ORFE Department

202 Sherrerd Hall

Princeton University

Princeton, NJ 08544

mykhaylo at princeton.edu

Currently, I am an
Assistant Professor in the ORFE Department and an Affiliated Faculty Member of the Bendheim Center for Finance at
Princeton
University, after being an Assistant Professor in the Math
Department at Princeton. Before coming to Princeton I was a
Postdoctoral Fellow at UC
Berkeley and MSRI
mentored by David Aldous. My PhD is from the Math Department at Stanford University
where my advisor was Amir Dembo.

At the moment, I am studying interacting particle systems arising in mathematical finance, mathematical physics, and neuroscience using tools from stochastic analysis and PDE/SPDE . More broadly, my interests include a variety of topics in probability theory and PDEs: random operators, integrable probability, models of random growth, concentration of measure, large deviations, and probabilistic approaches to hyperbolic and parabolic PDEs.

At the moment, I am studying interacting particle systems arising in mathematical finance, mathematical physics, and neuroscience using tools from stochastic analysis and PDE/SPDE . More broadly, my interests include a variety of topics in probability theory and PDEs: random operators, integrable probability, models of random growth, concentration of measure, large deviations, and probabilistic approaches to hyperbolic and parabolic PDEs.

My research is partially supported by NSF grant DMS-1506290.

Here you can find activities I am involved in, my PhD students,

publications, recent and upcoming talks, collaborators, and CV.

Activities:

- Oxford/Princeton Workshop
on Financial Math & Stochastic Analysis

- Princeton-Rutgers Math Finance Day

- Columbia-Princeton
Probability Day

- Princeton Probability Seminar
- Princeton Financial Mathematics Seminar

PhD students:

Jiacheng Zhang, Pierre Yves Gaudreau Lamarre, Levon Avanesyan, Praveen Kolli (defended 04/2018)

Publications:

1. Shkolnikov,
M.
(2007). Affine
matrix-valued diffusions. Diploma
thesis. University of Munich.

2. Shkolnikov, M. (2009). Competing particle systems evolving by i.i.d. increments. Electron. J. Probab. 14, 728-751.

3. Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21, 1911-1932.

4. Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stoch. Proc. Appl. 122, 1730-1747.

5. Pal, S., Shkolnikov, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24, 1482-1508.

6. Farinelli, S., Shkolnikov, M. (2012). Two models for stochastic loss given default. J. Credit Risk 8, paper 4.

7. Shkolnikov, M. (2013). Large volatility-stabilized markets. Stoch. Proc. Appl. 123, 212-228.

8. Ichiba, T., Pal, S., Shkolnikov, M. (2013). Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156, 415-448.

9. Karatzas, I., Shiryaev, A. N., Shkolnikov, M. (2011). On the one-sided Tanaka equation with drift. Electron. Commun. Probab. 16, 664-677.

10. Ichiba, T., Karatzas, I., Shkolnikov, M. (2013). Strong solutions to stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156, 229-248.

11. Shkolnikov, M. (2011). Competing particle systems and their applications. PhD thesis, Stanford University.

12. Shkolnikov, M. (2013). Some universal estimates for reversible Markov chains. Electron. J. Probab. 18, article 11.

13. Shkolnikov, M. (2012). On a non-linear transformation between Brownian martingales. Submitted.

14. Gorin, V., Shkolnikov, M. (2015). Limits of multilevel TASEP and related processes. Ann. Inst. Henri Poincaré Probab. Stat. 51, 18-27.

15. Gerhold, S., Kleinert, M., Porkert, P., Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics 87, 723-746.

16. Karatzas, I., Pal, S., Shkolnikov, M. (2016). Systems of Brownian particles with asymmetric collisions. Ann. Inst. Henri Poincaré Probab. Stat. 52, 323-354.

17. Aldous, D., Shkolnikov, M. (2013). Fluctuations of martingales and winning probabilities of game contestants. Electron. J. Probab. 18, article 47.

18. Dembo, A., Shkolnikov, M., Varadhan, S.R.S., Zeitouni, O. (2012). Large deviations for diffusions interacting through their ranks. Comm. Pure Appl. Math. 69, 1259-1313.

19. Racz, M.Z., Shkolnikov, M. (2015). Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Probab. 25, 1155-1188.

20. Ichiba, T., Shkolnikov, M. (2013). Large deviations for interacting Bessel-like processes and applications to systemic risk. Submitted.

21. Pal, S., Shkolnikov, M. (2013). Intertwining diffusions and wave equations. Submitted.

22. Shkolnikov, M., Karatzas, I. (2013). Time-reversal of reflected Brownian motions in the orthant. Submitted.

23. Gorin, V., Shkolnikov, M. (2015). Multilevel Dyson Brownian motions via Jack polynomials. Probab. Theory Related Fields 163, 413-463.

24. Gorin, V., Shkolnikov, M. (2017). Interacting particle systems at the edge of multilevel Dyson Brownian motions. Adv. Math. 304, 90-130.

25. Shkolnikov, M., Sircar, R., Zariphopoulou, T. (2016). Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations. SIAM J. Financial Math. 7, 588-618.

26. Shkolnikov, M. (2015). A construction of infinite Brownian particle systems. Submitted.

27. Gorin, V., Shkolnikov, M. (2016). Stochastic Airy semigroup through tridiagonal matrices. To appear in Ann. Probab.

28. Kolli, P., Shkolnikov, M. (2016). SPDE limit of the global fluctuations in rank-based models. To appear in Ann. Probab.

29. Ramanan, K., Shkolnikov, M. (2016). Intertwinings of beta-Dyson Brownian motions of different dimensions. To appear in Ann. Inst. Henri Poincaré Probab. Stat.

30. Nadtochiy, S., Shkolnikov, M. (2017). Particle systems with singular interaction through hitting times: application in systemic risk modeling. To appear in Ann. Appl. Probab.

31. Gaudreau Lamarre, P. Y., Shkolnikov, M. (2017). Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Submitted.

32. Almada Monter, S. A., Shkolnikov, M., Zhang, J. (2018). Dynamics of observables in rank-based models and performance of functionally generated portfolios. Submitted.

33. Avanesyan, L., Shkolnikov, M., Sircar, R. (2018). Construction of forward performance processes in stochastic factor models and an extension of Widder's theorem. Submitted.

Recent and upcoming talks:

2018: The Hong Kong Polytechnic University, University of Toronto, Columbia University, Duke University, Seminar on Stochastic Processes: Brown University, Casa Mathemática Oaxaca: Workshop on Stochastic Analysis and its Applications, Institut Mittag-Leffler

2017: London Mathematical Finance Seminar, Columbia University, Oberwolfach: Mathematics of Quantitative Finance, Western Conference in Mathematical Finance, University of Michigan, Oxford University: Oxford-Princeton Workshop on Financial Mathematics & Stochastic Analysis, Edinburgh: Workshop on BSDEs, SPDEs and their Applications, Montreal: Mathematical Congress of the Americas (Probability Theory session), SIAM-LMS Conference on Mathematical Modelling in Finance, Stony Brook, University of Vienna, UT Austin

2016: Kavli Institute for Theoretical Physics: Non-equilibrium dynamics of stochastic and quantum integrable systems, UC Santa Barbara, Rutgers University, MIT, University of Michigan: Byrne Workshop in Stochastic Analysis in Finance and Insurance, International Centre for Mathematical Sciences, Edinburgh: At the Frontiers of Quantitative Finance

2015: Canadian Mathematical Society conference: Montréal, University of Chicago, Northeast Probability Seminar: NYU, 9th Oxford-Princeton Workshop on Financial Mathematics and Stochastic Analysis, Morgan Stanley, TU Munich, University of Tokyo, Stochastic Portfolio Theory conference: Columbia University, Mathematical Finance and Partial Differential Equations conference: Rutgers University, CUNY, AMS sectional meeting: DC, Rutgers University, Princeton University

2014: Brown University, University of Tokyo, Johns Hopkins, Harvard University, University of Pennsylvania/Temple University, Princeton University, University of Connecticut, Columbia University, University of Southern California, University of British Columbia, University of Maryland, Georgia Tech

2013: University of Bonn, University of Minnesota, Carnegie Mellon University, Cornell University, Conference on Stochastic Processes and Their Applications (SPA): University of Colorado, Stanford University, University of Michigan, Columbia University, MIT, University of Washington, UC Berkeley, AMS meetings: San Diego

2012: UC Santa Barbara, Conference in honor of Ioannis Karatzas: Columbia University, Mathematical Sciences Research Institute, UC Berkeley, MAN Institute at Oxford University, ETH Zurich, University of Technology Vienna

2011: QMF 2011 conference: Sydney, UCLA, UC Irvine, UC Berkeley, New York University, Princeton University, University of North Carolina/Duke University, UC San Diego, Stanford University, University of Washington

2010: Columbia University, École d'Été de Probabilités: Saint-Flour, EPFL Lausanne, UC Berkeley, Stanford University

Collaborators:

Levon Avanesyan, David Aldous, Sergio A. Almada Monter, Amir Dembo, Simone Farinelli, Pierre Yves Gaudreau Lamarre, Stefan Gerhold, Vadim Gorin, Tomoyuki Ichiba, Ioannis Karatzas, Max Kleinert, Praveen Kolli, Sergey Nadtochiy, Soumik Pal, Piet Porkert, Miklos Z. Racz, Kavita Ramanan, Albert N. Shiryaev, Ronnie Sircar, S.R.Srinivasa Varadhan, Thaleia Zariphopoulou, Ofer Zeitouni, Jiacheng Zhang

2. Shkolnikov, M. (2009). Competing particle systems evolving by i.i.d. increments. Electron. J. Probab. 14, 728-751.

3. Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21, 1911-1932.

4. Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stoch. Proc. Appl. 122, 1730-1747.

5. Pal, S., Shkolnikov, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24, 1482-1508.

6. Farinelli, S., Shkolnikov, M. (2012). Two models for stochastic loss given default. J. Credit Risk 8, paper 4.

7. Shkolnikov, M. (2013). Large volatility-stabilized markets. Stoch. Proc. Appl. 123, 212-228.

8. Ichiba, T., Pal, S., Shkolnikov, M. (2013). Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156, 415-448.

9. Karatzas, I., Shiryaev, A. N., Shkolnikov, M. (2011). On the one-sided Tanaka equation with drift. Electron. Commun. Probab. 16, 664-677.

10. Ichiba, T., Karatzas, I., Shkolnikov, M. (2013). Strong solutions to stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156, 229-248.

11. Shkolnikov, M. (2011). Competing particle systems and their applications. PhD thesis, Stanford University.

12. Shkolnikov, M. (2013). Some universal estimates for reversible Markov chains. Electron. J. Probab. 18, article 11.

13. Shkolnikov, M. (2012). On a non-linear transformation between Brownian martingales. Submitted.

14. Gorin, V., Shkolnikov, M. (2015). Limits of multilevel TASEP and related processes. Ann. Inst. Henri Poincaré Probab. Stat. 51, 18-27.

15. Gerhold, S., Kleinert, M., Porkert, P., Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics 87, 723-746.

16. Karatzas, I., Pal, S., Shkolnikov, M. (2016). Systems of Brownian particles with asymmetric collisions. Ann. Inst. Henri Poincaré Probab. Stat. 52, 323-354.

17. Aldous, D., Shkolnikov, M. (2013). Fluctuations of martingales and winning probabilities of game contestants. Electron. J. Probab. 18, article 47.

18. Dembo, A., Shkolnikov, M., Varadhan, S.R.S., Zeitouni, O. (2012). Large deviations for diffusions interacting through their ranks. Comm. Pure Appl. Math. 69, 1259-1313.

19. Racz, M.Z., Shkolnikov, M. (2015). Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Probab. 25, 1155-1188.

20. Ichiba, T., Shkolnikov, M. (2013). Large deviations for interacting Bessel-like processes and applications to systemic risk. Submitted.

21. Pal, S., Shkolnikov, M. (2013). Intertwining diffusions and wave equations. Submitted.

22. Shkolnikov, M., Karatzas, I. (2013). Time-reversal of reflected Brownian motions in the orthant. Submitted.

23. Gorin, V., Shkolnikov, M. (2015). Multilevel Dyson Brownian motions via Jack polynomials. Probab. Theory Related Fields 163, 413-463.

24. Gorin, V., Shkolnikov, M. (2017). Interacting particle systems at the edge of multilevel Dyson Brownian motions. Adv. Math. 304, 90-130.

25. Shkolnikov, M., Sircar, R., Zariphopoulou, T. (2016). Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations. SIAM J. Financial Math. 7, 588-618.

26. Shkolnikov, M. (2015). A construction of infinite Brownian particle systems. Submitted.

27. Gorin, V., Shkolnikov, M. (2016). Stochastic Airy semigroup through tridiagonal matrices. To appear in Ann. Probab.

28. Kolli, P., Shkolnikov, M. (2016). SPDE limit of the global fluctuations in rank-based models. To appear in Ann. Probab.

29. Ramanan, K., Shkolnikov, M. (2016). Intertwinings of beta-Dyson Brownian motions of different dimensions. To appear in Ann. Inst. Henri Poincaré Probab. Stat.

30. Nadtochiy, S., Shkolnikov, M. (2017). Particle systems with singular interaction through hitting times: application in systemic risk modeling. To appear in Ann. Appl. Probab.

31. Gaudreau Lamarre, P. Y., Shkolnikov, M. (2017). Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Submitted.

32. Almada Monter, S. A., Shkolnikov, M., Zhang, J. (2018). Dynamics of observables in rank-based models and performance of functionally generated portfolios. Submitted.

33. Avanesyan, L., Shkolnikov, M., Sircar, R. (2018). Construction of forward performance processes in stochastic factor models and an extension of Widder's theorem. Submitted.

Recent and upcoming talks:

2018: The Hong Kong Polytechnic University, University of Toronto, Columbia University, Duke University, Seminar on Stochastic Processes: Brown University, Casa Mathemática Oaxaca: Workshop on Stochastic Analysis and its Applications, Institut Mittag-Leffler

2017: London Mathematical Finance Seminar, Columbia University, Oberwolfach: Mathematics of Quantitative Finance, Western Conference in Mathematical Finance, University of Michigan, Oxford University: Oxford-Princeton Workshop on Financial Mathematics & Stochastic Analysis, Edinburgh: Workshop on BSDEs, SPDEs and their Applications, Montreal: Mathematical Congress of the Americas (Probability Theory session), SIAM-LMS Conference on Mathematical Modelling in Finance, Stony Brook, University of Vienna, UT Austin

2016: Kavli Institute for Theoretical Physics: Non-equilibrium dynamics of stochastic and quantum integrable systems, UC Santa Barbara, Rutgers University, MIT, University of Michigan: Byrne Workshop in Stochastic Analysis in Finance and Insurance, International Centre for Mathematical Sciences, Edinburgh: At the Frontiers of Quantitative Finance

2015: Canadian Mathematical Society conference: Montréal, University of Chicago, Northeast Probability Seminar: NYU, 9th Oxford-Princeton Workshop on Financial Mathematics and Stochastic Analysis, Morgan Stanley, TU Munich, University of Tokyo, Stochastic Portfolio Theory conference: Columbia University, Mathematical Finance and Partial Differential Equations conference: Rutgers University, CUNY, AMS sectional meeting: DC, Rutgers University, Princeton University

2014: Brown University, University of Tokyo, Johns Hopkins, Harvard University, University of Pennsylvania/Temple University, Princeton University, University of Connecticut, Columbia University, University of Southern California, University of British Columbia, University of Maryland, Georgia Tech

2013: University of Bonn, University of Minnesota, Carnegie Mellon University, Cornell University, Conference on Stochastic Processes and Their Applications (SPA): University of Colorado, Stanford University, University of Michigan, Columbia University, MIT, University of Washington, UC Berkeley, AMS meetings: San Diego

2012: UC Santa Barbara, Conference in honor of Ioannis Karatzas: Columbia University, Mathematical Sciences Research Institute, UC Berkeley, MAN Institute at Oxford University, ETH Zurich, University of Technology Vienna

2011: QMF 2011 conference: Sydney, UCLA, UC Irvine, UC Berkeley, New York University, Princeton University, University of North Carolina/Duke University, UC San Diego, Stanford University, University of Washington

2010: Columbia University, École d'Été de Probabilités: Saint-Flour, EPFL Lausanne, UC Berkeley, Stanford University

Collaborators:

Levon Avanesyan, David Aldous, Sergio A. Almada Monter, Amir Dembo, Simone Farinelli, Pierre Yves Gaudreau Lamarre, Stefan Gerhold, Vadim Gorin, Tomoyuki Ichiba, Ioannis Karatzas, Max Kleinert, Praveen Kolli, Sergey Nadtochiy, Soumik Pal, Piet Porkert, Miklos Z. Racz, Kavita Ramanan, Albert N. Shiryaev, Ronnie Sircar, S.R.Srinivasa Varadhan, Thaleia Zariphopoulou, Ofer Zeitouni, Jiacheng Zhang