Mykhaylo Shkolnikov

Department of Mathematical Sciences

Center for Nonlinear Analysis

6128 Wean Hall, Carnegie Mellon University
Pittsburgh, PA 15213
mshkolni at andrew.cmu.edu

Currently, I am a Full Professor in the Department of Mathematical Sciences and a Faculty Member of the Center for Nonlinear Analysis at Carnegie Mellon University. Before joining Carnegie Mellon in 2024, I was an Associate Professor (with tenure) in the ORFE Department at Princeton University.

At the moment, I am studying interacting particle systems arising in mathematical finance and mathematical physics using tools from stochastic analysis and PDE. More broadly, my interests include a variety of topics in probability theory and related fields: moving interfaces, probabilistic approaches to PDEs, neural networks, stochastic PDEs, large deviations, random operators, and integrable probability.

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

Activities:

Associate Editor for Annals of Applied Probability
Associate Editor for Mathematical Finance
Associate Editor for Applied Mathematical Finance
Eastern Conference on Mathematical Finance, October 24-25, 2025
AMS-UMI International Joint Meeting: Special Session on Stochastic Analysis, July 25-26, 2024
IMSI workshop: Laplacian Growth Models, June 26-30, 2023
Carnegie Mellon Probability/Math Finance Seminar
Carnegie Mellon Math Colloquium

Postdocs and PhD students:

Anton Ullrich (postdoc, 2025-present), Lane Chun Yeung (postdoc, 2023-2025),
Benjamin Budway (PhD, expected: 05/2027), Jou-Hua Lai (PhD, expected: 05/2027), Yucheng Guo (PhD, expected: 05/2026), Scander Mustapha (PhD, defended: 2024), Graeme Baker (PhD, defended: 2023), Jiacheng Zhang (PhD, defended: 2021),
Levon Avanesyan (PhD, defended: 2020), Pierre Yves Gaudreau Lamarre (PhD, defended: 2020),
Praveen Kolli (PhD, defended: 2018)

Publications and submitted papers:

Shkolnikov, M. (2007). Affine matrix-valued diffusions. Diploma thesis. University of Munich.
Shkolnikov, M. (2009). Competing particle systems evolving by i.i.d. increments. Electron. J. Probab. 14, 728-751.
Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21, 1911-1932.
Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stoch. Proc. Appl. 122, 1730-1747.
Pal, S., Shkolnikov, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24, 1482-1508.
Farinelli, S., Shkolnikov, M. (2012). Two models for stochastic loss given default. J. Credit Risk 8, paper 4.
Shkolnikov, M. (2013). Large volatility-stabilized markets. Stoch. Proc. Appl. 123, 212-228.
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.
Karatzas, I., Shiryaev, A. N., Shkolnikov, M. (2011). On the one-sided Tanaka equation with drift. Electron. Commun. Probab. 16, 664-677.
Ichiba, T., Karatzas, I., Shkolnikov, M. (2013). Strong solutions to stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156, 229-248.
Shkolnikov, M. (2011). Competing particle systems and their applications. PhD thesis, Stanford University.
Shkolnikov, M. (2013). Some universal estimates for reversible Markov chains. Electron. J. Probab. 18, article 11.
Gorin, V., Shkolnikov, M. (2015). Limits of multilevel TASEP and related processes. Ann. Inst. Henri Poincaré Probab. Stat. 51, 18-27.
Gerhold, S., Kleinert, M., Porkert, P., Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics 87, 723-746.
Karatzas, I., Pal, S., Shkolnikov, M. (2016). Systems of Brownian particles with asymmetric collisions. Ann. Inst. Henri Poincaré Probab. Stat. 52, 323-354.
Aldous, D., Shkolnikov, M. (2013). Fluctuations of martingales and winning probabilities of game contestants. Electron. J. Probab. 18, article 47.
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.
Racz, M.Z., Shkolnikov, M. (2015). Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Probab. 25, 1155-1188.
Gorin, V., Shkolnikov, M. (2015). Multilevel Dyson Brownian motions via Jack polynomials. Probab. Theory Related Fields 163, 413-463.
Gorin, V., Shkolnikov, M. (2017). Interacting particle systems at the edge of multilevel Dyson Brownian motions. Adv. Math. 304, 90-130.
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.
Gorin, V., Shkolnikov, M. (2018). Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab. 46, 2287-2344.
Kolli, P., Shkolnikov, M. (2018). SPDE limit of the global fluctuations in rank-based models. Ann. Probab. 46, 1042-1069.
Ramanan, K., Shkolnikov, M. (2018). Intertwinings of beta-Dyson Brownian motions of different dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 54, 1152-1163.
Nadtochiy, S., Shkolnikov, M. (2019). Particle systems with singular interaction through hitting times: application in systemic risk modeling. Ann. Appl. Probab. 29, 89-129.
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.
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.
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.
Nadtochiy, S., Shkolnikov, M. (2020). Mean field systems on networks, with singular interaction through hitting times. Ann. Probab. 48, 1520-1556.
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.
Lacker, D., Shkolnikov, M., Zhang, J. (2020). Inverting the Markovian projection, with an application to local stochastic volatility models. Ann. Probab. 48, 2189-2211.
Baker, G., Shkolnikov, M. (2022). Zero kinetic undercooling limit in the supercooled Stefan problem. Ann. Inst. Henri Poincaré Probab. Stat. 58, 861-871.
Lacker, D., Shkolnikov, M., Zhang, J. (2023). Superposition and mimicking theorems for conditional McKean-Vlasov equations. J. Eur. Math. Soc. 25, 3229-3288.
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.
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.
Baker, G., Shkolnikov, M. (2024). A singular two-phase Stefan problem and particles interacting through their hitting times. Ann. Appl. Probab. 34, 4493-4512.
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.
Mustapha, S., Shkolnikov, M. (2024). Well-posedness of the supercooled Stefan problem with oscillatory initial conditions. Electron. J. Probab. 29, 1-21.
Guo, Y., Nadtochiy, S., Shkolnikov, M. (2024). Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry. Arch. Ration. Mech. Anal. 248, article 90.
Shkolnikov, M., Soner, H. M., Tissot-Daguette, V. (2024). Deep level-set method for Stefan problems. J. Comput. Phys. 503.
Budway, B., Shkolnikov, M. (2024). Multilevel Dyson Brownian motions via the superposition principle. To appear in Ann. Inst. Henri Poincaré Probab. Stat.
Shkolnikov, M., Yeung, L. C. (2026). From rank-based models with common noise to pathwise entropy solutions of SPDEs. Ann. Appl. Probab. 36, 319-354.
Guo, Y., Nadtochiy, S., Shkolnikov, M. (2024). Cascade equation for the discontinuities in the Stefan problem with surface tension. Submitted.
Budway, B., Pal, S., Shkolnikov, M. (2025). Intertwining diffusions and wave equations. Submitted.
Lai, J.-H., Shkolnikov, M., Soner, H. M. (2025). Relative arbitrage problem under eigenvalue lower bounds. Submitted.
Shkolnikov, M., Yeung, L. C. (2026). Universal Central Limit Theorem for non-exchangeable interacting diffusions. Submitted.