Mykhaylo Shkolnikov

ORFE Department

202 Sherrerd Hall
Princeton University

Princeton, NJ 08544

mykhaylo at princeton.edu


Currently, I am an Assistant Professor in the ORFE Department at Princeton University,
after being 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 Mathematics Department at Stanford University where my
advisor was Amir Dembo.

At the moment, I am applying tools from interacting particle systems to problems in random matrix theory and mathematical finance. More broadly, my interests include a variety of topics in probability theory and PDEs: integrable probability, models of random growth, SPDEs, concentration of measure, large deviations, random operators, probabilistic approach to hyperbolic and parabolic PDEs.

My research is partially supported by NSF grant DMS-1506290.

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


Activities:

PhD students:


Pierre Yves Gaudreau Lamarre, Levon Avanesyan, Praveen Kolli



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. (2014). Interacting particle systems at the edge of multilevel Dyson Brownian motions. To appear in Adv. Math.
25. Shkolnikov, M., Sircar, R., Zariphopoulou, T. (2015). Forward performance processes in incomplete markets and ill-posed HJB equations. To appear in SIAM J. Financial Math.
26. Shkolnikov, M. (2015). A construction of infinite Brownian particle systems. Submitted.
27. Gorin, V., Shkolnikov, M. (2016). Stochastic Airy semigroup through tridiagonal matrices. Submitted.
28. Kolli, P., Shkolnikov, M. (2016). SPDE limit of the global fluctuations in rank-based models. To appear in Ann. Probab.
29. Ramanan, K., Shkolnikov, M. (2016). Intertwinings of beta-Dyson Brownian motions of different dimensions. To appear in Ann. Inst. Henri Poincaré Probab. Stat.
30. Nadtochiy, S., Shkolnikov,  M. (2017). Particle systems with singular interaction through hitting times: application in systemic risk modeling. Submitted.
31. Gaudreau Lamarre, P. Y., Shkolnikov, M. (2017). Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Submitted.


Recent and upcoming talks:

2018: Seminar on Stochastic Processes: Brown University, Casa Mathemática Oaxaca: Workshop on Stochastic Analysis and its Applications

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


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:

David Aldous, Amir Dembo, Simone Farinelli, Pierre Yves Gaudreau Lamarre, Stefan Gerhold, Vadim Gorin, Tomoyuki Ichiba, Ioannis Karatzas, Max Kleinert, Praveen Kolli, Sergey Nadtochiy, Soumik Pal, Piet Porkert, Miklos Z. Racz, Kavita Ramanan, Albert N. Shiryaev, Ronnie Sircar, S.R.Srinivasa Varadhan, Thaleia Zariphopoulou, Ofer Zeitouni