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Mathematical Economics, Mathematical Finance, Random dynamical systems, Stochastic optimization and games.
Current research projects
Behavioral and Evolutionary Finance. Joint work with R. Amir, Economics Department, University of Arizona, T. Hens, Swiss Institute of Banking, University of Zurich, and K.R. Schenk-Hoppé, School of Mathematics and Leeds University Business School, University of Leeds.
Von Neumann-Gale dynamical systems with applications in Economics and Finance. Joint work with K.R. Schenk-Hoppé, School of Mathematics and Leeds University Business School, University of Leeds, and S.A. Pirogov, IITP, Russian Academy of Sciences.
Controlled random fields: theory and applications. Joint work with S.A. Pirogov, IITP, Russian Academy of Sciences, and S.D. Flåm, Department of Economics, University of Bergen.
Current organizational initiatives
Organizer (with R. Amir and K.R. Schenk-Hoppé): Trimester Program "Stochastic Dynamics in Economics and Finance", Hausdorff Research Institute for Mathematics, Bonn, May - August 2013. Poster
Guest Editor (with K.R. Schenk-Hoppé and W.T. Ziemba): Special issue of the Annals of Finance "Behavioral and Evolutionary Finance", to appear in 2013.
Guest Editor (with A. Bensoussan and M. Steffensen): Special issue of Stochastics "Stochastic Control with Applications in Finance, Insurance and Operations Management: In memory of Michael Taksar", to appear in 2013.
Organizer (with Goran Peskir): Stochastic Economics and Finance 2012, Annual Workshop, University of Manchester, 24-25 November 2012.
Teaching
Mathematical Economics, Mathematical Finance, Financial Economics, Game Theory, Operations Research, Mathematics for Economists.
Monographs
1. Stochastic models of control and economic dynamics. Academic Press, London, 1987 (with V.I. Arkin).
2. Markov fields over countable partially ordered sets: Extrema and splitting. Memoirs of Amer. Math. Soc., v. 112 (537), 1994 (with P.E. Greenwood).
Papers accepted for publication
3. Asset market games of survival: A synthesis of evolutionary and dynamic games. To appear in the Annals of Finance, 2013. Electronic publication (online first): October 2012. (With R. Amir and K.R. Schenk-Hoppé.) Pdf file
4. Growth-optimal investments and numeraire portfolios under transaction costs. To appear in: Handbook of the Fundamentals of Financial Decision Making. ( L.C. MacLean and W.T. Ziemba, eds.), 2012. Working paper No 09-23 of the Mathematics Department of the University of Loughborough, September 2009; also pre-published electronically as arXiv:0909.4730 (With W. Bahsoun and M.I. Taksar.)
Published papers
5.
Evolutionary finance and dynamic
games, 2011, Mathematics and Financial Economics, v. 5, 161-184. (With R.
Amir, T.
Hens and L. Xu.)
6. Local stability analysis of a stochastic evolutionary
financial market model with a risk-free asset, 2011, Mathematics and Financial Economics, v. 5, 185-202. (With T.
Hens and K.R. Schenk-Hoppé.)
7. Linearization and local stability of random
dynamical systems, 2011, Proceedings of the American
Mathematical Society, v. 139, 1061-1072 (with K.R. Schenk-Hoppé
and S.A. Pirogov). 8. Almost sure Nash equilibrium strategies in
evolutionary models of asset markets, Mathematical Methods of
Operations Research, 2011, v. 73, 235-250 (with W.
Bahsoun and L. Xu). 9. Survival and evolutionary stability of the Kelly
rule, 2011, in: The Kelly Capital Growth Investment Criterion:
Theory and Practice (L.C. MacLean, E.O. Thorp and
10. Growing wealth with fixed-mix strategies, 2011, in: The Kelly Capital Growth Investment Criterion: Theory and Practice (L.C. MacLean, E.O. Thorp and W.T. Ziemba, eds.), World Scientific, Singapore, pp. 427-455 (with M.A.H. Dempster and K.R. Schenk-Hoppé).
11. Stochastic nonlinear Perron-Frobenius theorem, 2010, Positivity, v. 14, 43-57 (with S.A. Pirogov).
12. Arbitrage in stationary markets, 2009, Decisions in Economics and Finance, v. 32, 5-12 (with D. Kapoor).
13. Dynamic interaction models of economic equilibrium, 2009, Journal of Economic Dynamics and Control, v. 33, 166-182 (with M.I. Taksar).
14. Evolutionary finance, 2009, in: Handbook of Financial Markets: Dynamics and Evolution (T. Hens and K.R. Schenk-Hoppé, eds.), a volume in the Handbooks in Finance series, W.T. Ziemba, ed., Elsevier, Amsterdam, pp. 507-566 (with T. Hens and K.R. Schenk-Hoppé).
15. Globally evolutionarily stable portfolio rules, 2008, Journal of Economic Theory, v. 140, 197-228 (with T. Hens and K.R. Schenk-Hoppé).
16. Rapid paths in von Neumann-Gale dynamical systems, 2008, Stochastics, v. 80, 129-142 (with W. Bahsoun and M.I. Taksar.)
17. Financial markets. The joy of volatility, 2008, Quantitative Finance, v. 8, 1-3 (with M.A.H. Dempster and K.R. Schenk-Hoppé).
18. Stochastic equilibria in von Neumann-Gale dynamical systems, 2008, Transactions of the American Mathematical Society, v. 360, 3345-3364 (with K.R. Schenk-Hoppé).
19. A stochastic contraction principle, 2007, Random Operators and Stochastic Equations, v. 15, 155-162 (with S. A. Pirogov).
20. Volatility-induced financial growth, 2007, Quantitative Finance, v. 7, 151-160 (with M.A.H. Dempster and K.R. Schenk-Hoppé). Reprinted in: Quantitative Fund Management, M.A.H. Dempster, G. Mitra and G. Pflug, eds., Chapman and Hall / CRC Financial Mathematics Series (Taylor and Francis Group), 2009, 67-84.
21. Pure and randomized equilibria in the stochastic von Neumann-Gale model, 2007, Journal of Mathematical Economics, v. 43, 871-887 (with K.R. Schenk-Hoppé).
22. Asset pricing and hedging in financial markets with transaction costs: An approach based on the von Neumann-Gale model, 2006, Annals of Finance, v. 2, 327-355 (with M.A.H. Dempster and M.I. Taksar).
23. The von Neumann-Gale growth model and its stochastic generalization, 2006, in: Handbook on Optimal Growth, R.-A. Dana, C. Le Van, T. Mitra and K. Nishimura, eds., Springer, New York, 337-383 (with K.R. Schenk-Hoppé).
24. Evolutionary Stable Stock Markets, 2006, Economic Theory, v. 27, 449-468 (with T. Hens and K.R. Schenk-Hoppé).
25. Market selection and survival of investment strategies, 2005, Journal of Mathematical Economics, Special Issue on Evolutionary Finance, v.41, 105-122 (with R. Amir, T. Hens and K.R. Schenk-Hoppé).
26. On the Fundamental Theorem of Asset Pricing: Random constraints and bang-bang no-arbitrage criteria, 2004, Mathematical Finance, v.14, 201-221 (with K. Schürger and M.I. Taksar).
27. Noncooperative versus cooperative R&D with endogenous spillover rates, 2003, Games and Economic Behavior, v. 42, 183-207 (with R. Amir and J. Wooders).
28. Exponential growth of fixed-mix strategies in stationary asset markets, 2003, Finance and Stochastics, v. 7, 263-276 (with M.A.H. Dempster and K.R. Schenk-Hoppé).
29. From rags to riches: On constant proportions investment strategies, 2002, International Journal of Theoretical and Applied Finance, v. 5, 563-573 (with K.R. Schenk-Hoppé).
30. Equilibrium states of random economies with locally interacting agents and solutions to stochastic variational inequalities, 2002, Annals of Operations Research, Special Issue ''Stochastic Equilibrium Problems in Economics and Game Theory'', v. 114, 145-165 (with M.I. Taksar).
31. Market selection of financial trading strategies: Global stability, 2002, Mathematical Finance, v. 12, 329-339 (with T. Hens and K.R. Schenk-Hoppé).
32. Convex stochastic duality and the ''biting lemma'', 2002, Journal of Convex Analysis, v. 9, 237-244 (with S.D. Flåm).
33. Sharing nonconvex costs, 2001, Journal of Global Optimization, Special Issue on Applications to Economics, v. 20, 257-271 (with S.D. Flåm).
34. Convex stochastic optimization for random fields on graphs: A method of constructing Lagrange multipliers, 2001, Mathematical Methods of Operations Research, v. 54, 217-237 (with M.I. Taksar).
35. Rapid growth paths in convex-valued random dynamical systems, 2001, Stochastics and Dynamics, v. 1, 493-509 (with M.I. Taksar).
36. Stochastic programming: Non-anticipativity and Lagrange multipliers, in: Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht, 2001, v. 4, 332-338 (with S.D. Flåm).
37. On Dynkin's model of economic equilibrium under uncertainty, 2001, Economics Bulletin, v. 3, 1-8 (with G. Carlier).
38. A functional central limit theorem for equilibrium paths of economic dynamics, 2000, Journal of Mathematical Economics, v. 33, 81-99 (with R. Amir).
39. Turnpike theorems for positive multivalued stochastic operators, 2000, Advances in Mathematical Economics, v. 2, 1-20 (with S. Anoulova and V.M. Gundlach).
40. Robust insurance mechanisms and the shadow prices of information constraints, 1999, Journal of Applied Mathematics and Decision Sciences, v. 3, 85-128 (with W.K. Klein Haneveld and L.J. Mirman).
41. Convex-valued random dynamical systems: A variational principle for equilibrium states, 1999, Random Operators and Stochastic Equations, v. 7, 23-38 (with L. Arnold and V.M. Gundlach).
42. Stochastic version of Polterovich's model: Exponential turnpike theorems for equilibrium paths, 1999, Macroeconomic Dynamics, v. 3, n. 2, 149-166 (with R. Amir).
43. Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators, 1998, Set-Valued Analysis, v. 6, 61-82 (with S.D. Flåm).
44. Balanced states in stochastic economies with locally interacting agents, 1998, Stochastics, v. 64, 235-253 (with M.A.H. Dempster and S.A. Pirogov).
45. The turnpike property and the central limit theorem in stochastic models of economic dynamics, 1997, in: Statistics and Control of Stochastic Processes, The Liptser Festschrift, Yu.M. Kabanov, B.L. Rozovskii, and A.N. Shiryaev, eds., World Scientific, Singapore - New Jersey - London, 63-101 (with S.D. Flåm).
46. Noncooperative games in networks: Stability and sensitivity of equilibria, 1997, in: Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.-S. Pang, eds., SIAM, Philadelphia, 62-75 (with S.D. Flåm).
47. Metonymy and cross section demand, 1997, Journal of Math. Economics, v. 28, 397-414 (with W. Hildenbrand and M. Jerison).
Selected publications prior to 1997
48. Turnpike theorems for stochastic equilibria on graphs, 1996, in: Operations Research Proceedings 1995, P. Kleinschmidt et al., eds., Springer-Verlag, Berlin - Heidelberg - N.Y., 241-245 (with M.A.H. Dempster and S.A. Pirogov).
49. The shortest path around an island outside the shallows, 1995, Markov Processes and Related Fields, v. 1, 407-418.
50. Stochastic equilibria on graphs, 1995, II, Journal of Math. Economics, v. 24, 383-406 (with M.I. Taksar).
51. Stochastic equilibria on graphs, 1994, I, Journal of Math. Economics, v. 23, 401-433 (with M.I. Taksar).
52. A limit theorem for random matrices with a multiparameter, 1994, Stochastic Processes and their Appl., v. 52, 65-74 (with K. Schürger ).
53. Regular conditional expectations and the continuum hypothesis, in: The Dynkin Festschrift, Birkhäuser, Boston, 1994, 85-93.
54. Stochastic extrema, splitting random elements and models of crack formation, in: System Modelling and Optimization, Proc. of the 16th IFIP-TC7 Conf. (Compiègne, July 5-9, 1993), Lecture Notes in Control and Inform. Sci., No. 197, Springer-Verlag, London, 1994, 315-319 (with P.E. Greenwood).
55. Controlled random fields on graphs and stochastic models of economic equilibrium, 1991, in: New Trends in Probability and Statistics (Volume in Honour of Yu. V. Prohorov), VSP, Utrecht, 391-412.
56. The method of monotone operators in the stochastic theory of economic equilibrium, 1990, Matekon, v. 26, n. 3, Sharpe Publ., N.Y., 21-43.
57. Stochastic extremal problems and the strong Markov property of random fields, 1988, Uspekhi Matem. Nauk (Russian Math. Surveys), v. 43, n. 2, 3-41.
58. Measurable images of compact spaces and selectors of analytic sets, 1988, Doklady AN SSSR (Soviet Math. Dokl.), v. 299, n. 3, 538-541.
59. Controlled random fields on a directed graph, 1988, Teor. Ver. i Primen. (Theory of Probab. and Appl.), v. 33, n. 3, 465-479.
60. Random sets and Markov fields (Invited paper), 1987, in: Proc. of the 1st World Congress of the Bernoulli Soc.,Tashkent, Sept. 1986. Utrecht, VNU, 339-344.
61. Measurable selection theorems and stochastic control models in general topological spaces, 1986, Mat. Sbornik (Math. USSR Sbornik), v. 131, n. 1, 27-39.
62. Regular conditional expectations of random variables depending on parameters, 1986, Teor. Ver. i Primen. (Theory of Probab. and Appl.), v. 31, n. 3, 586-589.
63. Equilibrium states of monotone operators and equilibrium trajectories in stochastic economic models, 1986, Lect. Notes Control Inf. Sci., n. 81, 334-338.
64. Stochastic models of economic dynamics and equilibrium: Asymptotic behaviour of equilibrium paths, 1984, in: Proc. of the Seventh Conf. on Probab. Theory (Brasov 1982), Bucharest, 149-158 (with P.K. Katyshev).
65. An optimality principle and an equilibrium theorem for controlled random fields on a directed graph, 1984, Doklady AN SSSR (Soviet Math. Dokl.), v. 274, n. 4, 782-786.
66. Strong Markov property and splitting elements for random fields over a partially ordered set, 1983, Teor. Ver. i Primen. (Theory of Probab. and Appl.), v. 28, n. 4, 801-802.
67. Probabilistic variant of the turnpike theorem for homogeneous convex controllable models, 1983, Math. Zametki (Math. Notes), v. 33, n. 1, 185-194 (with S. E. Kuznetsov).
68. Models of economic growth taking account of random factors, 1982, Econ. i Mat. Metody, v. 18, n. 6, 1125-1132 (with S. E. Kuznetsov). (In Russian.)
69. A non-stationary stochastic analog of the von Neumann-Gale model of a developing economy: a theorem on a characteristic, 1981, Optimizatsya, v. 27, 34-43. (In Russian.)
70. Equilibrium paths in probabilistic models of economic dynamics, 1982, Teor. Ver. i Primen. (Theory of Probab. and Appl.), v. 27, n. 1, 120- 128 (with P.K. Katyshev).
71. Homogeneous convex models in the theory of controlled random processes, 1980, Doklady AN SSSR (Soviet Math. Dokl.), v. 253, n. 3, 524-527.
72. Probabilistic modification of the von Neumann-Gale model, 1980, Uspekhi Matem. Nauk (Russian Math.Surveys), v. 35, n. 4, 185-186 (with Yu.M. Kabanov).
73. Bounds for prices in models of economic development, 1980, Econ. i Mat. Metody, v. 16, n. 3, 592-593. (In Russian.)
74. Measurable choice and the continuum axiom, 1978, Doklady AN SSSR (Soviet Math. Dokl.), v. 238, n. 1, 11-14.
75. ''Splitting times'' for random fields, 1978, Teor.Ver. i Primen. (Theory of Probab. And Appl.), v. 23, n. 2, 433-438 (with A.I. Ovseevich).
76. Liftings and conditional distributions, 1977, in: Selected Topics in Probab.Theory and Math. Economics, Moscow, CEMI, 92-113. (In Russian.)
77. ''Markov times'' for random fields, 1977, Teor.Ver. i Primen. (Theory of Probab. and Appl.), v. 22, n. 3, 575-581.
78. Turnpike theorems in stochastic models of economic dynamics, 1976, Mat. Zametki (Math. Notes), v. 19, n. 2, 279-290.
79. Optimal economic planning under production uncertainty, 1976, Publ. Économétr., v. 9, n. 1, 81-98.
80. Lagrange multipliers for the problems of stochastic programming, 1976, Lect. Notes Econ. Math. Syst., n. 133, 34-48.
81. Measurable selection and dynamic programming, 1976, Math. Oper. Res., v. 1, n. 3, 52-55.
82. Regular conditional expectations of correspondences, 1976, Teor. Ver. i Primen.(Theory of Probab. and Appl.), v. 21, n. 2, 333-347 (with E.B. Dynkin).
83. The space 2X and Markov fields, 1976, Doklady AN SSSR (Soviet Math. Dokl.), v. 230, n. 1, 22-25.
84. Models of economic dynamics taking account of indeterminacy in the operation of an industrial process, 1975, Doklady AN SSSR (Soviet Math. Dokl.), v. 223, n. 3, 537-540.
85. Positive matrix-valued cocycles over dynamical systems, 1974, Uspekhi Matem. Nauk (Russian Math. Surveys), v. 29, n. 5, 219-220.
86. Optimal stochastic programs and their supporting prices, 1974, in: Mathematical Models in Economics, Amsterdam, North Holland, 219-252.
87. Optimal economic planning taking account of stationary random factors, 1972, Doklady AN SSSR (Soviet Math. Dokl.), v. 206, n. 5, 1040-1042.
88. Markov chains on a matrix group, 1971, Mat. Zametki (Math. Notes), v. 10, n. 2, 181-186.
Editorial and organisational activities
Member of the International Programme Committees of the 15th, 16th, and 17th Conferences, "System Modelling and Optimization'', IFIP (Zurich 1991, Compiegne 1993, Prague 1995).
Member of the Programme Committee of the Conference "Stochastic Optimization in Economics and Finance'' (Colchester, 1991).
Member of the Editorial Board of the Journal of Mathematical Economics: from 1992 to 2009.
Member of the Editorial Advisory Board of the journal "Random Operators and Stochastic Equations" (since 1998).
Member of the Editorial Board of Economic Theory (since 2009).
Member of the IFIP Working Group 7.7 ''Stochastic Optimization'' (since 1991).
Guest Editor (with S.D. Flåm) and L. J. Mirman) of the special volume of the Annals of Operations Research ''Stochastic Equilibrium Problems in Economics and Game Theory'' (appeared as v. 114 of Ann. Oper. Res. in December 2002).
Organiser of the session ''Evolutionary Economics and Finance'' at the Conference of the Society for Advancement of Economic Theory, Kos, Greece, June 2007.
Guest Editor (with N.H. Bingham) of the special volume of Stochastics ''A Festschrift for Priscilla Greenwood''; appeared: Stochastics, v. 80, No. 2-3, 2008.
Co-organiser (with K.R. Schenk-Hoppé) of the Manchester-Leeds Conference on Mathematical Economics and Finance, School of Social Sciences, University of Manchester, 30 May - 1 June 2008.
Organiser of the session "Stochastic Optimization and Games with Applications to Finance" at the Conference of the Society for Advancement of Economic Theory, Ischia, Italy, June-July 2009.
Co-organiser (with K.R. Schenk-Hoppé) of the session ''Evolutionary Finance'' at the Conference of the Society for Advancement of Economic Theory, Faro, Portugal, June-July 2011.
Co-organiser (with R. Amir, S.D. Flåm and K.R. Schenk-Hoppé) of the International Conference "Stochastic Economics and Finance", Department of Economics, University of Bergen, Norway, September 2011.
Selected invited lectures
1. 1st World Congress of the Bernoulli Society, Tashkent, USSR, 1986. "Random sets and Markov fields''.
2. International Conference "Stochastic Optimization in Economics and Finance'', Colchester, England, 1991. "Controlled random fields on graphs and stochastic models of economic equilibrium''.
3. 15th IFIP Conference, Zurich, Switzerland, 1991. "Dynamic decision problems involving random fields''.
4. Conference on Operations Research, Lunteren, The Netherlands, 1992. a) "Monotone demand and supply systems'' (joint work with W. Hildenbrand and R. John). b) "Stochastic equilibria on graphs'' (joint work with M.I. Taksar).
5. Isaac Newton Institute and Bank of England Conference Mathematics of Finance: Models, Theories, and Computation, Cambridge, England, 1995. "Robust insurance mechanisms and the shadow price of information'' (joint work with W. K. Klein Haneveld and L.J. Mirman).
6. International Conference "Social Interactions and Aggregate Economic Activity'', The Santa Fe Institute, Santa Fe, New Mexico, US, 1996. "Economic equilibrium models with locally interacting agents''.
7. International Conference "The Mathematics of Dynamic Economic Models", University of Copenhagen, Denmark, 2000. "Stochastic equilibrium dynamics with local interactions: turnpike properties and stationary equilibria".
8. International Conference "Evolutionary Finance", University of Zurich, Switzerland, 2002. "The evolution of a financial market with short-lived assets: Asymptotic results" (joint work with R. Amir, T. Hens and K.R. Schenk-Hoppé).
9. International Conference "Game Theory and Mathematical Economics", Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland, 2004. ''Evolutionary Finance: Survival and extinction of portfolio rules'' (joint work with R. Amir, T. Hens and K.R. Schenk-Hoppé).
10. International Conference "Microscopic stochastic dynamics in economics", University of Bielefeld, Germany, 2005. "Controlled random fields on graphs and stochastic models of economic equilibrium".
11. International Conference "Behavioural Models in Economics and Finance", University of Vienna, Austria, 2005. "Globally evolutionarily stable portfolio rules in a financial market with long-lived dividend-paying assets" (joint work with T. Hens and K.R. Schenk-Hoppé).
12. NBER-NSF-CEME Conference on General Equilibrium Theory, Department of Economics, Kansas University, Lawrence, Kansas, US, 2006. "The von Neumann-Gale growth model: New applications and solutions to old problems" (joint work with M.A.H. Dempster, K.R. Schenk-Hoppé and M.I. Taksar).
13. International Conference ''Nonlinear Economic Dynamics'', University of Bielefeld, Germany, 2007. ''Evolutionary dynamics and temporary equilibrium in financial markets'' (joint work with T. Hens and K.R. Schenk-Hoppé).
14. Workshop "Finance, Stochastics, Insurance", University of Bonn, Germany, 2008. "Evolutionary finance: Discrete time models" (joint work with T. Hens and K.R. Schenk-Hoppé).
15. Workshop "Evolution and Market Behavior in Economics and Finance". Sant' Anna School of Advanced Studies, Pisa, Italy, 2009. "Evolutionary finance and dynamic games'' (joint work with R. Amir, T. Hens and K.R. Schenk-Hoppé).
16. Workshop on the Probabilistic Impulse behind Modern Economic Theory, Institute for Mathematical Sciences, National University of Singapore, 2011. "Asset market games of survival: a synthesis of evolutionary and dynamic games'' (joint work with R. Amir, T. Hens and K.R. Schenk-Hoppé).
17. International Conference "General Equilibrium Analysis", Higher School of Economics, Moscow, Russia, 2011. "Behavioral equilibrium and evolutionary dynamics: A way to a non-Walrasian world'' (joint work with R. Amir, T. Hens and K.R. Schenk-Hoppé).
18. International Conference "Stochastic Economics and Finance", Department of Economics, University of Bergen, Norway, 2011. "Von Neumann-Gale dynamical systems and their applications in finance" (joint work with M.A.H. Dempster, K.R. Schenk-Hoppé and M.I. Taksar).