Ilya Molchanov
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0017
- Subject:
- Mathematics, Geometry / Topology
This chapter surveys several examples where random sets appear in mathematical finance and econometrics: trading with transaction costs, risk measures, option prices, and partially identified ...
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This chapter surveys several examples where random sets appear in mathematical finance and econometrics: trading with transaction costs, risk measures, option prices, and partially identified econometric models.Less
This chapter surveys several examples where random sets appear in mathematical finance and econometrics: trading with transaction costs, risk measures, option prices, and partially identified econometric models.
Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691190716
- eISBN:
- 9780691193717
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691190716.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical ...
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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.Less
This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
Michael Harris
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175836
- eISBN:
- 9781400885527
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175836.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
The explosion of finance mathematics, and its implication in the 2008 financial crisis, has had the welcome, but unintended, consequence of establishing a common border between mathematics and ...
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The explosion of finance mathematics, and its implication in the 2008 financial crisis, has had the welcome, but unintended, consequence of establishing a common border between mathematics and morality. This chapter does not aim to assign responsibility for the 2008 crash and certainly not to imply that mathematics professors are specifically to blame. Nor does this chapter aim to change anyone's mind about fundamental questions of economic policy. Its primary purpose is to explain some of the context for a debate that is actually taking place, within and around mathematics, in connection with the growth of mathematical finance. The tensions between the internal and external goods involved in the creation of mathematics are well illustrated by this debate. The secondary purpose of this chapter is to provide a very brief introduction to the mathematical modeling of reality.Less
The explosion of finance mathematics, and its implication in the 2008 financial crisis, has had the welcome, but unintended, consequence of establishing a common border between mathematics and morality. This chapter does not aim to assign responsibility for the 2008 crash and certainly not to imply that mathematics professors are specifically to blame. Nor does this chapter aim to change anyone's mind about fundamental questions of economic policy. Its primary purpose is to explain some of the context for a debate that is actually taking place, within and around mathematics, in connection with the growth of mathematical finance. The tensions between the internal and external goods involved in the creation of mathematics are well illustrated by this debate. The secondary purpose of this chapter is to provide a very brief introduction to the mathematical modeling of reality.
Alex Preda
- Published in print:
- 2009
- Published Online:
- February 2013
- ISBN:
- 9780226679310
- eISBN:
- 9780226679334
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226679334.003.0004
- Subject:
- Sociology, Culture
This chapter examines the emergence and consequences of a vernacular “science of financial investments.” While many eighteenth-century writers saw financial knowledge as devilish and destructive ...
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This chapter examines the emergence and consequences of a vernacular “science of financial investments.” While many eighteenth-century writers saw financial knowledge as devilish and destructive (centered upon the bodily and verbal skills required by street transactions), these new authors set out to build a science of investments grounded in observation and calculation. Among the main outcomes of this process are the rationalization of investor behavior and the representation of financial markets as supra-individual, quasi-natural entities, which cannot be controlled by any group. It is the latter notion which allowed for the shift to price behavior as the core actor of abstract market models. The effort to transform investment knowledge into a science is crowned by the formulation of basic views of the random walk hypothesis. The first mathematical formulation of the random walk hypothesis plays a decisive role in the development of mathematical finance (more specifically, of the options pricing theory). The main tenet of the random walk hypothesis is that securities prices move independently of each other, and that future movements do not depend on past movements. One of the most important implications of this hypothesis is that in the long run, the market cannot be controlled by any group or person.Less
This chapter examines the emergence and consequences of a vernacular “science of financial investments.” While many eighteenth-century writers saw financial knowledge as devilish and destructive (centered upon the bodily and verbal skills required by street transactions), these new authors set out to build a science of investments grounded in observation and calculation. Among the main outcomes of this process are the rationalization of investor behavior and the representation of financial markets as supra-individual, quasi-natural entities, which cannot be controlled by any group. It is the latter notion which allowed for the shift to price behavior as the core actor of abstract market models. The effort to transform investment knowledge into a science is crowned by the formulation of basic views of the random walk hypothesis. The first mathematical formulation of the random walk hypothesis plays a decisive role in the development of mathematical finance (more specifically, of the options pricing theory). The main tenet of the random walk hypothesis is that securities prices move independently of each other, and that future movements do not depend on past movements. One of the most important implications of this hypothesis is that in the long run, the market cannot be controlled by any group or person.
Charles L. Epstein and Rafe Mazzeo
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.001.0001
- Subject:
- Mathematics, Probability / Statistics
This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as ...
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This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.Less
This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.
