Author: Anders SzepessyPublisher:
ISBN:
Category :
Languages : en
Pages : 26
Book Description
Adaptive Weak Approximation of Stochastic Differential Equations
Author: Anders SzepessyWeak Approximation of Itô Stochastic Differential Equations and Related Adaptive Algorithms
Author: Adaptive Concepts for High-dimensional Stochastic Differential Equations
Author: Fabian MerlePublisher:
ISBN:
Category :
Languages : en
Pages : 0
Book Description
The objective of this thesis is the efficient approximation of high-dimensional stochastic differential equations (SDE's) via newly developed, theoretical-based adaptive methods. The thesis is split into two parts, which motivate and discuss the (temporal) approximation of high-dimensional SDE's from different aspects. Conceptually, the derivation of the corresponding adaptive methods follows the same principle: finding an appropriate scheme for the approximation of the underlying SDE, derivation of a (weak) a posteriori error estimate, and an implementation of an adaptive method based on it. In the first part of this thesis we mainly consider SDE systems emerging from a spatial discretization of a given semilinear stochastic partial differential equation (SPDE). The corresponding adaptive method consists of the semi-implicit Euler scheme and a local refinement/coarsening strategy of the temporal mesh based on a computable error estimator, and generates time step sizes as well as iterates, such that the resulting (weak) error is always less or equal than a prescribed tolerance. The (computable) error estimator directly comes from the related a posteriori error estimate, which is derived by means of the Kolmogorov equation. In this regard, we (globally) bound derivatives of the solution of Kolmogorov's equation via (probabilistic) variation equations independently of the dimension and in terms of derivatives of the underlying test function. At this juncture, the use of the Clark-Ocone formula reduces the complexity of the derivatives to be bounded. Furthermore, the approximation via the semi-implicit Euler scheme allows for stability bounds which are independent of the dimension, and which, in particular, contribute to bound the error estimator. The combination of the above concepts enables an error analysis of the a posteriori estimate resp.~the estimator, which is independent of the dimension, and, in particular, is the key for convergence of the adaptive method, as well as its applicability in high dimensions. Computational experiments compare adaptive meshes with uniform meshes and show a considerable gain in efficiency of the adaptive method. The second part can conceptually be regarded as an extension of the first one and considers SDE systems, which arise from the probabilistic reformulation of an underlying boundary value problem, i.e., of an elliptic/parabolic partial differential equation (PDE) on a bounded domain. Opposed to the setting in the first part, the solution of the SDE here takes values in a bounded domain, which, in particular, involves a convenient exposure to stopping in an approximative framework when the (approximated) solution process is about to leave the domain. To this end, we use an already existing scheme in the literature (slightly modified), which, among other things, replaces unbounded Wiener increments in the generation of (explicit) Euler iterates by bounded ones having the same distribution, and which thus allows to properly control the dynamics of the (approximated) solution process up to the boundary of the domain. Based on this scheme, we derive an a posteriori error estimate from which three error estimators emerge, where each of them captures different dynamics concerning the distance of the approximated process to the boundary. These dynamics are especially reflected in the choice of the local time step size selection (up to the boundary) of the adaptive method, which approximates the solution of the underlying boundary value problem at a fixed point. The choice of the local time step sizes is complemented by a suitable temporal weight factor within the related refinement/coarsening strategy, which, aside from stability results concerning stopping dynamics, ensures the (optimal) convergence of the method with respect to a given tolerance parameter. Computational experiments illustrate a stable application of the method even for violated data requirements, and a substantial gain in efficiency through adaptive (time) mesh generation.
Applied Stochastic Differential Equations
Author: Simo SärkkäPublisher: Cambridge University Press
ISBN: 1316510085
Category : Business & Economics
Languages : en
Pages : 327
Book Description
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
Optimal approximation of stochastic differential equations by adaptive step size control
Author: Norbert HofmannWeak Approximation of Stochastic Differential Equations by Discrete Time Series
Author: Numerical Solution of Stochastic Differential Equations with Jumps in Finance
Author: Eckhard PlatenPublisher: Springer Science & Business Media
ISBN: 364213694X
Category : Mathematics
Languages : en
Pages : 868
Book Description
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.
Theory and Numerics of Differential Equations
Author: James BloweyPublisher: Springer Science & Business Media
ISBN: 9783540418467
Category : Mathematics
Languages : en
Pages : 336
Book Description
A compilation of detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to quickly gain an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research is given.
Weak approximation for stochastic differential equations with small noises
Author: Grigorij N. MilʹstejnPublisher:
ISBN:
Category :
Languages : de
Pages : 49
Book Description
Recent Advances in Adaptive Computation
Author: Zhongci ShiPublisher: American Mathematical Soc.
ISBN: 0821836625
Category : Computers
Languages : en
Pages : 394
Book Description
There has been rapid development in the area of adaptive computation over the past decade. The International Conference on Recent Advances in Adaptive Computation was held at Zhejiang University (Hangzhou, China) to explore these new directions. The conference brought together specialists to discuss modern theories and practical applications of adaptive methods. This volume contains articles reflecting the invited talks given by leading mathematicians at the conference. It is suitable for graduate students and researchers interested in methods of computation.