Philip Protter - US grants

The principal investigator will work on two projects in collaboration with many US and foreign scientist. Both projects relate to stochastic integrals and stochastic differential equations. The many types of stochastic integrals that exist at the moment have developed on an ad hoc basis, to fit the particular needs of the situation. The principal investigator will develop a general unifying theory based on semi-martingale integrators. It is hoped that in the process, the traditional assumption of asymptotic quasi-left continuity can be dropped. This will make the theory much more applicable. A particular application of interest is in finance theory for the extension of the Black-Scholes model. The second part of the project will deal with defining stochastic integrals with anticipating integrands. In traditional definition of stochastic integrals, the integrand (non-anticipating) is assumed to be dependent only on the past of the process with respect to which the integral is carried out. This makes the convergence properties of the integrals easy to prove. However, in many applications, such as in Volterra equations in Physics, the integrand cannot be assumed to be non- anticipating. The work will use many technical results that the principal investigator has developed in the past.

9401108 Protter This three-year continuing grant supports U.S.-France cooperative research in numerical analysis of stochastic differential equations (SDE). The investigators are Philip Protter of Purdue University, Thomas Kurtz of the University of Wisconsin, Denis Talay, French National Institute for Research in Computer Science and Applied Mathematics and Jean Jacod, University of Paris VI. The investigators have identified six topics for exploration, including studies of the discretization of stochastic differential equations and their flows; applications to numerical analysis of parabolic partial differential equations and to mathematical finance; and numerical analysis of interacting particle systems. Stochastic differential equations deal with the modeling of random dynamical systems and have been applied to models for statistical communication theory, finance theory, control theory among others. With the advent of computers, it is now important to analyze such models numerically. The project takes advantage of unique expertise of the INRIA researchers, who are world leaders in stochastic differential equations. The results of this work will further understanding of other numerical problems and advance our understanding of the special nature of stochastic differential equations. SDEs could eventually replace, for example, current financial models with more efficient and faster computational models. ***

This two-year US-France award supports the participation of US investigators in research workshops addressing numerical analysis of stochastic differential equations. The workshops, which take place in the US and France in 1998 and 1999 respectively, will be organized by Philip Protter of Purdue University and Denis Talay of INRIA (the French National Institute for Research in Informatics and Applied Mathematics). The workshops are aimed at recent developments in simulation methods, approximation and numerical schemes. The area of stochastic models has a variety of applications in economics (finance), interacting particle systems, population genetics, and mathematical biology. The proposed workshop will advance our knowledge of the use of stochastic modeling processes in these and other disciplines in physics and engineering.

9804750 Carriquiry This Americas Program award will fund a Latin American workshop on "Probability and Mathematical Statistics", to be held in Cordoba,Argentina, September 21-25, 1998, in conjunction with the Iberoamerican Congress of Statistics. Organizers are Drs. Alicia Carriquiry, Iowa State University, and Philip Protter, Purdue University, in collaboration with Dr. Victor Yohai, Universidad de Buenos Aires. The workshop will focus on the importance of statistical science and probability as fundamental components of scientific and social inquiry. The main objectives will be to discuss new developments and identify common research areas for international and Latin American researchers In order to achieve these objectives participants will address topics in statistics and probability with wide application in other areas of research. Coinciding with the improvement of economic conditions in Latin America during the past ten or fifteen years, the basic sciences have experienced a markedly improvement in quantity and quality in the region. Probability and Mathematical Statistics are at the forefront of this forward movement, mostly due to the regional and international collaborations resulting from challenging problems in areas such as environmental statistics, econometrics and social statistics. While not disregarding the more theoretical aspects of probability and statistics, the emphasis of this activity is on the application of the mathematical sciences to other areas of scientific and social inquiry, highlighting methodology and interdisciplinary collaboration. ***

The principal investigators will study various issues involving stochastic differential equations (hereafter SDEs). These include numerical approximations of expectations of both functions and functionals of solutions of SDEs, whether they be diffusions or more generally Markov processes with jumps. Models inspired by Stochastic Finance theory will receive attention; one example is forward-backward SDEs, where a general weak existence and uniqueness theorem is sought. Further a special emphasis is devoted to filtering theory where a new Monte Carlo approach is developed, and also a martingale problem approach will be tried to treat the infinite dimensional case. Stochastic partial differential equations will be studied via a new method of viscosity solutions, hopefully yielding existence, uniqueness and stability results in the fully nonlinear case. Quasi-linear backward SDEs will be studied with an eye to applications in Stochastic Finance theory; in addition several problems in Finance theory will be treated directly, including complete markets with jumps and competitive price equilibria in incomplete markets.
The principal investigators of this project study stochastic differential equations. A differential equation models phenomena that vary with time, dealing with rates of change. A stochastic differential equation includes the possibility that the change comes from random forces. Examples of applications can be found in modeling radio and x-ray transmissions, and cellular telephone signals, where the randomness comes from static noise; biological examples where growth of viruses and plants have a random component; and also in banking and finance where interest rate models, commodity prices, and securities prices are modeled with random components. The stochastic differential equations are complicated and sophisticated and cannot be solved with explicit solutions: the desired quantities need to be approximated by using a high speed computer. Efficient algorithms are needed and these methods are not yet well understood; by consequence the practitioner has a choice to use models that are too simple but with methods that work, or models that better approximate the true state of nature but for which methods work poorly, if at all. We propose to remedy this situation to a large extent by developing algorithms that work for the better models, and in addition we propose to quantify to what extent the models are effective.

A workshop on the future directions of probability will be held in Arlington, Virginia, May 29-31, 2002. The purpose of the workshop is to discuss the future of probability, both as a discipline and as a component of research in other areas of science and engineering. Participants in the workshop will include prominent researchers in probability theory, as well as researchers at the boundaries of probability with other areas. A general interest paper is expected to be generated by the workshop participants and disseminated widely.

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Protter
The principal investigator and co-principal investigator will investigate several topics in probability within the subfields of stochastic calculus and Monte Carlo simulation. The problems selected are motivated primarily by applications in mathematical finance, but the results will be of more general theoretical significance. In stochastic calculus, new results about stopping times will lead to better models of credit risk and a deeper understanding of what makes possible successful hedging of financial risks. New progress in stochastic calculus will help extend theories of incomplete markets, i.e. markets containing risks that can not be perfectly hedged, as is true in practice. One Monte Carlo issue to be addressed is the optimal use of an algorithm that has recently become popular in option pricing. Another line of research will involve improving numerical techniques for solutions of stochastic differential equations in connection with simulation. A third Monte Carlo topic is the optimal application of some variance reduction techniques that are well suited to problems in nuclear physics, estimation of rare events probabilities, and some option pricing problems. The recent development of a new model for stock prices that incorporates sizes of trade will lead to new pricing technology for financial derivatives.
The application of probability to finance has revolutionized an industry. In the past 20 years the creation of multi-trillion dollar derivative security markets has facilitated the world-wide flow of capital and thereby enhanced international commerce and productivity. Without the mathematical models which provide reliable pricing of derivative securities (e.g., stock options) and guide the management of their associated risk, these markets could not exist. The underlying theme of the mathematical success has been to compute precisely the price of financial derivatives which enable companies to lay off risk by buying financial instruments that protect them from unlikely but possibly disastrous events. Equally if not more important has been the description of a recipe for the seller of the instrument to follow in order to protect himself from the risk he accepts through the sale. A complete market is one in which the theory explains how to do this in principle, and in such a market the theory often provides an explicit guide to implementation of this recipe. In other words, in complete markets a new type of insurance has been created, and this has been made possible by existing probability theory. This type of "risk insurance" generated the revolution mentioned above. A real problem, however, is that in reality markets are not complete, and thus new mathematical techniques are needed to extend the theory and to make it more truly applicable. This has already begun, but it is in its infancy, and this extension of the theory will be a large focus of the proposed project. In addition, recently new models have been proposed to better incorporate liquidity issues and market frictions (such as transaction costs when implementing stock trades), in part by the PI himself. These models will continue to be developed, calibrated, and statistically verified.

In recent years, there has been a tremendous expansion in the use of probability models in finance, geosciences, neuroscience, artificial intelligence and communication networks in addition to an increase in its
use in traditional application areas in engineering disciplines, physics, ecology, genetics, and various fields of mathematics. This has created a strong demand for researchers trained in probability to develop new
methodologies and to work in an interdisciplinary context. We propose a variety of activities to meet these needs. Graduate fellowships will enhance the training of new researchers at Cornell, where a cohort of more
than 20 Ph.D. students exists and 4-6 students receive their Ph.D.'s in this subject each year. On a national scale, two yearly 2.5 day hot topics conferences will bring to Cornell a small group of established researchers and young investigators (in equal numbers). The conference will feature talks describing recent developments, and the young researchers will have a unique opportunity to discuss their research and open problems with the leaders in the field. A two week summer school featuring 4-6 lectures by three prominent researchers, two series of 1-3 on interdisciplinary opportunities, and a limited number of short talks by selected participants, will benefit a large number of graduate students and researchers (young and old) throughout the country.

This project will be under the direction of six probabilists from Math and Operations Research at Cornell (Durrett, Lawler, Protter, Resnick, Saloff-Coste, and Samorodnitsky). Their combined research covers a wide variety of topics in probability and its applications. However, to ensure that this is truly a national resource and covers all aspects of modern probability, they will receive advice from a nationwide committee of prominent researchers that represent a wide variety of specialties and many of the major probability groups throughout the country: David Aldous (U.C. Berkeley), Thomas Kurtz (U. of Wisconsin, Madison), Claudia Neuhauser (U. of Minnesota), Charles Newman (Courant Institute), Yuval Peres (U. C. Berkeley), Simon Tavare' (U. of Southern California).

Mathematical Finance not only has relevance to asset pricing, credit risk,
and the term structure of interest rates (for example), but also presents
to mathematicians a new way of looking at research in probability, since
the types of questions that need to be answered are no longer motivated by
physics or subjects such as electrical engineering. Building on the
foundation of the First Cornell Seminar in Finance, the Second will again
bring together leading researchers from the Northeast and selected points
beyond for an exchange of ideas in seminar format. The basic themes will
be asset pricing and hedging, viscosity solution based control theory,
credit risk, interest rates, and numerical methods and data analysis using
modern statistical techniques.

Fundamental research in finance has helped to change the world of commerce
during the last 30 years, and has arguably contributed to the explosive
growth in wealth. The key idea is the transfer of risk: modern financial
tools allow one party to sell an aspect of a risky exposure to another
party, and to do so at a fair price. Often a way for the buyer of the risk
to hedge against the risk (a hedging strategy) can also be provided in
terms of mathematical formulae. A problem is, however, that while the
models have led to huge advances, they are still often crude approximations
of reality, and also most of the advances have come from the creation of
financial derivatives in markets such as the stock market. Of current
interest is the development of analogous tools one could use in other
markets, such as credit risk. For example, two companies issuing similar
bonds may command different prices due to the markets' differing
assessments of their risk of not repaying the bonds. How to model this
risk mathematically is as yet poorly understood, and the subject of intense
research efforts. This conference will bring together leading scholars,
young researchers, and advanced graduate students for an intellectual
exchange involving both formal presentations of research advances and also
private discussions of special research topics. It will be held at Cornell
University, one of the nation's leading institutions of higher learning.

In this proposal we propose to study the mathematical framework of incomplete markets in Mathematical Finance Theory, and in particular in the equity markets (and markets with the same mathematical structure). Incomplete markets are characterized by there being an infinite number of risk neutral measures, and therefore there is not a unique price, for the typical contingent claim. Various methods have been proposed over the last 20 years or so to give a reasonable method to choose one risk neutral measure over the others, but they have always been unsatisfying, and ultimately arbitrary. A current popular approach is to use "indifference pricing," which involves economics arguments when choosing (still arbitrarily) risk neutral measure. We propose instead to try to "complete" incomplete markets, using a modification of the idea of Heath, Jarrow and Morton, who did an analogous feat for the bond market. The idea is to use the continuous time pricing of financial derivatives, such as options, which are traded in the markets, and which (by the theory) have to be local martingales. Therefore we need to show that there exist risk neutral measures that simultaneously make the underlying and its traded derivatives all martingales (or local martingales, or even sigma martingales); this can get quite technical, especially in cases where the maturity of the derivative is before the trading horizon.

In the trading of equities, such as stocks, an element which has become of fundamental importance are financial derivatives, such as options. An option allows one to transfer risk, such as (for example) portfolio exposure, or foreign currency risk, from one party to another, for a fee. In this way, it is like fire insurance on a home, where the home owner transfers the financial risk of losing his home to an insurance company willing to bear the risk, for of course, a fee. However unlike fire insurance or life insurance, it is quite complicated mathematically to calculate a fair price for a financial derivative. (It is also difficult to price some types of home insurance too, at times, such as storm insurance for homes in Florida and along the Gulf Coast of the US, and insurance companies are increasingly relying on "re-insurance," which can often be modeled in ways analogous to option pricing.) The models of Black, Scholes, and Merton, for which the latter two were awarded a Nobel prize, explain how to calculate a fair price in simple situations. We now know how to calculate fair prices in slightly more sophisticated models, known as "complete markets", however it is widely believed that the world is more complicated than the complete market case, and is in fact "incomplete." All of the many methods proposed to date to calculate a fair price in incomplete markets have been arbitrary, and in general not accepted either by academics, nor practitioners. This proposal might go a long way towards solving that problem, by using the market prices of the options themselves, together with the underlying market stock prices, essentially to make an incomplete market into a complete one.

This project will support research on probability problems that arise from a wide variety of fields. Topics from genetics include regulatory sequence evolution, gene duplication, and the analysis of data generated by cancer genome projects. In finance, understanding bubbles in markets, swing options, and credit risk associated with subprime mortgages require sophisticated mathematical ideas from control theory, backward stochastic differential equations, and enlargement of filtration. The study of financial markets, insurance risk, and communication and computer networks lead to stochastic models that exhibit long range dependence and heavy tails. The analysis of these processes and the new phenomena they present, involves techniques much different from the classical theory of independent random variables, making use of ideas from a wide variety of sources including ergodic theory. In addition, there is a fertile interaction between probability and a variety of topics that arise from algebra, geometry, and analysis, including the use of analytic techniques to study convergence rates of Markov chains.

Probability theory, born in the analysis of gambling games, now plays an important role in biology, physics, economics, finance, insurance, communication networks, and in many topics within mathematics. The main aim of this proposal is to train more researchers in probability to tackle the many important problems that arise from its applications. Cornell is an ideal place for these activities because of the strong traditions of research in probability and interdisciplinary work in applied mathematics. Support for summer research projects will show undergraduates that probability is an interesting research area with many important applications. At the graduate level, increased fellowship support will enhance the training of our Ph.D. students. Postdoctoral positions will help new graduates expand their horizons, while hot topics conferences and annual summer schools will be a national resource for broadening the education of researchers (young and old). In all of these activities, the focus will be on developing theory to treat problems that arise from applications.

The Principal Investigator Protter uses the idea that a filtration of sigma-algebras can model a history of available information. In applications, it is sometimes of importance to have different levels of available information, and this is reflected in containment relations among these sigma algebras (or histories of observable events). In economics, for example, there is a mathematical condition which indicates an absence of arbitrage opportunities (the possibility to make a profit without taking any risk), and this condition can hold or not, depending on the fine structure of the sigma algebras. This phenomenon is shown to happen, and will be investigated systematically. In a second part of the proposal, the P.I. proposes to study discretization procedures for the statistical estimation of various aspects of stochastic processes. These techniques have led to some delicate results, such as statistical tests to see whether or not dynamically evolving data arrives in a continuous stream, or has intrinsic jumps.
This will be a massive study, resulting in the publication of a book on the subject.

The P.I. proposes to develop statistical tests, using modern techniques as developed and explained in the recent book Discretization of Processes, Springer, 2012 that he co-authored with Jean Jacod, to determine if stochastic volatility models are superior to local volatility models, and for which kind of risky assets that might be true. While there is much indirect evidence this is the case, the P.I. proposes systematically to examine the question. In addition, the P.I. proposes to use recently developed techniques in the theory of the expansion of filtrations to study questions concerning mathematical models of insider trading. It is hoped that such an analysis could be of benefit to regulators trying to ensure equitable financial markets, by showing how insider trading affects the calculation of the risk neutral measure of the insider, and renders it different (thereby affecting option prices) from the risk neutral measure of the traditionally informed market.

Mathematical models of the evolution of stock prices are widely used on "Wall Street." While the models are justified by economic reasoning, there is a wide variety of them, and practitioners try to use models that they think correspond to reality. This is a difficult procedure, and mathematical/statistical techniques to check to see if one class of models is better than an alternative class currently do not exist in any comprehensive form. It is the purpose of this grant to develop systematically such procedures. This should lead to more accurate modeling not just for practitioners of the financial industry, but also it should benefit government regulators (such as the SEC, the CFTC, and the Federal Reserve) in their attempts to minimize excesses and corrupt practices. A second goal of this research is to provide a workable mathematical model of insider trading activity. In principle this should lead to the ability to detect insider trading activity as it happens in real time.

The award will support the PI's research on mathematical models of financial bubbles and of insider trading in the stock market. Using mathematical models one can obtain insights not normally available to mere intuitive knowledge of the markets, in rough analogy to how one can see more details and learn new things by looking at the heavens through a telescope rather than with the naked eye. The PI will treat models of credit risk, and methods currently in use by practitioners (for example banks and investment houses) to calculate the risk involved. The research plan makes the informed conjecture that when bubbles are present, the standard approximations used by financial modelers in the US and around the world are in fact significantly worse than is currently believed. The research will draw on delicate techniques in probability theory which have to do with the availability and flow of information under uncertainty; as a consequence, new mathematical results will be established which deepen our understanding of probability. The broader impact which will apply to finance will potentially result in tools available to bankers, investors, and regulators to understand and therefore increase national and global financial stability.

The classical way of looking at mathematical models of financial risk uses reduced-form models and attempts to approximate the hazard rate, which gives the likelihood of imminent default at a given time. The PI intends to investigate how fast numerical approximation methods converge for these reduced-form models in the presence of financial bubbles. This involves the numerical analysis of solutions of stochastic differential equations when the coefficients are neither Lipschitz-continuous, nor have linear growth in the space variable, creating technical challenges which should have significant numerical implications.

The subject of speculative pricing in financial markets, leading to what is commonly known as bubbles, is a topic of current concern. Its importance is underscored by the huge housing market bubble, which crashed in 2008, and which thereby caused extensive financial suffering. This proposal aims to continue a study of the mathematical modeling of bubbles in financial markets. The mathematical basis for modeling speculative pricing is that it provides the opportunities to quantify when bubble pricing is occurring and how big (in an appropriate sense) the bubble is, and perhaps even more importantly, to identify when a bubble is occurring, or not. Early steps in this direction were burdened by rather severe restrictions in the generality of the mathematical models. In this research, the PI will continue the analysis in a more general setting. The key is to drop the standard restriction of what is known as a "complete market" in favor of the more realistic situation involving "incomplete markets."

In the study of bubbles in incomplete markets, the role of strict local martingales will continue to be of paramount importance. The PI plans to tackle the issue of identifying models that lead to strict local martingales within incomplete markets, thus abandoning the wonderful but too simple framework first begun with the work of Delbaen and Shirakawa. The plan is to begin with the models of M. Musiela and P.L Lions, but then progress to multidimensional strict local martingales, using the theory of Lyapunov exponents as developed in the work of Narita, Khasminskii, Stroock, and Varadhan. The PI will also tackle some thorny numerical analysis issues created by the lack of linear growth, a problem inherent in the framework of strict local martingale models.

A major problem in financial circles, since 2008, is that two big banks ("too big to fail") actually can fail at the same time. We provide mathematical models to detect when this could happen. To do this we need to create new theory, beyond the traditional models for credit risk. It turns out that such mathematical models can easily be modified to model certain issues in the propagation of epidemics (such as the current COVID-19 pandemic). In particular, imagine that a group of people attend a super spreader event. Assuming more than a few will contract the disease, with a subset needing hospitalization, then - from the standpoint of health control and hospital capacity control - one might want to know the probability of two or more people getting the disease at once. It is important to note that two people exposed to the disease at the same event will contract the disease at different times (if at all), and the progress of the disease within their bodies will depend on a large number of factors, many of which are unknown, or impossible to quantify; hence the need for random modeling. The project will provide training opportunities and support for graduate students to be involved in the research.

In Credit Risk Theory, default times are typically modeled via a Cox construction, and for two different companies a standard assumption is that the stopping times are conditionally independent, give the underlying filtration of observable events. Such models do not allow, however, for simultaneous defaults, due to the use of independent exponential random variables used in the Cox constructions. We propose to replace the independent exponentials with multivariate exponentials, using (for example) the form proposed in 1967 by Marshall and Olkin. We will then use martingale orthogonality in place of conditional independence to make the desired calculations of different properties of the default times. This extension should be especially useful when modeling catastrophic credit events, such as the simultaneous default of two banks, both of them being "too big to fail." The other class of problems we propose to study is the modeling of the development of COVID-19 (or other epidemics) on an individual level. A key example is that if two people attend a "super spreader" event, what are the times after simultaneous exposure to the development of disease? Perhaps surprisingly this can be modeled in a near perfect analogy with the credit risk issues discussed above. Such models could be useful for, for example, hospital preparedness in a given locality.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.