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Professor Vassili Kolokoltsov

Contact: email:, tel. 02476574806, Office D0.11 Zeeman Building

Office hours winter term 2018: Monday and Thursday 11.30-12.30

General research interests: probability and stochastic processes, optimization and games with applications to business, biology and finances, mathematical physics, differential equations and functional analysis.

Applications from perspective PhD students are welcome.

Publications: more than 100 papers and several monographs, see selected earlier reprints below and the list of main publications in CV, (on the right side of this page).

The drafts of most of the recent publications can be found on ArXiv:

Teaching 2006-: 'Probability A and B' for the first year students, see ProbBlectnotes; 'Advanced probability' for the third year, 'Brownian motion', see BMlectnotes, and 'Stochastic methods in Finances', see FinanceExercises for final years MMORSE.

My talks in 2009-2011: Talks My books: monographs

Talks on the Conferences in 2015:

2015 (11 Feb) Mathematics Colloquium, University of Birmingham. Talk:
Evolutionary game-theoretic modelling of cyber-security, crime prevention, inspection and corruption.

2015 Semester Program 'Geometric Mechanics, Variational and Stochastic Methods'
01.01.15 - 30.06.15 at the Centre Interfacultaire Bernoulli (CIB) - EPFL, Switzerland.
Talk (20.02.15) 'Generalized mass action law and thermodynamics of nonlinear Markov processes'

2015 (2 - 6 March) 'New Perspectives in Analysis and Probability', Dep. Maths, University of Sussex.
Talk: Probabilistic methods for solving fractional differential equations.

2015 (10-12 June) 'Mean-Field Games and Related Topics - 3', Paris, Institute Henry Poincare.
Talk 'On mean-field games with common noise' was presented on my behalf
by P. Caines (as I was unable to attend).

2015 (29 Jun - 04 Jul) 'Modern trends in controlled stochastic processes: theory and applications', Liverpool.
Talk: Games of Pressure and resistance.

2015 (8 - 10 July) SIAM Control Conference, Paris. Two talks presented by myself and two talks on joint research presented by my co-authors.

2015 (16-17 July) ISDG2015, University of Strathclyde, Glasgow. Talk: Mean-field game model of corruption.

2015 (16-18 Dec) 'Stochastic Limit Analysis for Reacting Particle Systems (SLARPS)', Weierstrasse Inst. Berlin.
Mini lecture course (three two-hour lectures): Dynamic law of large numbers and CLT, with the rates of convergence, for interacting particle systems including Smoluchovskii coagulation and Boltzmann collisions.

I received  the St. Petersburg University award 'Research work in 2011' for the book 'Understanding Game Theory' (jointly with O. A. Malafeyev)

Current Membership in Editorial Boards:

Member of the Editorial Board of the 'Dynamic Games and Applications' (2013 -)

Member of the Editorial Board of the Open Journal of Optimization OJO, (2012 -)

Member of the Edtorial Board of Statistics, Optimization & Information Computing SOIC (2013 -)

Member of the Edtorial Board of Fractional Differential Calculus (FDC), see (2014 -)

Member of the Editorial Board Asia Pacific Journal of Mathematics (2014 - ), see
Member of the Editorial Board 'Issues of Analysis' (2013 - ), see

Current PhD students: D. Santiago (Nonlinear Markov processes), Alexei Pak (SDEs with the coefficients depending on VAR), E. Hernandez-Hernandez (Fractional calculus), Stamatios Katsikas (Inspection-Corruption Games), Lorenzo Toniazzi on 'Fractional calculus and duality'

Recent conference organization:

The fifth Warwick School-Conference on Control and Games, Warwick 29th Apr - 2 May 2014

The fourth Warwick School-Conference on Control and Games, Warwick 7-10 May 2013

 'From Mean-Field Control to Weak KAM theory' , Warwick 7 - 10 May 2012 , 2010,

Slides and pictures: 

'SIAM conference on Control and its Applications', July 2011:

Satellite 'Trends in game theory' to ECCS 2011:

Some preprints:

Vassili Kolokoltsov. Stochastic monotonicity and duality of $k$th order with application to put-call symmetry of powered options.

To appear: Journal of Applied Probability 52:1 (March 2015)

Vassili Kolokoltsov and Wei Yang
Existence of solutions to path-dependent kinetic equations and related forward - backward systems.
Open Journal of Optimization 2:2, 39-44, June 2013, see
Also on archive

V. N. Kolokoltsov Nonlinear diffusions and stable-like processes with coefficients depending on the median or VaR Applied Mathematics and Optimization 68:1 (2013), 85-98.

V. Kolokoltsov and R. Lee Stochastic duality of Markov processes: a study via generators (2013)
arXiv:1304.1688. Oublished in 'Stochastic Analysis and Applications' 31 :6 (2013), 992-1023.

V. N. Kolokoltsov. Nonlinear Markov games on a finite state space (mean-field and binary interactions)
International Journal of Statistics and Probability 1:1 (2012), 77-91

V. Kolokoltsov and Wei Yang. The turnike theorems for Markov games (2012). Dynamic Games and Applications 2: 3 (2012), 294-312.

V. N. Kolokotsov and M. Veretennikova. Controlled CTRW and Fractional HJB equations (2012).

V. N. Kolokoltsov and O. A. Malafeyev. On some models of many agent systems with competition and cooperation (2012).

Vassili N.Kolokoltsov, Jiajie Li and Wei Yang (2011). Mean Field Games and Nonlinear Markov Processes.

V. N. Kolokoltsov (2011) Game theoretic analysis of incomplete markets: emergence of probabilities,
nonlinear and fractional Black-Scholes equations. ``Risk and Decision Analysis'' 4 (2013), 131-161.

V. N. Kolokoltsov (2011) Nonlinear Levy and nonlinear Feller processes, Published in De Gruyter ``Mathematics and Life Sciences'' 2013, p. 45-70.

V. N. Kolokoltsov (2010)  Stochastic monotonicity and duality for one-dimensional Markov processes.

arXiv:1002.4773. Mathematical Notes 89:5 (2011), 652-660.

V. Kolokoltsov (2008). The Levy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups 

arxive Prob. Theory Related Fields 151 (2011), 95-123.

V. Kolokoltsov (2007). The Central Limit Theorem for the Smoluchowski Coagulation Model

arXiv:0708.0329v1 [math.PR]. Probability Theory Related Fields 146: 1 (2010), 87-153.

V. Kolokoltsov (2007). Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics

arXiv:0706.1928v1[math.PR]. Theory of Probability and its Applications 53:4 (2009), 594-609.

V. Kolokoltsov. Nonlinear Markov Semigroups and Interacting Levy Type Processes.

Journ. Stat. Physics 126:3 (2007), 585-642.

V. Kolokoltsov. Kinetic equations for the pure jump models of k -nary interacting particle systems.

 Markov Processes and Related Fields 12 (2006), 95-138.

Z. Hucki, V. Kolokoltsov. Pricing of rainbow options: game theoretic approach.

  International Game Theory Review 9:2 (2007), 215-242.

V. Kolokoltsov. On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel.

Advanced Stud. Contemp. Math. 12:1 (2006), 9-38.

V.N. Kolokoltsov, A.E. Tyukov. On boundary Value Problems for Hamiltonian Systems and Absolute Minimizers in Calculus of Variations. Electronic J. Dif. Eqns. 2006 (2006), n. 90, 1-21.

V.N. Kolokoltsov. Path integration: connecting pure jump and Wiener processes. In: Waymire, J. Duan (Eds.).

"Probability and Partial Differential Equations in Modern Applied Mathematics" (2005), p. 163-180;

Some other papers:

(1) V. Kolokoltsov. Nonexpansive maps and option pricing. Kibernetika 34:6 (1998), 713-724.

(2) V. Kolokoltsov. Idempotent Structures in Optimization. Journal Math. Sci. New York 104:1 (2001), 847-880.

(3) V.P. Belavkin, V.N. Kolokoltsov. Stochastic evolution as interaction representation of a boundary value problem for Dirac type equation.

Infinite Dimensional Analysis, Quantum Probability and Related Fields 5:1 (2002), 61-92.

(4) V.P. Belavkin, V.N. Kolokoltsov. On general kinetic equation for many particle systems with interaction, fragmentation and coagulation.

Proc. Royal Society London A 459 (2003), issue 2031, 727-748.

(5) V.N. Kolokoltsov, R.L. Schilling, A.E. Tyukov. Estimates for Multiple Stochastic Integrals and stochastic Hamilton-Jacobi equations.

 Revista Matematika Iberoamerikana 20 (2004), 333-380;

(6) V. Kolokoltsov. On Extensions of Mollified Boltzmann and Smoluchovski Equations to Particle Systems with a k-ary Interaction.

Russian Journal of Mathematical Physics 10:3 (2003), 268-295.

(7) V.N. Kolokoltsov. On Markov processes with decomposable pseudo-differential generators.

 Stochastics and Stochastics Reports 76:1 (2004), 1-44.

(8) V. Kolokoltsov. Measure-valued limits of interacting particle systems with k-ary interaction I . One-dimensional limits.

Probability Theory Related Fields 126 (2003), 364-394.

(9) V. Kolokoltsov. Measure-valued limits of interacting particle systems with k -nary interaction II. Finite-dimensional limits .

 Stochastics and Stochastics Reports 76:1 (2004), 45-58;

(10) V. Kolokoltsov. Hydrodynamic limit of coagulation-fragmentation type models of k -nary interacting particles.

 Journ. Stat. Physics 115, 5/6 (2004), 1621-1653;

(11) M. Akian, S. Gaubert, V.N. Kolokoltsov. Set Coverings and the Invertibility of Functional Galois Connections.

 In: "Idempotent Mathematics in Mathematical Physics" (Eds. G.L. Litvinov and V.P. Maslov), Contemporary Mathematics Series v. 377, AMS (2005), 19-51;

(12) V.N. Kolokoltsov, R.L. Schilling, A.E. Tyukov. Transience and non-explosion of certain stochastic Newtonian systems.

Electronic J. of Probability 7 (2002), Paper 19.

(13) V.N. Kolokoltsov. Small diffusion and fast dying out asymptotics for superprocesses as non-Hamiltonian quasi-classics for evolution equations.

Electronic J. of Probability 6 (2001), paper 21.

Prof Vassilli