*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/9780199657063.003.0007
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

The martingale problem due to Stroock and Varadhan provides another way to define a solution of a stochastic differential equation. It is a concept that is unique to stochastic differential equation ...
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The martingale problem due to Stroock and Varadhan provides another way to define a solution of a stochastic differential equation. It is a concept that is unique to stochastic differential equation in the sense that it has no counterpart in the theory of ordinary and partial differential equations. Under this approach, existence and uniqueness of solutions of stochastic differential equations can be proved under milder conditions. First, we obtain equivalent formulations of martingale problems, and then proceed to establish existence of a solution to the martingale problem. Uniqueness of solutions is shown using certain analytical tools and Laplace transforms. Further extensions and the Markov property of solutions are discussed.Less

The martingale problem due to Stroock and Varadhan provides another way to define a solution of a stochastic differential equation. It is a concept that is unique to stochastic differential equation in the sense that it has no counterpart in the theory of ordinary and partial differential equations. Under this approach, existence and uniqueness of solutions of stochastic differential equations can be proved under milder conditions. First, we obtain equivalent formulations of martingale problems, and then proceed to establish existence of a solution to the martingale problem. Uniqueness of solutions is shown using certain analytical tools and Laplace transforms. Further extensions and the Markov property of solutions are discussed.

*Gopinath Kallianpur and P Sundar*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.001.0001
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local ...
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Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.Less

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.

*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.