Steven E. Shreve
Orion Hoch University Professor of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213-3890

Wean Hall 6216
(412) 268-8484
Fax: (412) 268-6380
E-mail: shreve at the address

Curriculum Vita pdf file.


Essays on the Financial Crisis

Don't Blame the Quants, Forbes, October 7, 2008.

Model Risk, Analytics, April 2009.

Response to Pablo Triana's article "The Flawed Math of Financial Models", published by


Stochastic Calculus for Finance
Volume I: The Binomial Asset Pricing Model
Volume II: Continuous-Time Models

by Steven E. Shreve
Springer-Verlag, New York

Errata for 2004 printing of Volume II, September 2006

More errata for 2004 printing of Volume II, July 2007

More errata for 2004 printing of Volume II, February 2008

Errata for 2008 printing of Volume I, July 2011

Errata for 2008 printing of Volume II, July 2011

Methods of Mathematical Finance

by Ioannis Karatzas and Steven E. Shreve
Springer-Verlag, New York

Mathematical Finance

Mark H. A. Davis, Darrell Duffie, Wendell Fleming and Steven E. Shreve, editors
IMA Volumes in Mathematics and its Applications 65
Springer-Verlag, New York

Brownian Motion and Stochastic Calculus

by Ioannis Karatzas and Steven E. Shreve
Springer-Verlag, New York
Second Edition, 1991.

Stochastic Optimal Control: The Discrete Time Case

by Dimitri P. Bertsekas and Steven E. Shreve
Academic Press, Orlando
Reprinted by Athena Scientific Publishing, 1995, and is available for free download at


Diffusion limit of Poisson limit-order book models

by C. Almost, J. Lehoczky, S. Shreve and X. Yu.

Recently published papers

Matching an Ito Process by a Solution of a Stochastic Differential Equation

by G. Brunick and S. Shreve
Annals Applied Probability 23 (2013), 1584--1628

"Mimicking an Ito Process" pdf file.


Given a multi-dimensional Ito 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 Ito process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Ito 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 modelled by the original Ito process or the mimicking process that solves the stochastic differential equation.

Utility Maximization Trading Two Futures with Transaction Costs

by M. Bichuch and S. Shreve
SIAM J. Financial Math 4 (2013), 26--85.

"Utility Maximization" pdf file.


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.

Optimal Execution of a General One-Sided Limit-Order Book

by S. Predoiu, G. Shaikhet and S. Shreve
SIAM J. Financial Math 2 (2011), 183--212.


"Optimal Execution" pdf file.

We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero.

Heavy Traffic Analysis for EDF Queues with Reneging

by L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve
Annals of Applied Probability 35 (2007), 1740--1768.

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, which are validated by simulation results.

"Reneging" pdf file.

Futures Trading with Transaction Costs

by K. Janecek and S. Shreve
Illinois Journal of Mathematics, 54 (2010), 1239-1284.

A model for optimal consumption and investment is posed whose solution is provided by the classical Merton analysis when there is zero transaction cost. A probabilistic argument is developed to identify the loss in value when a proportional transaction cost is introduced. There are two sources of this loss. The first is a loss due to "displacement'' that arises because one cannot maintain the optimal portfolio of the zero-transaction-cost problem. The second loss is due to "transaction,'' a loss in capital that occurs when one adjusts the portfolio. The first of these increases with increasing tolerance for departure from the optimal portfolio in the zero-transaction-cost problem, while the second decreases with increases in this tolerance. This paper balances the marginal costs of these two effects. The probabilistic analysis provided here complements earlier work on a related model that proceeded from a viscosity solution analysis of the associated Hamilton-Jacobi-Bellman equation.

"Futures Trading" pdf file.

Double Skorokhod map and reneging real-time queues

by L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve
in Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz,
S. Ethier, J. Feng and R. Stockbridge, eds.,
Institute of Mathematical Statistics Collections, Vol. 4, pp. 169-193.

An explicit formula for the Skorokhod map $\Gamma_{0,a}$ on $[0,a]$ for $a>0$ is provided and related to similar formulas in the literature. Specifically, it is shown that on the space $D[0,\infty)$ of right-continuous functions with left limits taking values in $\mathbb{R}$, $$ \Gamma_{0,a}(\psi)(t) = \psi (t) -\left[\big(\psi(0)-a\big)^+ \wedge\inf_{u\in[0,t]}\psi(u)\right] \vee \sup_{s \in [0,t]} \left[ (\psi(s) - a) \wedge \inf_{u \in [s,t]} \psi(u)\right] $$ is the unique function taking values in $[0,a]$ that is obtained from $\psi$ by minimal ``pushing'' at the endpoints $0$ and $a$. An application of this result to real-time queues with reneging is outlined.

"Double Skorokhod" pdf file.

An Explicit Formula for the Skorohod Map on [0,a]

by L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve
Annals of Probability
2007, Vol. 35, No. 5, 1740-1768

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $\Gamma_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $D[0,\infty)$ of right-continuous functions with left limits taking values in R, $\Gamma_{0,a} = \Lambda_a \circ \Gamma_0$, where $\Lambda_a$ mapping $D[0,\infty)$ into itself is defined by $$ \Lambda_a(\phi)(t) =\phi(t)-\sup_{s\in[0,t]}[(\phi(s)-a)^+ \wedge \inf_{u\in [s,t]}\phi(u)] $$ and $\Gamma_0$ mapping $D[0,\infty)$ into itself is the Skorokhod map on $[0,\infty)$. In addition, properties of $\Lambda_a$ are developed and comparison properties of $\Gamma_{0,a}$ are established.

A Two-Person Game for Pricing Convertible Bonds

by M. Sirbu and S. Shreve
SIAM J. Control and Optimization
2006 Vol. 45, No. 4, pp. 1508-1639

A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, the bond can be called by the issuing firm, which presumably acts to maximize the equity value of the firm by minimizing the bond value. This creates a two-person game. We show that if the coupon rate is below the interest rate times the call price, then conversion should precede call. On the other hand, if the dividend rate times the call price is below the coupon rate, call should precede conversion. In either case, the game reduces to a problem of optimal stopping.

"Finite Maturity Convertible Bonds" pdf file.

Satisfying Convex Risk Limits by Trading

by K. Larsen, T. Pirvu, S. Shreve and R. Tutuncu
Finance and Stochastics
2005 Vol. 9, No. 2, pp. 177-195.

A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line {x;x\geq a}. Methods for computing a are provided, and the application of these ideas to derivative security pricing is developed.

Perpetual Convertible Bonds

by M. Sirbu, I. Pikovsky and S. Shreve
SIAM J. Control and Optimization
2004 Vol. 43, No. 1, pp. 58-85

A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The firm may at any time call the bond. Because calls and conversions usually occur far from maturity, we model this situation with a perpetual convertible bond, i.e, a convertible coupon-paying bond without maturity. This model admits a relatively simple solution, under which the value of the perpetual convertible bond, as a function of the value of the underlying firm, is determined by a nonlinear ordinary differential equation.

"PerpetualConvertibleBonds" pdf file.

Accuracy of State Space Collapse for Earliest-Deadline-First Queues

by L. Kruk, J. Lehoczky and S. Shreve
Annals of Applied Probability
2006, Vol. 16, No. 2, 516-581

This paper presents a second-order heavy traffic analysis of a single server queue that processes customers having deadlines using the earliest-deadline-first scheduling policy. For such systems, referred to as {\em real-time queueing systems}, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures such as customer delay, queue length, or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the earlier heavy traffic analysis of such systems that provided approximations to the system's behavior. The main result of this paper is the development of a second-order analysis that gives the accuracy of the approximations and the rate of convergence of the sequence of real-time queueing systems to its heavy traffic limit.

"Accuracy of State Space Collapse" pdf file.

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