Analysis Of Markov Chain Approximation For Diffusion Models With Non Smooth Coefficients
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Author | : Gongqiu Zhang |
Publisher | : |
Total Pages | : |
Release | : 2019 |
Genre | : |
ISBN | : |
Diffusion models with non-smooth coefficients often appear in financial applications, with examples including but not limited to threshold models for financial variables, the pricing of occupation time derivatives and shadow rate models for interest rate dynamics. To calculate the expected value of a discounted payoff under general state-dependent discounting and monitoring of barrier crossing, continuous time Markov chain (CTMC) approximation can be applied. In a recent work, Zhang and Li (2018, Operations Research, forthcoming) established sharp convergence rates of CTMC approximation for diffusion models with smooth coefficients but non-smooth payoff functions, and proposed grid design principles to ensure nice convergence behaviors. However, their theoretical analysis fails to obtain sharp convergence rates when model coefficients lack smoothness. Moreover, it is unclear how to design the grid of CTMC to remedy the inferior convergence behaviors resulting from non-smooth model coefficients. In this paper, we introduce new ways for the theoretical analysis of CTMC approximation for general diffusion models with non-smooth coefficients. We prove that convergence of option price is only first order in general. However, strikingly, if all the discontinuous points of the model coefficients and the payoff function are in the midway between two grid points, second order convergence in the maximum norm is restored and in this case, delta and gamma have second order convergence at almost all grid points except those next to the discontinuous points. Numerical experiments are conducted that confirm the validity of our theoretical results. We also compare the CTMC approximation approach with properly designed grids to a classical numerical PDE scheme for diffusion models with non-smooth coefficients, where the finite difference method is applied separately in each region with smooth coefficients and continuous pasting of the value function is enforced at the discontinuities. We show that our approach is superior to the latter in terms of both the convergence rate and the simplicity of implementation.
Author | : Daniel W. Stroock |
Publisher | : Springer |
Total Pages | : 338 |
Release | : 2007-02-03 |
Genre | : Mathematics |
ISBN | : 3540289992 |
From the reviews: "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik
Author | : Jean-Dominique Deuschel |
Publisher | : |
Total Pages | : 40 |
Release | : 2011 |
Genre | : Markov processes |
ISBN | : |
Author | : Amarjit Budhiraja |
Publisher | : Springer |
Total Pages | : 577 |
Release | : 2019-08-10 |
Genre | : Mathematics |
ISBN | : 1493995790 |
This book presents broadly applicable methods for the large deviation and moderate deviation analysis of discrete and continuous time stochastic systems. A feature of the book is the systematic use of variational representations for quantities of interest such as normalized logarithms of probabilities and expected values. By characterizing a large deviation principle in terms of Laplace asymptotics, one converts the proof of large deviation limits into the convergence of variational representations. These features are illustrated though their application to a broad range of discrete and continuous time models, including stochastic partial differential equations, processes with discontinuous statistics, occupancy models, and many others. The tools used in the large deviation analysis also turn out to be useful in understanding Monte Carlo schemes for the numerical approximation of the same probabilities and expected values. This connection is illustrated through the design and analysis of importance sampling and splitting schemes for rare event estimation. The book assumes a solid background in weak convergence of probability measures and stochastic analysis, and is suitable for advanced graduate students, postdocs and researchers.
Author | : Harold J. Kushner |
Publisher | : |
Total Pages | : 52 |
Release | : 1979 |
Genre | : |
ISBN | : |
Many problems in communication theory involve approximations of a Markov type to outputs of non-linear (feedback or not) systems, often so that Fokker-Planck techniques can be used. A general and powerful method is presented for getting diffusion approximations to outputs of systems with wide band inputs. The input is parameterized by epsilon and as epsilon approaches 0 the band width goes to infinity (loosely speaking). It is proved, under reasonable conditions on the systems and noise, that the sequence of system output processes converges weakly to a Markov diffusion process, which is characterized completely. Many communication systems fit the model of the paper and, in order to make mathematical sense out of many common developments of system properties, assumptions such as those of this paper are often required. The usefulness and relative ease of use of the method is illustrated by application to three examples: (a) phase locked loop, where a Markov diffusion approximation of the error process is developed, (b) adaptive antenna system, where an asymptotic analysis of the equations for the system is given, (c) di-diffusion approximation to the output of a hard limiter followed by a band pass filter; input-output S/N ratios are developed (a version of a classical problem of Davenport). Difficulties with the usual heuristic approaches to (a), (b) are discussed. The method is versatile and the models quite general. Since weak convergence methods are used, the approximate limits yield approximations to many types of functionals of the actual system. (Author).
Author | : Vassili N. Kolokoltsov |
Publisher | : Cambridge University Press |
Total Pages | : 394 |
Release | : 2010-07-15 |
Genre | : Mathematics |
ISBN | : 1139489739 |
A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.
Author | : Dominique Bakry |
Publisher | : |
Total Pages | : 574 |
Release | : 2013-10-31 |
Genre | : |
ISBN | : 9783319002286 |
Author | : David Freedman |
Publisher | : Springer |
Total Pages | : 168 |
Release | : 1983 |
Genre | : Mathematics |
ISBN | : |
A long time ago I started writing a book about Markov chains, Brownian motion, and diffusion. I soon had two hundred pages of manuscript and my publisher was enthusiastic. Some years and several drafts later, I had a thousand pages of manuscript, and my publisher was less enthusiastic. So we made it a trilogy: Markov Chains Brownian Motion and Diffusion Approximating Countable Markov Chains familiarly - MC, B & D, and ACM. I wrote the first two books for beginning graduate students with some knowledge of probability; if you can follow Sections 10.4 to 10.9 of Markov Chains, you're in. The first two books are quite independent of one another, and completely independent of this one, which is a monograph explaining one way to think about chains with instantaneous states. The results here are supposed to be new, except when there are specific disclaimers. It's written in the framework of Markov chains; we wanted to reprint in this volume the MC chapters needed for reference. but this proved impossible. Most of the proofs in the trilogy are new, and I tried hard to make them explicit. The old ones were often elegant, but I seldom saw what made them go. With my own, I can sometimes show you why things work. And, as I will argue in a minute, my demonstrations are easier technically. If I wrote them down well enough, you may come to agree.
Author | : Kushner |
Publisher | : Academic Press |
Total Pages | : 263 |
Release | : 1977-04-14 |
Genre | : Computers |
ISBN | : 0080956386 |
Probability Methods for Approximations in Stochastic Control and for Elliptic Equations
Author | : Alexei Kulik |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 268 |
Release | : 2017-11-20 |
Genre | : Mathematics |
ISBN | : 3110458934 |
The general topic of this book is the ergodic behavior of Markov processes. A detailed introduction to methods for proving ergodicity and upper bounds for ergodic rates is presented in the first part of the book, with the focus put on weak ergodic rates, typical for Markov systems with complicated structure. The second part is devoted to the application of these methods to limit theorems for functionals of Markov processes. The book is aimed at a wide audience with a background in probability and measure theory. Some knowledge of stochastic processes and stochastic differential equations helps in a deeper understanding of specific examples. Contents Part I: Ergodic Rates for Markov Chains and Processes Markov Chains with Discrete State Spaces General Markov Chains: Ergodicity in Total Variation MarkovProcesseswithContinuousTime Weak Ergodic Rates Part II: Limit Theorems The Law of Large Numbers and the Central Limit Theorem Functional Limit Theorems