About

Mykhaylo Shkolnikov is 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, he was an Associate Professor (with tenure) in the ORFE Department at Princeton University. His research focuses on interacting particle systems arising in mathematical finance and mathematical physics, utilizing tools from stochastic analysis and PDE. His broader interests include probability theory and related fields such as moving interfaces, probabilistic approaches to PDEs, neural networks, stochastic PDEs, large deviations, random operators, and integrable probability.

Research topics

• Mathematics
• Computer Science
• Mathematical analysis
• Physics
• Statistics
• Thermodynamics
• Econometrics
• Applied mathematics
• Algorithm
• Pure mathematics
• Geology

Selected publications

• Relative arbitrage problem under eigenvalue lower bounds

  arXiv (Cornell University) · 2025-12-19

  preprintOpen access

  We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix.

  PublisherDOI

• Relative arbitrage problem under eigenvalue lower bounds

  ArXiv.org · 2025-12-19

  articleOpen access

  We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix.

  PublisherOA PDF

• Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2024-02-01 · 2 citations

  articleOpen access

  Nous considérons (une variante) du processus d’agrégation limitée par diffusion externe multi-particule (MDLA) de Rosenstock et Marquardt dans le plan. Sur la base des résultats de (Ann. Probab. 24 (1996) 559–598, Arch. Ration. Mech. Anal. 233 (2019) 643–699, Delarue, Nadtochiy and Shkolnikov (2019)) en dimension un, il est naturel de conjecturer que la limite d’échelle de l’agrégat en croissance dans un tel modèle est donnée par la phase solide en croissance dans une formulation “probabiliste” appropriée du problème de Stefan de surfusion monophasé pour l’équation de la chaleur. Pour répondre à cette conjecture, nous prouvons d’abord que les points limites des systèmes MDLA à échelle diffusive sont bien définis et décrits par des mouvements browniens absorbés. Puis, nous montrons que ces points limites satisfont l’équation qui caractérise le taux de croissance de la phase solide dans le problème de Stefan de surfusion avec une inégalité, qui peut être stricte en général. Ce résultat fournit la première réponse rigoureuse à une question qui a reçu beaucoup d’attention dans la littérature physique. Au cours de la preuve, nous établissons deux résultats supplémentaires intéressants en soi : (i) la stabilité d’une “propriété de croisement” du mouvement brownien plan et (ii) une connexion rigoureuse entre les solutions probabilistes du problème de Stefan surfusion et ses solutions classiques et faibles.

  PublisherOA PDFDOI

• From rank-based models with common noise to pathwise entropy solutions of SPDEs

  arXiv (Cornell University) · 2024-06-11

  preprintOpen access1st authorCorresponding

  We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19, 16, 17].

  PublisherOA PDFDOI

• Deep level-set method for Stefan problems

  Journal of Computational Physics · 2024-02-07 · 5 citations

  article1st author
  PublisherDOI

• Cascade equation for the discontinuities in the Stefan problem with surface tension

  arXiv (Cornell University) · 2024-10-20

  preprintOpen accessSenior author

  The Stefan problem with surface tension is well known to exhibit discontinuities in the associated moving aggregate (i.e., in the domain occupied by the solid), whose structure has only been understood under translational or radial symmetry so far. In this paper, we derive an auxiliary partial differential equation of second-order hyperbolic type, referred to as the cascade equation, that captures said discontinuities in the absence of any symmetry assumptions. Specializing to the one-phase setting, we introduce a novel (global) notion of weak solution to the cascade equation, which is defined as a limit of mean-field game equilibria. For the spatial dimension two, we show the existence of such a weak solution and prove a natural perimeter estimate on the associated moving aggregate.

  PublisherOA PDFDOI

• Multilevel Dyson Brownian motions via the superposition principle

  arXiv (Cornell University) · 2024-03-15

  preprintOpen accessSenior author

  Multilevel Dyson Brownian motions (MDBMs) combine Dyson Brownian motions of different dimensions into a single process in a canonical way. This paper completes the theory of MDBMs for $β\ge2$. Specifically, we use the superposition principle of Figalli and Trevisan to construct the MDBMs for all $β>2$ in a unified manner. This also extends their stochastic differential equation representation, first discovered by Gorin and Shkolnikov, to all $β>2$ and proves the uniqueness of the MDBMs for all $β>2$. Finally, we show that their limit as $β\downarrow2$ is given by the $β=2$ MDBM, commonly referred to as the Warren process.

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• A singular two-phase Stefan problem and particles interacting through their hitting times

  The Annals of Applied Probability · 2024-09-27

  articleSenior author

  We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean–Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension. As for the one-phase problems, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not necessarily monotone, so that the large system limit is obtained by a novel argument.

  PublisherDOI

• Stefan Problem with Surface Tension: Uniqueness of Physical Solutions under Radial Symmetry

  Archive for Rational Mechanics and Analysis · 2024-09-23 · 1 citations

  articleOpen accessSenior author

  Abstract We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [21]. The paper in hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own right.

  PublisherOA PDFDOI

• Well-posedness of the supercooled Stefan problem with oscillatory initial conditions

  Electronic Journal of Probability · 2024-01-01 · 1 citations

  articleOpen accessSenior author
  PublisherDOI

Recent grants

• Investigation of Interacting Particle Systems by Stochastic Analysis Methods

  NSF · $174k · 2015–2018

• Large-Scale Behavior of Interacting Particle Systems

  NSF · $210k · 2018–2021

Frequent coauthors

• Sergey Nadtochiy

  23 shared

• Vadim Gorin

  15 shared

• Ioannis Karatzas

  13 shared

• Soumik Pal

  University of Washington

  13 shared

• Jiacheng Zhang

  Guangdong Ocean University

  10 shared

• Thaleia Zariphopoulou

  The University of Texas at Austin

  8 shared

• Ronnie Sircar

  8 shared

• Simone Farinelli

  8 shared

Labs

• Mykhaylo Shkolnikov LabPI

Education

• Ph.D.

  Stanford University

Awards & honors

• Early Career Prize by the Activity Group on Financial Mathem…
• Erlang Prize by the Applied Probability Society of the Insti…
• E. Lawrence Keyes, Jr./Emerson Electric Co. Faculty Advancem…
• Princeton Engineering Commendation List for Outstanding Teac…