ORFE Department

202 Sherrerd Hall

Princeton University

Princeton, NJ 08544

mykhaylo at princeton.edu

Currently,
I am an Associate Professor in the ORFE
Department, an Affiliated Faculty Member of the Bendheim
Center for Finance and an Associated Faculty Member with the
Program
in Applied & Computational Math at Princeton
University. Before
joining ORFE
I was 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 PDEs.

My research is partially supported by the NSF grant DMS-2108680.

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 PDEs.

My research is partially supported by the NSF grant DMS-2108680.

Here you can find activities I am involved in, my PhD students, publications, talks, collaborators, and CV.

Activities:

- Associate Editor for Annals of Applied Probability
- Associate
Editor for Mathematical
Finance

- Associate Editor for Applied Mathematical Finance
- Princeton Financial Mathematics Seminar
- Princeton
Probability
Seminar

- Oxford/Princeton Workshop on Financial Math & Stochastic Analysis
- Columbia-Princeton Probability Day
- Princeton-Rutgers
Math Finance Day

PhD students:

Scander Mustapha (expected 05/2024), Graeme Baker (expected 05/2023), Jiacheng Zhang (defended: 05/2021), Levon Avanesyan (defended: 12/2020), Pierre Yves Gaudreau Lamarre (defended: 05/2020), 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. (2018). Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab. 46, 2287-2344.

28. Kolli, P., Shkolnikov, M. (2018). SPDE limit of the global fluctuations in rank-based models. Ann. Probab. 46, 1042-1069.

29. Ramanan, K., Shkolnikov, M. (2018). Intertwinings of beta-Dyson Brownian motions of different dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 54, 1152-1163.

30. Nadtochiy, S., Shkolnikov, M. (2019). Particle systems with singular interaction through hitting times: application in systemic risk modeling. Ann. Appl. Probab. 29, 89-129.

31. Gaudreau Lamarre, P. Y., Shkolnikov, M. (2019). Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Ann. Inst. Henri Poincaré Probab. Stat. 55, 1402-1438.

32. Almada Monter, S. A., Shkolnikov, M., Zhang, J. (2019). Dynamics of observables in rank-based models and performance of functionally generated portfolios. Ann. Appl. Probab. 29, 2849-2883.

33. Avanesyan, L., Shkolnikov, M., Sircar, R. (2020). Construction of forward performance processes in stochastic factor models and an extension of Widder's theorem.*Finance Stoch.*
24,
981-1011.

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. (2018). Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab. 46, 2287-2344.

28. Kolli, P., Shkolnikov, M. (2018). SPDE limit of the global fluctuations in rank-based models. Ann. Probab. 46, 1042-1069.

29. Ramanan, K., Shkolnikov, M. (2018). Intertwinings of beta-Dyson Brownian motions of different dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 54, 1152-1163.

30. Nadtochiy, S., Shkolnikov, M. (2019). Particle systems with singular interaction through hitting times: application in systemic risk modeling. Ann. Appl. Probab. 29, 89-129.

31. Gaudreau Lamarre, P. Y., Shkolnikov, M. (2019). Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Ann. Inst. Henri Poincaré Probab. Stat. 55, 1402-1438.

32. Almada Monter, S. A., Shkolnikov, M., Zhang, J. (2019). Dynamics of observables in rank-based models and performance of functionally generated portfolios. Ann. Appl. Probab. 29, 2849-2883.

33. Avanesyan, L., Shkolnikov, M., Sircar, R. (2020). Construction of forward performance processes in stochastic factor models and an extension of Widder's theorem.

34.
Nadtochiy, S., Shkolnikov, M. (2020). Mean
field systems on networks, with singular interaction through hitting
times. Ann.
Probab. **48**,
1520-1556.

35. Delarue, F., Nadtochiy, S., Shkolnikov, M. (2019). Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness. To appear in Probab. Math. Phys.

36. Lacker, D., Shkolnikov, M., Zhang, J. (2020). Inverting the Markovian projection, with an application to local stochastic volatility models. Ann. Probab.**48**,
2189-2211.

35. Delarue, F., Nadtochiy, S., Shkolnikov, M. (2019). Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness. To appear in Probab. Math. Phys.

36. Lacker, D., Shkolnikov, M., Zhang, J. (2020). Inverting the Markovian projection, with an application to local stochastic volatility models. Ann. Probab.

37.
Baker, G., Shkolnikov, M. (2020). Zero
kinetic undercooling limit in the supercooled Stefan problem. To
appear in
Ann.
Inst. Henri Poincaré Probab. Stat.

38.
Lacker, D., Shkolnikov, M., Zhang, J. (2020). Superposition
and mimicking theorems for conditional McKean-Vlasov equations.
Submitted.

39.
Kaushansky, V., Reisinger, C., Shkolnikov, M., Song, Z. Q. (2020). Convergence
of a time-stepping scheme to the free boundary in the supercooled
Stefan problem. To
appear in Ann.
Appl. Probab.

40.
Nadtochiy, S., Shkolnikov, M., Zhang, X. (2021). Scaling
limits of external multi-particle DLA on the plane and the supercooled
Stefan problem. *Submitted.
*

41.
Baker, G., Shkolnikov, M. (2022). A
singular two-phase Stefan problem and particles interacting through
their hitting times. *Submitted.
*

42.
Nadtochiy, S., Shkolnikov, M. (2022). Stefan
problem with surface tension: global existence of physical solutions
under radial symmetry. *Submitted.
*

2022: USC (online), Carnegie Mellon University (online)

2021:
Harvard University
(online), University of Leeds (online), Berlin Probability Colloquium
(online), Stanford University (online), SIAM Meeting on Financial
Mathematics & Engineering (online), Centre
International de Recontres Mathematiques (Luminy): Advances in
Stochastic Analysis for Handling Risks in Finance and Insurance
(online), USC (online)

2020: Oxford University (online), Indiana University (online), Princeton University (online), Columbia University (online), Rutgers University (online)

2019:
TU
Vienna, London School of Economics, Imperial College London, Rutgers
University, 4th Berlin-Princeton-Singapore Workshop on Quantitative
Finance: National University of Singapore, MIT, Centre International de
Recontres Mathematiques (Luminy): Integrability and Randomness in
Mathematical Physics and Geometry, Southeastern Probability Conference:
Duke University, SIAM Conference on Financial Mathematics &
Engineering: Toronto, Berlin Mathematical Finance Seminar, Annals
Probability Meeting: Nice, NYU, University of Münster

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: Gaussian fields in random matrix theory, Illinois Institute of Technology, AMS sectional meeting: University of Delaware, Program in Applied & Compuational Mathematics colloquium: Princeton University, Eastern Conference on Mathematical Finance: Illinois Institute of Technology, Advances in Asymptotic Probability (in honor of Amir Dembo's 60th birthday): Stanford University, TU Munich

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, Graeme Baker, Francois Delarue, Amir Dembo, Simone Farinelli, Pierre Yves Gaudreau Lamarre, Stefan Gerhold, Vadim Gorin, Tomoyuki Ichiba, Ioannis Karatzas, Vadim Kaushansky, Max Kleinert, Praveen Kolli, Daniel Lacker, Sergey Nadtochiy, Soumik Pal, Piet Porkert, Miklos Z. Racz, Kavita Ramanan, Christoph Reisinger, Albert N. Shiryaev, Ronnie Sircar, Zhuo Qun Song, S.R.Srinivasa Varadhan, Thaleia Zariphopoulou, Ofer Zeitouni, Jiacheng Zhang, Xiling Zhang