# 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:

- Associate Editor for Annals of Applied Probability
- Associate Editor for Mathematical Finance
- Associate Editor for Applied Mathematical Finance
- IMSI workshop: Laplacian Growth Models
- Carnegie Mellon Probability/Math Finance Seminar

## PhD students:

Benjamin Budway (expected: 05/2027), Jou-Hua Lai (expected: 05/2026), Yucheng Guo (expected: 05/2026), Scander Mustapha (expected: 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:

- 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.
*Finance Stoch.***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. - Shkolnikov, M. (2012). On a non-linear transformation between Brownian martingales.
- 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. - Ichiba, T., Shkolnikov, M. (2013). Large deviations for interacting Bessel-like processes and applications to systemic risk.
- Pal, S., Shkolnikov, M. (2013). Intertwining diffusions and wave equations.
- Shkolnikov, M., Karatzas, I. (2013). Time-reversal of reflected Brownian motions in the orthant.
- 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. - Shkolnikov, M. (2015). A construction of infinite Brownian particle systems.
- 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. (2021). Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem. To appear in
*Ann. Inst. Henri Poincaré Probab. Stat.* - Baker, G., Shkolnikov, M. (2022). A singular two-phase Stefan problem and particles interacting through their hitting times.
*Submitted*. - 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. (2023). Well-posedness of the supercooled Stefan problem with oscillatory initial conditions.
*Submitted*. - Guo, Y., Nadtochiy, S., Shkolnikov, M. (2023). Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry.
*Submitted*. - Shkolnikov, M., Soner, H. M., Tissot-Daguette, V. (2023). Deep level-set method for Stefan problems.
*Submitted*.

## 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
- Daniel Lacker
- Sergey Nadtochiy
- Soumik Pal
- Piet Porkert
- Miklos Z. Racz
- Kavita Ramanan
- Christoph Reisinger
- Albert N. Shiryaev
- Ronnie Sircar
- H. Mete Soner
- Valentin Tissot-Daguette
- S.R.Srinivasa Varadhan
- Thaleia Zariphopoulou
- Ofer Zeitouni
- Jiacheng Zhang
- Xiling Zhang