Numerical Solution of Stochastic Differential Equations

Numerical Solution of Stochastic Differential Equations

This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations.

Author: Peter E. Kloeden

Publisher: Springer Science & Business Media

ISBN: 9783662126165

Category: Mathematics

Page: 636

View: 457

The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. From the reviews: "The authors draw upon their own research and experiences in obviously many disciplines... considerable time has obviously been spent writing this in the simplest language possible." --ZAMP
Categories: Mathematics

Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach.

Author: Eckhard Platen

Publisher: Springer Science & Business Media

ISBN: 9783642136948

Category: Mathematics

Page: 856

View: 265

In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.
Categories: Mathematics

Continuous Time Random Walks for the Numerical Solution of Stochastic Differential Equations

Continuous Time Random Walks for the Numerical Solution of Stochastic Differential Equations

[64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] D. J. Higham, X. Mao, and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, ...

Author: Nawaf Bou-Rabee

Publisher: American Mathematical Soc.

ISBN: 9781470431815

Category: Random walks (Mathematics)

Page: 124

View: 396

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs.
Categories: Random walks (Mathematics)

Numerical Solution of SDE Through Computer Experiments

Numerical Solution of SDE Through Computer Experiments

This book provides an easily accessible, computationally-oriented introduction into the numerical solution of stochastic differential equations using computer experiments.

Author: Peter Eris Kloeden

Publisher: Springer Science & Business Media

ISBN: 9783642579134

Category: Mathematics

Page: 294

View: 259

This book provides an easily accessible, computationally-oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations. It also creates an intuitive understanding of the necessary theoretical background. Software containing programs for over 100 problems is available online.
Categories: Mathematics

A Brief Analysis of Certain Numerical Methods Used to Solve Stochastic Differential Equations

A Brief Analysis of Certain Numerical Methods Used to Solve Stochastic Differential Equations

Since there has been an expansion in the range and volume of interest raterelated products being traded in the international financial markets in the past decade, it has become important for investment banks, other financial institutions, ...

Author: Nadrajh Govender

Publisher:

ISBN: OCLC:145811886

Category: Mathematical models

Page: 230

View: 955

Categories: Mathematical models

From Elementary Probability to Stochastic Differential Equations with MAPLE

From Elementary Probability to Stochastic Differential Equations with MAPLE

... S.: From Elementary Probability to Stochastic Differential Equations with MAPLE Kloeden, P. E.; Platen; E.; Schurz, H.; Numerical Solution of SDE Through Computer Experiments Kostrikin, A.I.: Introduction to Algebra Krasnoselskii, ...

Author: Sasha Cyganowski

Publisher: Springer Science & Business Media

ISBN: 9783642561443

Category: Mathematics

Page: 310

View: 466

This is an introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. Based on measure theory, which is introduced as smoothly as possible, it provides practical skills in the use of MAPLE in the context of probability and its applications. It offers to graduates and advanced undergraduates an overview and intuitive background for more advanced studies.
Categories: Mathematics

Numerical Solutions of Stochastic Differential Equations

Numerical Solutions of Stochastic Differential Equations

At last, a recent progress of the strong convergence of the numerical solutions of stochastic differential equations driven by Lévy processes under non-globally Lipschitz conditions is also presented.

Author: Liguo Wang

Publisher:

ISBN: OCLC:981462351

Category: Brownian motion processes

Page: 141

View: 350

In this dissertation, we consider the problem of simulation of stochastic differential equations driven by Brownian motions or the general Lévy processes. There are two types of convergence for a numerical solution of a stochastic differential equation, the strong convergence and the weak convergence. We first introduce the strong convergence of the tamed Euler-Maruyama scheme under non-globally Lipschitz conditions, which allow the polynomial growth for the drift and diffusion coefficients. Then we prove a new weak convergence theorem given that the drift and diffusion coefficients of the stochastic differential equation are only twice continuously differentiable with bounded derivatives up to order 2 and the test function are third order continuously differentiable with all of its derivatives up to order 3 satisfying a polynomial growth condition. We also introduce the multilevel Monte Carlo method, which is efficient in reducing the total computational complexity of computing the expectation of a functional of the solution of a stochastic differential equation. This method combines the three sides of the simulation of stochastic differential equations: the strong convergence, the weak convergence and the Monte Carlo method. At last, a recent progress of the strong convergence of the numerical solutions of stochastic differential equations driven by Lévy processes under non-globally Lipschitz conditions is also presented.
Categories: Brownian motion processes