characterised the unique solvability of equations involving Moreau conjugacies over an infinite state space, in terms of the minimality of certain coverings of the state space by generalised subdifferentials.

math.OC↗

arXiv2007

The Central Limit Theorem for the Smoluchovski Coagulation Model

The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN). A rather precise rate of convergence is given both for LLN and CLT.

math.PR↗

arXiv2005

Solutions of max-plus linear equations and large deviations

We generalise the Gartner-Ellis theorem of large deviations theory. Our results allow us to derive large deviation type results in stochastic optimal control from the convergence of generalised logarithmic moment generating functions. They rely on the characterisation of the uniqueness of the solutions of max-plus linear equations. We give an illustration for a simple investment model, in which logarithmic moment generating functions represent risk-sensitive values.

math.PR↗

arXiv2004

Set coverings and invertibility of Functional Galois Connections

We consider equations of the form Bf=g, where B is a Galois connection between lattices of functions. This includes the case where B is the Legendre-Fenchel transform, or more generally a Moreau conjugacy. We characterise the existence and uniqueness of a solution f in terms of generalised subdifferentials. This extends a theorem of Vorobyev and Zimmermann, relating solutions of max-plus linear equations and set coverings. We give various illustrations.

math.FA↗