About

Ioannis Karatzas is the Eugene Higgins Professor of Applied Probability at the Department of Mathematics and the Department of Statistics at Columbia University. His research interests encompass Probability and Mathematical Statistics, Random Processes, Stochastic Analysis, Optimization, and Mathematical Economics and Finance. He is involved in teaching a variety of courses related to probability, stochastic processes, and financial mathematics, reflecting his expertise in these areas. His academic work includes supervising doctoral students and contributing to the fields of applied probability and stochastic analysis, with a focus on their applications in finance and economics.

Research topics

• Financial economics
• Applied mathematics
• Geometry
• Statistical physics
• Thermodynamics
• Mathematical analysis
• Physics
• Statistics
• Mathematics
• Economics
• Mathematical economics

Selected publications

• Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

  Open MIND · 2026-02-26

  preprint

  The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.

  DOI

• Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

  arXiv (Cornell University) · 2026-02-26

  articleOpen access

  The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.

  PublisherOA PDF

• Drift control with discretionary stopping for a diffusion

  The Annals of Applied Probability · 2025-06-01 · 1 citations

  articleSenior author

  We consider stochastic control with discretionary stopping for the drift of a diffusion process over an infinite time horizon. The objective is to choose a control process and a stopping time to minimize the expectation of a convex terminal cost in the presence of a fixed operating cost and a control-dependent running cost per unit of elapsed time. Under appropriate conditions on the coefficients of the controlled diffusion, an optimal pair of control and stopping rules is shown to exist. Moreover, under the same assumptions, it is shown that the optimal control is a constant which can be computed fairly explicitly; and that it is optimal to stop the first time an appropriate interval is visited. We consider also a constrained version of the above problem, in which an upper bound on the expectation of available stopping times is imposed; we show that this constrained problem can be reduced to an unconstrained problem with some appropriate change of parameters and, as a result, solved by similar arguments.

  PublisherDOI

• Drift Control with Discretionary Stopping for a Diffusion

  arXiv (Cornell University) · 2024-01-18

  preprintOpen accessSenior author

  We consider stochastic control with discretionary stopping for the drift of a diffusion process over an infinite time horizon. The objective is to choose a control process and a stopping time to minimize the expectation of a convex terminal cost in the presence of a fixed operating cost and a control-dependent running cost per unit of elapsed time. Under appropriate conditions on the coefficients of the controlled diffusion, an optimal pair of control and stopping rules is shown to exist. Moreover, under the same assumptions, it is shown that the optimal control is a constant which can be computed fairly explicitly; and that it is optimal to stop the first time an appropriate interval is visited. We consider also a constrained version of the above problem, in which an upper bound on the expectation of available stopping times is imposed; we show that this constrained problem can be reduced to an unconstrained problem with some appropriate change of parameters and, as a result, solved by similar arguments.

  PublisherOA PDFDOI

• Invariant measure of gaps in degenerate competing three-particle systems

  arXiv (Cornell University) · 2024-01-19

  preprintOpen access

  We study the gap processes in a degenerate system of three particles interacting through their ranks. We obtain the Laplace transform of the invariant measure of these gaps, and an explicit expression for the corresponding invariant density. To derive these results, we start from the basic adjoint relationship characterizing the invariant measure, and apply a combination of two approaches: first, the invariance methodology of W. Tutte, thanks to which we compute the Laplace transform in closed form; second, a recursive compensation approach which leads to the density of the invariant measure as an infinite convolution of exponential functions. As in the case of Brownian motion with reflection or killing at the endpoints of an interval, certain Jacobi theta functions play a crucial role in our computations.

  PublisherOA PDFDOI

• A strong law of large numbers for positive random variables

  Illinois Journal of Mathematics · 2023-08-29 · 1 citations

  article1st authorCorresponding

  In the spirit of the famous Komlós (1967) theorem, every sequence of nonnegative, measurable functions {fn}n∈N on a probability space contains a subsequence which—along with all its subsequences—converges a.e. in Cesàro mean to some measurable f∗:Ω→[0,∞]. This result of von Weizsäcker (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of Delbaen and Schachermayer (1994), replacing general convex combinations by Cesàro means.

  PublisherDOI

• A Weak Law of Large Numbers for Dependent Random Variables

  Theory of Probability and Its Applications · 2023-11-01 · 4 citations

  article1st authorCorresponding

  \bad Each sequence $f_1,f_2,\dots$ of random variables satisfying $\lim_{M\to \infty}(M\sup_{k\in \mathbf N}\mathbf{P}(|f_k|>M))=0$ contains a subsequence $f_{k_1},f_{k_2},\dots$ which, along with all its subsequences, satisfies the weak law of large numbers $\lim_{N\to\infty}\bigl((1/N) \sum^N_{n=1} f_{k_n}- D_N\bigr)=0$ in probability. Here, $D_N$ is a ``corrector” random variable with values in $[-N,N]$ for each $N\in\mathbf{N}$; these correctors are all equal to zero if, in addition, $\lim \inf_{n\to\infty}\mathbf{E}(f^2_n \mathbf{1}_{\{|f_n|\le M\}})=0$ for every $M\in(0,\infty)$.

  PublisherDOI

• A weak law of large numbers for dependent random variables

  Теория вероятностей и ее применения · 2023-01-01 · 1 citations

  article1st authorCorresponding

  Любая последовательность $f_1, f_2, …$ случайных величин, удовлетворяющая условию $\lim_{M\to\infty}(M \sup_{k\in \mathbf N} \mathbf{P}(|f_k|> M))=0$, содержит подпоследовательность $f_{k_1}, f_{k_2}, …$, которая вместе со всеми своими подпоследовательностями удовлетворяет слабому закону больших чисел $\lim_{N\to\infty} ((1/N) \sum^N_{n=1} f_{k_n}- D_N)=0$ по вероятности. Здесь $D_N$ является "корректирующей" случайной величиной со значениями в $[-N,N]$ для каждого $N\in\mathbf{N}$. Все корректоры равны нулю при условии, что $\lim \inf_{n\to\infty}\mathbf{E}(f^2_n \mathbf{1}_{\{|f_n|\le M\}})=0$ для каждого $M\in (0,\infty)$.

  PublisherDOI

• A sequential estimation problem with control and discretionary stopping

  Probability Uncertainty and Quantitative Risk · 2022-01-01 · 2 citations

  articleOpen accessSenior author

  <p style='text-indent:20px;'>We show that “full-bang” control is optimal in a problem which combines features of (i) sequential least-squares <i>estimation</i> with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded <i>control</i> of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary <i>stopping</i>. We develop also the optimal filtering and stopping rules in this context.</p>

  PublisherOA PDFDOI

• A Trajectorial Approach to the Gradient Flow Properties of Langevin--Smoluchowski Diffusions

  Theory of Probability and Its Applications · 2022 · 6 citations

  1st authorCorresponding
  • Mathematics
  • Statistical physics
  • Applied mathematics

  Article DataHistorySubmitted: 19 May 2021Published online: 03 February 2022Keywordsrelative entropy, Wasserstein distance, Fisher information, optimal transport, gradient flow, diffusion processes, time reversal, functional inequalitiesPublication DataISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied MathematicsCODEN: tprbau

  PublisherDOI

Recent grants

• Stochastic Controls, Games and Portfolios

  NSF · $628k · 2009–2015

• Stochastic Portfolios, Controls, and Interacting Particles

  NSF · $600k · 2020–2025

• Stochastic Controls, Portfolios, and Competing Particle Systems

  NSF · $590k · 2014–2019

• Topics in Stochastic Analysis and Optimization

  NSF · $300k · 2006–2009

Frequent coauthors

• Steven E. Shreve

  65 shared

• William D. Sudderth

  29 shared

• Tomoyuki Ichiba

  28 shared

• Robert Fernholz

  27 shared

• Martín Shubik

  25 shared

• John Geanakoplos

  Santa Fe Institute

  15 shared

• Walter Schachermayer

  15 shared

• Johannes Ruf

  15 shared

Education

• Ph.D., Probability, Mathematical Finance

  Columbia University

  1980