# Steven E. Shreve

## Orion Hoch University Professor of Mathematical Sciences

**Address:**

6216 Wean Hall

Department of Mathematical Sciences

Carnegie Mellon University

5000 Forbes Avenue

Pittsburgh, PA 15213

**P:** 412-268-8484**F:** 412-268-6380

# Education

M.S. in Electrical Engineering, University of Illinois Urbana-Champaign

Ph.D. in Mathematics, University of Illinois Urbana-Champaign

# Research

My recent research has followed two tracks. The first has been 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 of how a bank should compensate its traders. In all these cases, continuous-time models using stochastic calculus are constructed and analyzed.

A second activity has been modeling of queueing systems in heavy traffic when tasks have deadlines for completion. Although queues are intrinsically discrete-event systems, when in heavy traffic, the queue lengths can profitably be approximated by diffusions.

If tasks have attributes, such as lead times until deadlines expire, the approximation is a measure-valued diffusion. In a series of papers with multiple co-authors, I have determined the limiting measure-valued diffusion processes obtained in a variety of queueing systems.

These two threads have recently come together in the construction of diffusion approximations for limit-order books that govern trading on electronic exchanges. This is ongoing work with John Lehoczky in the CMU Department of Statistics and Data Science and with Ph.D. advisees Christopher Almost and Xiaofeng Yu.

# Select Publications

Bertsekas, D. P. and Shreve, S. E. Stochastic Optimal Control: The Discrete Time Case. *Academic Press*, (1979). *Republished by Athena Scientific*, (1996).

Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. *Springer-Verlag*.

Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. *Springer-Verlag*.

Shreve, S. E. (2004). Stochastic Calculus for Finance. *Volume 1: The Binomial Asset Pricing Model. Volume II: Continuous Time Models. Springer-Verlage.*

Karatzas, I. and Shreve, S. E. Connections between optimal stopping and singular stochastic control. *Part I, SIAM Journal Control and Optimization *22 (1984), 856-877. *Part II, SIAM Journal Control and Optimization* 23 (1985), 433-451.

Karatzas, I., Lehoczky, J., and Shreve, S. E. (1987). Optimal portfolio consumption decisions for a "small investor" on a finite horizon. *SIAM Journal Control and Optimization.* 25 , 1557-1586.

Karatzas, I., Lehoczky,J., Shreve, S. E. and Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. *SIAM Journal Control and Optimization*, 29, 702-730.

Shreve, S. E. and Xu, G.-L., A duality method for optimal consumption and investment under short-selling prohibition. *Part I, Annals of Applied Probability* 2 (1992), 87-112. *Part II, Annals of Applied Probability* 2 (1992), 314-328.

Soner, H. M. and Shreve, S. E. (1994). Optimal investment and consumption with transaction costs. *Annals of Applied Probability,* 4, 609-692.

Doytchinov, B., Lehoczky, J., and Shreve, S. E. (2001). Real-time queues in heavy traffic with earliest-deadline-first queue discipline. *Annals of Applied Probability,* 11, 332-378.

Kruk, L., Lehoczky, J., Ramanan, K., and Shreve, S. E. (2011). Heavy traffic analysis for EDF queues with reneging. *Annals of Applied Probability*, 21, 484-545.

Brunick, G. and Shreve, S. E. (2013). Mimicking an Ito process by a solution of a stochastic differential equation. *Annals of Applied Probabilty,* 23, 1584-1628.