scholarly journals The asymptotic error of chaos expansion approximations for stochastic differential equations

2019 ◽  
pp. 145-165 ◽  
Author(s):  
Tony Huschto ◽  
Mark Podolskij ◽  
Sebastian Sager
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chaos Expansion ◽  
Asymptotic Error

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

2015 ◽  
Vol 52 (01) ◽  
pp. 149-166 ◽  
Author(s):  
Hanchao Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Euler Scheme ◽  
Asymptotic Error ◽  
Error Distributions ◽  
Itô Semimartingale

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

scholarly journals The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

2015 ◽  
Vol 52 (1) ◽  
pp. 149-166 ◽  
Author(s):  
Hanchao Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Euler Scheme ◽  
Asymptotic Error ◽  
Error Distributions ◽  
Itô Semimartingale

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

scholarly journals Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
M. A. Alshanskiy
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Calculus Of Variations ◽  
Hilbert Spaces ◽  
Malliavin Calculus ◽  
Random Variables ◽  
Random Variable ◽  
Chaos Expansion

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

scholarly journals Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part II

2011 ◽  
Vol 90 (104) ◽  
pp. 85-98 ◽  
Author(s):  
Tijana Levajkovic ◽  
Dora Selesi
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Chaos Expansion ◽  
Malliavin Derivative ◽  
White Noise Space

We solve stochastic differential equations involving the Malliavin derivative and the fractional Malliavin derivative by means of a chaos expansion on a general white noise space (Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise space). There exist unitary mappings between the Gaussian and Poissonian white noise spaces, which can be applied in solving SDEs.

THE GROSS DERIVATIVE AND GENERALIZED RANDOM VARIABLES

1999 ◽  
Vol 02 (03) ◽  
pp. 381-396 ◽  
Author(s):  
FRED ESPEN BENTH
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Random Variable ◽  
Representation Formula ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Chaos Expansion ◽  
Schwartz Distributions

We extend the Gross derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of the properties of the Gross derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the Hida derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochastic differential equations of Wick type.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law
Sign in / Sign up

Export Citation Format

Share Document