research Interests

Optimal Stopping with Applications 2009, Symposium, 23 - 26 June 2009, Abo/Turku, Finland, Slide Presentation,

Third World Conference of the Game Theory Society - July 13-17, 2008, Evanston, USA
Session 134: Dynamics and convexity, July 16, 2008

Slide Presentation, May/June 2008, Chiba Univ., Japan / Petrozavodsk, Russia [pdf]

D1 (joint with Alexander Gordon). The expected number of intersections of a four valued bounded martingale with any level may be infinite, appear in "Optimality and Risk - Modern Trends in Mathematical Finance: The Kabanov Festschrift, eds. F. Delbaen, M. Rasonyi, and C. Stricker, Springer , 2009. [pdf]

D2 The Decomposition-Separation Theorem for Finite Nonhomogeneous Markov Chains and Related Problems, to appear in Markov Processes and Related Fields: A Festschrift in Honor of Thomas G. Kurtz, eds. S. Ethier, J. Feng and R.H. Stockbridge, IMS, 2008, v. 4, 1-15. [pdf]

R1 Gittins Type Index Theorem for Randomly Evolving Graphs, (joint with E. Presman), in: From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, Kabanov, Y; Lipster, R; Stoyanov, J (Eds.), Springer, 2006, XXXVIII, pp. 567-588.

R2 The Optimal Stopping of Markov Chain and Recursive Solution of Poisson and Bellman Equations, From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, Kabanov, Y; Lipster, R; Stoyanov, J (Eds.), Springer, 2006, XXXVIII, pp. 609-621.

My main general research interests are the theory of stochastic processes, theory of optimal stochastic control, and their applications to problems in economics, operations research, and finance. In a more detailed way they can be grouped as follows with the corresponding selected publications on these topics. The full list of publications including the link to the
Math Sci Net. and abstracts of conference presentations can be found on [ List of Publications ] .

It may seem surprising, but there is a theorem describing the asymptotic behavior of any finite nonhomogeneous Markov chain defined by a sequence of stochastic matrices without any assumptions on this sequence.
Papers: A3, A4, A8 - A12, B27.
A4 The asymptotic behavior of a general finite nonhomogeneous Markov chain (the decomposition-separation theorem). Statistics, probability and game theory, 337--346, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, 1996. [pdf]

Usually this area is understood in a narrow sense, i.e. arms considered as independent, ("Gittings index" theory). In our book we studied the generalization of the classical "two-armed bandit problem" and "one-armed bandit problem" solved correspondingly by D. Feldman and R. Bellman, where "arms" are dependent. One of our main results is a theorem that states that in a general case (m arms, N hypotheses) all matrices can be classified as F- or B-matrices, where loss function and suboptimal strategies are similar to the two above mentioned cases. Another topic which we studied in our book is a Poisonnian version of these problems in continuous time.
A7 Book: Sequential control with incomplete information. The Bayesian approach to multi-armed bandit problems, (with E.L. Presman). Academic Press, Inc., San Diego, CA, 1990.
Papers A14, A18, B10 - B12.

Secretary problem with unknown number of objects and game setting: Papers A22, B3 - B9,

The Elimination Algorithm - a new algorithm to solve optimal stopping problem for finite and countable Markov chains: Papers: A1, A2, B14, B31, B32, R1, R2.

A2 The elimination algorithm for the problem of optimal stopping. Math. Methods Oper. Res. 49 (1999), no. 1, 111 -123. [pdf]
A1 The state reduction and related algorithms and their applications to the study of Markov chains, graph theory, and the optimal stopping problem. Adv. Math. 145 (1999), no. 2, 159 - 188.
B32: (joint with John Thornton). Recursive Algorithm for the Fundamental/Group Inverse Matrix of a Markov Chain from an Explicit Formula, SIAM J. on Matrix Analysis and Appl. 23, (2001), no. 1, 209 - 224. [pdf]

Game theory

Papers: A20, A21, B5, B8, B9

B35 = W4. The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game "Shoot later, shoot first !" (joint with E. Presman), 185-205, International J. of Game Theory, v. 34, 2, August, Springer-Verlag, 2006. [pdf]

Markov decision processes

The structure of optimal strategies and algorithms: Papers A5, A9, A13, A15, A16, B18, B19

Economics, Finance and Operations Research

Growth rate and internal rates of return: It may seem surprising, but in a classical investment model " there are such turnpikes that an investor is doomed to stay on them forever because the financial obligations connected with previous investments can be met only on such turnpikes."

B29 Growth rate, internal rates of return and turnpikes in an investment model. Economic Theory 5 (1995), 383--400. [pdf]

It may seem surprising, but in a classical replacement model ...
A6: Increasing the reliability of a machine reduces the period of its work. J. Appl. Probab. 33 (1996), no. 1, 217--223. [pdf] .
Optimal investment and resource allocation under uncertainty, multistage parallel projects, optimal selection of projects having block structure, models of economic dynamics with R&D: Papers A26, B13, B15 - B17, B20 - B26, B27, B28.

Other Papers: A23, A24, B1, B2, B30.