Probability and Financial Mathematics

Probability is concerned with the mathematical analysis of random phenomena. It is younger than most other areas of Mathematics, as it originated with the study of games of chance by Pascal and Fermat (1654); since then it has become indispensable for dealing with randomness in almost all branches of science. Today, in addition to its strong position as a full-fledged area within mathematics -- and its interactions with analysis, partial differential equations, geometry, combinatorics, number theory and
mathematical physics -- probability forms the language of statistics and of the quantitative social sciences. It is essential in the discovery and study of macroscopic regularities that occur when large systems of particles, organisms, or agents, interact according to the laws of physics, biology, or economics; in the study of population genetics and of the genome; in the study of how signals are transmitted through a noisy channel, and then recovered; in the design and analysis of large-scale communication, neural or queueing networks, and of algorithms for combinatorial optimization, computerized tomography, signal processing, pattern recognition, and so on. A central role in several of these applications is played by the development of stochastic analysis over the last 60 years, starting with the pioneering work of K. Ito in 1942.

Probability has also been central in the study of finance, ever since Bachelier pioneered in 1900 the mathematical study of Brownian motion and understood its significance as a tool for the analysis of financial markets (five years before Einstein developed his physical theory of Brownian motion). The theory of finance, brought to worldwide attention with the award of the Nobel prize in economics to Markowitz, Sharpe, Miller and then to Merton and Scholes, tries to understand how financial markets work, how they can be made more efficient, how they should be regulated, and how they can help manage the risk inherent in various economic activities. Over the last 25 years this theory has become increasingly mathematical, to the extent that problems arising in finance are now not just drawing upon, but are also driving research in stochastic analysis, partial differential equations, and control theory. At the same time, the development of sophisticated analytical and numerical methods, often based on partial differential equations and on their numerical solution, has helped to increase the relevance of these developments in the everyday practice of finance.

The Columbia probability group has a strong interest in several areas of stochastic analysis and optimization, including stochastic control and optimal stopping, detection of change-points and of signals in noise, stochastic differential equations, stochastic differential geometry, statistical regularities in interacting particle systems. It focuses in particular on various aspects of mathematical finance. These include hedging and portfolio optimization; market frictions such as constraints, transaction costs, liquidity and size effects; and the study of qualitative properties of large equity and debt markets.