About

Steven Shreve is the Orion Hoch Professor of Mathematical Sciences at the Mellon College of Science and by courtesy, at the Tepper School of Business at Carnegie Mellon University. His recent research has focused on problems in financial mathematics, including models for derivative securities, utility maximization especially in the presence of transaction costs, optimal execution of large financial transactions, and the principal agent problem related to bank trader compensation. He constructs and analyzes continuous-time models using stochastic calculus in these areas. Additionally, Shreve has worked on modeling queueing systems in heavy traffic when tasks have deadlines for completion. His research involves approximating queue lengths by diffusions and measure-valued diffusions when tasks have attributes such as lead times until deadlines expire. Recently, his work has integrated these areas into the construction of diffusion approximations for limit-order books that govern trading on electronic exchanges, in collaboration with colleagues including John Lehoczky and Ph.D. advisee Christopher Almost.

Research topics

• Mathematics
• Computer Security
• Computer Science
• Statistics
• Economics
• Applied mathematics
• Mathematical analysis
• Statistical physics
• Econometrics
• Mathematical economics
• Geometry
• Business
• Quantum mechanics
• Physics

Selected publications

• Limits of Limit-Order Books

  Lecture notes in mathematics · 2022 · 2 citations

  • Mathematics
  • Applied mathematics
  • Statistical physics
  PublisherDOI

• Escrow and Clawback

  SIAM Journal on Financial Mathematics · 2022

  1st authorCorresponding
  • Computer Science
  • Computer Security
  • Computer Science

  Since the financial crisis of 2008, clawback provisions have been implemented by several high-profile banks and are also required by regulators in order to mitigate the cost of financial failures and to deter excessive risk taking. We construct a model to investigate the long-term effect on the bank's revenue of deferring (escrowing) a trader's bonuses to facilitate clawback. We formulate the question by setting up an infinite-horizon dynamic programming model. Within this model, the trader's optimal investment and consumption strategy, with and without bonus escrow, can be expressed by explicit analytic formulas. These formulas enable calculation and comparison of the bank's total expected revenue under the two bonus payout schemes. The results of the comparison depend on the parameters describing the trader's risk appetite, the discount factor, and the bank's level of patience, in addition to the market parameters. In particular, when the bank's total expected discounted revenue is finite under both types of bonus payment schemes and the bank is sufficiently patient, the bank benefits by escrowing the trader's bonus, although not escrowing the trader's bonus brings better short-term revenue.

  PublisherDOI

• Diffusion Limit of Poisson Limit-Order Book Models

  arXiv (Cornell University) · 2020

  • Statistical physics
  • Mathematics
  • Applied mathematics

  This ia a companion paper to Almost, Lehoczky, Shreve & Yu \cite{ALSY}, where the rationale for studying the diffusion limit of Poisson limit-order book models is explained and the results of a particular "representative" model are detailed. This paper contains the proofs and technical details cited in that work.

  PublisherOA PDFDOI

• Steven E. Shreve

  Authors group · 2019-12-06

  dataset1st authorCorresponding
  PublisherDOI

• A free boundary problem related to singular stochastic control

  Research Showcase @ Carnegie Mellon University (Carnegie Mellon University) · 2018-06-29 · 20 citations

  articleOpen access1st authorCorresponding

  Abstract: "It is desired to control a multi-dimensional Brownian motion by adding a (possibly singularly) continuous process to its n[superscript th] components so as to minimize an expected infinite-horizon discounted running cost. The Hamilton-Jacobi-Bellman characterization of the value function is a variational inequality which has a unique twice continuously differentiable solution. The optimal process is constructed by solving the Skorokhod problem of reflecting the Brownian motion along a free boundary in the (0,0,..., -1) direction."

  PublisherOA PDFDOI

• Equivalent martingale measures and optimal market completions

  Research Showcase @ Carnegie Mellon University (Carnegie Mellon University) · 2018-06-29 · 3 citations

  articleOpen accessSenior author

  Abstract: "Optimal fictitious completions of an incomplete financial market are shown to be associated with exponential martingales (not just local martingales) and, therefore, to 'an optimal equivalent martingale measure'. Results of independent interest, in the theory of continuous-time martingales, are derived as well."

  PublisherOA PDFDOI

• Heavy traffic analysis for EDF queues with reneging

  Research Showcase @ Carnegie Mellon University (Carnegie Mellon University) · 2018-06-29 · 51 citations

  articleOpen accessSenior author

  This paper presents a heavy-traffic analysis of the behavior of a single-server queue under an Earliest-Deadline-First (EDF) scheduling policy in which customers have deadlines and are served only until their deadlines elapse. The performance of the system is measured by the fraction of reneged work (the residual work lost due to elapsed deadlines) which is shown to be minimized by the EDF policy. The evolution of the lead time distribution of customers in queue is described by a measure-valued process. The heavy traffic limit of this (properly scaled) process is shown to be a deterministic function of the limit of the scaled workload process which, in turn, is identified to be a doubly reflected Brownian motion. This paper complements previous work by Doytchinov, Lehoczky and Shreve on the EDF discipline in which customers are served to completion even after their deadlines elapse. The fraction of reneged work in a heavily loaded system and the fraction of late work in the corresponding system without reneging are compared using explicit formulas based on the heavy traffic approximations. The formulas are validated by simulation results.

  PublisherOA PDFDOI

• A duality method for optimal consumption and investment under short-selling prohibition

  Research Showcase @ Carnegie Mellon University (Carnegie Mellon University) · 2018-06-29 · 54 citations

  articleOpen accessSenior author

  Abstract: "A continuous-time, consumption/investment problem on a finite horizon is considered for an agent seeking to maximize expected utility from consumption plus expected utility from terminal wealth. The agent is prohibited from selling stocks short, so the usual martingale methods for solving this problem do not directly apply. A dual problem is posed and solved, and the solution to the dual problem provides information about the existence and nature of the solution to the original problem. When the market coefficients are constant, the value functions for both problems are provided in terms of solutions to linear, second-order, partial differential equations. If, furthermore, the utility functions are of the power form, the solutions to these equations take a particularly simple form, as do the formulas for the optimal consumption and investment processes."

  PublisherOA PDFDOI

• Mimicking an Itô process by a solution of a stochastic differential equation

  The Annals of Applied Probability · 2013-06-21 · 111 citations

  articleOpen accessSenior author

  Given a multi-dimensional Itô process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Itô process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modeled by the original Itô process or the mimicking process that solves the stochastic differential equation.

  PublisherOA PDFDOI

• Utility Maximization Trading Two Futures with Transaction Costs

  SIAM Journal on Financial Mathematics · 2013-01-01 · 15 citations

  articleSenior author

  An agent invests in two types of futures contracts, whose prices are possibly correlated arithmetic Brownian motions, and invests in a money market account with a constant interest rate. The agent pays a transaction cost for trading in futures proportional to the size of the trade. She also receives utility from consumption. The agent maximizes expected infinite-horizon discounted utility from consumption. We determine the first two terms in the asymptotic expansion of the value function in the transaction cost parameter around the known value function for the case of zero transaction cost. The method of solution when the futures are uncorrelated follows a method used previously to obtain the analogous result for one risky asset. However, when the futures are correlated, a new methodology must be developed. It is suspected in this case that the value function is not twice continuously differentiable, and this prevents application of the former methodology.

  PublisherDOI

Recent grants

• Stochastic Analysis with Applications to Finance

  NSF · $651k · 2009–2016

• Mathematical Finance and Stochastic Networks

  NSF · $523k · 2004–2009

Frequent coauthors

• Ioannis Karatzas

  65 shared

• John P. Lehoczky

  Carnegie Mellon University

  63 shared

• Łukasz Kruk

  Maria Curie-Skłodowska University

  36 shared

• Shu-Ngai Yeung

  Carnegie Mellon University

  18 shared

• Kavita Ramanan

  14 shared

• Dimitri P. Bertsekas

  8 shared

• H. Meté Soner

  7 shared

• Suresh Sethi

  The University of Texas at Dallas

  6 shared

Education

• Ph.D., Mathematics

  University of California, Berkeley

  1983

• M.S., Mathematics

  University of California, Berkeley

  1979

• B.S., Mathematics

  University of California, Berkeley

  1977