*Gopinath Kallianpur and P. Sundar*

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.003.0008
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

The connection between stochastic differential equations and partial differential equations is discussed in this chapter. The Dirichlet problem is studied and several examples are presented. Next, ...
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The connection between stochastic differential equations and partial differential equations is discussed in this chapter. The Dirichlet problem is studied and several examples are presented. Next, the heat equation, and the Feynman-Kac formula are discussed. The Kolmogorov forward and backward equations are derived after proving the smoothness of solutions in the mean square sense. Applications to pricing of derivatives in finance theory are given.Less

The connection between stochastic differential equations and partial differential equations is discussed in this chapter. The Dirichlet problem is studied and several examples are presented. Next, the heat equation, and the Feynman-Kac formula are discussed. The Kolmogorov forward and backward equations are derived after proving the smoothness of solutions in the mean square sense. Applications to pricing of derivatives in finance theory are given.

*Daniel L. Hartl*

- Published in print:
- 2020
- Published Online:
- August 2020
- ISBN:
- 9780198862291
- eISBN:
- 9780191895074
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198862291.003.0006
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies, Evolutionary Biology / Genetics

Chapter 6 deals with the consequences of random genetic drift in finite populations and includes details of the diffusion approximations and their solutions as well as conditional diffusion ...
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Chapter 6 deals with the consequences of random genetic drift in finite populations and includes details of the diffusion approximations and their solutions as well as conditional diffusion processes. It includes probabilities of fixation and conditional times to fixation for neutral and nonneutral alleles. Various scenarios of mutation, migration, and selection are examined with regard to the stationary distributions of allele frequency. The Ewens sampling formula and its importance is discussed, as well as its implications for the distribution of the number of alleles in samples. An analysis of allozyme polymorphisms supports the hypothesis that most amino acid polymorphisms in natural populations are slightly deleterious.Less

Chapter 6 deals with the consequences of random genetic drift in finite populations and includes details of the diffusion approximations and their solutions as well as conditional diffusion processes. It includes probabilities of fixation and conditional times to fixation for neutral and nonneutral alleles. Various scenarios of mutation, migration, and selection are examined with regard to the stationary distributions of allele frequency. The Ewens sampling formula and its importance is discussed, as well as its implications for the distribution of the number of alleles in samples. An analysis of allozyme polymorphisms supports the hypothesis that most amino acid polymorphisms in natural populations are slightly deleterious.

*Tomas Björk*

- Published in print:
- 2019
- Published Online:
- February 2020
- ISBN:
- 9780198851615
- eISBN:
- 9780191886218
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851615.003.0005
- Subject:
- Economics and Finance, Econometrics

In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most ...
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In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.Less

In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.

*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.