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 ] .

One of  my latest publicationis : A zero-one law for Markov chains, (joint with M. Grabchak) : Stochastics, 2021,  DOI: 10.1080/17442508.2021.1980569, 19 pp. 2021,  [link]

The other: Some Nontrivial Properties of a Formula for Compound Interest, (joint with Mark Whitmeyer), Finance Research Letters, vol. 33,

March 2020,  link [pdf].

 My most recent topics of Research are two original models:  Locks, Bombs and Testing;  Banks as Tanks:

Locks, Bombs and Testing: the Case of Independent Locks, p. 247-264, in Modern Trends in Controlled Stochastic Processes, V.III, : - Theory and Applications, eds. Alexey B. Piunovskiy and Yi Zhang, Springer, 2021.  [link]

Bayesian Game of Locks, Bombs and Testing, joint with Konstantin Sonin, 2017 - 2020,  [link]

Parallel Computing for Markov chains with Islands and Ports, joint with A. Basnet, Annals of Operations Research, 2017, [pdf] , [link]

Elimination and Insertion Operations for Finite Markov Chains, joint with Constantine Steinberg, 2015, in Modern Trends in Controlled Stochastic Processes – Vol. II, Luniver Press, Frome,  2015, Ed. A. Piunovskiy,  pp.130 – 139, [pdf].

          The Decomposition-Separation (DS) theorem.

       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.          Two survey Papers:

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

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] 

One more paper related to DS Theorem:  The expected number of intersections of a four valued bounded martingale with any level may be infinite, 
 (joint with Alexander Gordon), in "Optimality and Risk - Modern Trends in Mathematical Finance, eds. F. Delbaen, M. Rasonyi, and C. Stricker, pp. 87-98, Springer, 2009. [pdf] 

Statistics & Probability Letters, Volume 78, Issue 12, 1 September 2008,  [2004], Pages 1526-1533
see also a modified version of this paper [pdf]

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]

Multi-armed bandit Problems  (sequential statistical analysis):

     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. This book is out of print and difficult to find. Some sections can be found below.

Ch. 1 [pdf]   [pdf]  [pdf]  [pdf]  [pdf] Ch. 2 [pdf]  [pdf]   [pdf]  [pdf] Ch. 3  [pdf]  Ch. 4 [pdf]   [pdf]  [pdf] Ch. 5 [pdf]   [pdf]  Ch. 6 [pdf]   [pdf]  

Ch. 7 [pdf]  Ref. [pdf]    Papers A14, A18, B10 - B12.

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 

A recent paper: Some Nontrivial Properties of a Formula for Compound Interest, joint with Mark Whitmeyer, Finance Research Letters, vol. 33,

March 2020,  link [pdf].

  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.    A couple of drafts: [pdf]  [pdf]

Published or Working Papers or Presentation on Conferences:

Conditional Expectations and Pouring Water from Full Cups to Empty, joint with Stanislav Molchanov, 2020, accepted in Arnold Mathematical Journal.

During my sabbatical semester Spring 2017 I gave talks at 11 Bachelier Colloquium, Metabief, Univ. of Birmingham, Univ. of Liverpool, Univ. of Warwick,

Uinv. of Oxford, INRIA Sophia Antipolis, France and a few other.

During 2016 I gave talks at: During my sabbatical semester Spring 2017 I gave talks at 11 Bachelier Colloquium, Metabief, Univ. of Birmingham, Univ. of Liverpool,

Univ. of Warwick, Uinv. of Oxford, INRIA Sophia Antipolis, France and a few other.

During 2016 I gave talks at:  1. International conference on stochastic methods, Abrau_Durso, Russia, 2016
 (joint with S. Molchanov). Conditional Expectations and Pouring Water from Full Cups to Empty.
 2 XVII th International Summer Conference on Probability and Statistics (ISCPS), Plovdiv, Bulgaria  2016
 Consensus Algorithms and Decomposition-Separation theorem,
 INFORMS Annual Meeting 2016 Nashville   Session WB76 - Applied Probability II 5 - A Parallel Computation Of Characteristics Of Markov Chains With “Islands” And "Ports”(joint with A. Basnet), the abstract published in Proceedings of the Conference.

Modern trends in controlled stochastic processes: theory and applications, Workshop,  The University of Liverpool, UK, Dept. of Math. Sciences, 29 June - 04 July 2015. Continue, Quit, Restart Probability Model,  (joint with Constantine Steinberg), Annals of Operations Research,  Ann. Oper. Res. 241 (2012, 2016), no. 1-2, 295–318. http://link.springer.com/article/10.1007/s10479-012-1089-2; 

Optimal Stopping of Markov Chains and Three Abstract Optimization Problems, Sonin, I.M.:  Stochastics, 83(4-6), 405–414 (2011).  [pdf]On Optimal Stopping of Random Sequences Modulated by Markov Chain,  (joint with Ernest Presman),  Theory of Probability and its Applications,  V. 54, Issue 3, pp. 375-551, 2010.  [pdf]