Mykhaylo Shkolnikov
Photo
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.


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


Activities:


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.
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. (2022). Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness.
Probab. Math. Phys. 3, 171-213.
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.
37. Baker, G., Shkolnikov, M. (2022). Zero kinetic undercooling limit in the supercooled Stefan problem. Ann. Inst. Henri Poincaré Probab. Stat. 58, 861-871.
38. Lacker, D., Shkolnikov, M., Zhang, J. (2020). Superposition and mimicking theorems for conditional McKean-Vlasov equations. To appear in J. Eur. Math. Soc.
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.


Talks:

2022: USC (online), Carnegie Mellon University (online), Joint Israeli Probability Seminar (online), Columbia University (online), Centro di Ricerca Matematica Ennio De Giorgi (Pisa): Advances in Mathematical Finance and Optimal Transport (in honor of Walter Schachermayer's birthday)

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

2018:
The Hong Kong Polytechnic University, University of Toronto, Columbia University, Duke University, Brown University: Seminar on Stochastic Processes, Casa Mathemática Oaxaca: Workshop on Stochastic Analysis and its Applications, Institut Mittag-Leffler: Gaussian fields in random matrix theory
, Illinois Institute of Technology, University of Delaware: AMS sectional meeting, Princeton University: Program in Applied & Compuational Mathematics colloquium, Illinois Institute of Technology: Eastern Conference on Mathematical Finance, Stanford University: Advances in Asymptotic Probability (in honor of Amir Dembo's birthday), 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:
Montréal: Canadian Mathematical Society conference, University of Chicago, NYU: Northeast Probability Seminar, 9th Oxford-Princeton Workshop on Financial Mathematics and Stochastic Analysis, Morgan Stanley, TU Munich, University of Tokyo, Columbia University: Stochastic Portfolio Theory conference, Rutgers University: Mathematical Finance and Partial Differential Equations conference, CUNY, DC: AMS sectional meeting, 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,
University of Colorado: Conference on Stochastic Processes and Their Applications (SPA), Stanford University, University of Michigan, Columbia University, MIT, University of Washington, UC Berkeley, San Diego: AMS meetings

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

2011:
Sydney: QMF 2011 conference, 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,
Saint-Flour: École d'Été de Probabilités, 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