Mykhaylo Shkolnikov

Mykhaylo Shkolnikov

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 adviser 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 current activities I am involved in, my PhD students, publications, collaborators, and CV.


Current activities:

PhD students:

Benjamin Budway (expected: 05/2027), Jou-Hua Lai (expected: 05/2026), Yucheng Guo (expected: 05/2026), Scander Mustapha (defended: 05/2024), Graeme Baker (defended: 06/2023), 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.
  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.
  21. Pal, S., Shkolnikov, M. (2013). Intertwining diffusions and wave equations.
  22. Shkolnikov, M., Karatzas, I. (2013). Time-reversal of reflected Brownian motions in the orthant.
  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.
  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. (2023). Superposition and mimicking theorems for conditional McKean-Vlasov equations. J. Eur. Math. Soc. 25, 3229-3288.
  39. Kaushansky, V., Reisinger, C., Shkolnikov, M., Song, Z. Q. (2023). Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem. Ann. Appl. Probab. 33, 274-298.
  40. Nadtochiy, S., Shkolnikov, M., Zhang, X. (2024). Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem. Ann. Inst. Henri Poincaré Probab. Stat. 60, 658-691.
  41. Baker, G., Shkolnikov, M. (2022). A singular two-phase Stefan problem and particles interacting through their hitting times. To appear in Ann. Appl. Probab.
  42. Nadtochiy, S., Shkolnikov, M. (2023). Stefan problem with surface tension: global existence of physical solutions under radial symmetry.  Probab. Theory Related Fields 187, 385-422.
  43. Mustapha, S., Shkolnikov, M. (2023). Well-posedness of the supercooled Stefan problem with oscillatory initial conditions. Submitted.
  44. Guo, Y., Nadtochiy, S., Shkolnikov, M. (2023). Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry. Submitted.
  45. Shkolnikov, M., Soner, H. M., Tissot-Daguette, V. (2023). Deep level-set method for Stefan problems. J. Comput. Phys. 503.
  46. Budway, B., Shkolnikov, M. (2024). Multilevel Dyson Brownian motions via the superposition principle. Submitted.
  47. Shkolnikov, M., Yeung, L. C. (2024). From rank-based models with common noise to pathwise entropy solutions of SPDEs. Submitted.

Collaborators: