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,

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]

Optimal stopping of Markov chain and computations for Markov chains

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].

Continue, Quit, Restart Probability Model, joint with Constantine Steinberg, 2016, Annals of Operations Research, Springer, vol. 241(1), pp. 295-318, [link]On Optimal Stopping of Random Squences Modulated by Markov chain, (joint with E. Presman), Th. of Probability and its Appl., (2009). [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,

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:

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]

Insertion - a New Operation in Markov Chains, 18th
INFORMS Applied Probability Conference, July 5th - 8th,
2015, Koc University, Istanbul, Turkey.

Optimal
Stopping of Seasonal Observations and Projection of a Markov
Chain, Isaac
M. Sonin, in: *Inspired by Finance, The Musiela Festschrift.*
Editors: Yuri
Kabanov, Marek
Rutkowski, Thaleia Zariphopoulou, Springer 2014, pp. 535 -
543. [pdf]
A simplified version [pdf] .

**Independent
Events in a Simple Random Experiment and the Meaning of Independence.**
Isaac M. Sonin. math.PR
(math.CO). May 2012

___________________________________________________________Talks_______________________

Optimal Stopping of Markov Chains and Related Problems, Third Applied Prob. Conference, Rutgers Univ., June 2014,

Edinburg Univ., [pdf] , Pushkino (St. Petersburg), June - July 2014.

Elimination and Insertion Operations for Finite Markov Chains and their Application in Probability Models, Metabief, France, 8th Bachelier Colloquium, Jan. 2014 and 2nd Applied Prob. Conference, Rutgers Univ., Dec. 2013.

Optimal Stopping of Markov chain, Gittins Index and Related Optimization Problems, six similar talks: Berlin, Paris, Kiel, Brussel, BerSheva, Technion, Febr., March 2012.

Third International Workshop in Sequential Methodologies, Stanford University, June 2011;

16th Applied
Probability
Conference,
INFORMS, Stockholm, Sweden, July 2011 ;

Columbia University
and CUNY, Risk seminar, Sept. 2011. [pdf]

Optimal
Stopping of Seasonal Observations and Calculation of Related
Fundamental Matrices, The Fifth Bachelier Colloquium on
Mathematical Finance and

Stochastic Calculus, January 16-23, 2011, Metabief, France. Slide
Presentation [pdf]

Gittins Theory, Index and Theorem in a General
Form, Probability and Computational Finance Seminar, Dept. of Mathematical Sciences,

Carnegie Mellon University, Sept. 27, 2010,
www.math.cmu.edu/CCF/Seminars

6TH WORLD CONGRESS OF THE BACHELIER FINANCE SOCIETY, June 22 - 26, 2010, Toronto, Canada, Optimal Stopping of Markov Chain and Three Abstract Optimization Problems [pdf]

Stochmod 10 - 3rd meeting of the EURO Group on Stochastic
Modeling, June 7 - 9, 2010, Nafplio, Greece.

Optimal Stopping of Markov Chain, Gittins Index and Related
Optimization Problems

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

extended version [pdf]

Session 134: Dynamics and convexity, July 16, 2008

Nash Equilibrium Points in a Game of ''Seasonal'' Stopping [pdf]

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

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]

http://www.springerlink.com/content/v4859r0kk4781488/ [pdf]

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.

http://www.springerlink.com/content/k72177471g325426/ [pdf] [ef]