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.

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 ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350005
Author(s):  
SOLESNE BOURGUIN ◽  
JEAN-CHRISTOPHE BRETON
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Independent Random Variables ◽  
Asymptotic Decomposition ◽  
Type Decomposition

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250018 ◽  
Author(s):  
ALBERTO LANCONELLI ◽  
LUIGI SPORTELLI
Keyword(s):  
Gaussian Measure ◽  
Type Inequality ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Probabilistic Interpretation ◽  
Chi Square ◽  
Wick Calculus

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

scholarly journals An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

2020 ◽  
Vol 5 (1) ◽  
pp. 337-348 ◽  
Author(s):  
Nihal İnce ◽  
Aladdin Shamilov
Keyword(s):  
Probability Density Function ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Density Function ◽  
Model Fitting ◽  
Random Variable ◽  
Optimization Methods ◽  
Fixed Time ◽  
New Method

AbstractIn this study, a new method to obtain approximate probability density function (pdf) of random variable of solution of stochastic differential equations (SDEs) by using generalized entropy optimization methods (GEOM) is developed. By starting given statistical data and Euler–Maruyama (EM) method approximating SDE are constructed several trajectories of SDEs. The constructed trajectories allow to obtain random variable according to the fixed time. An application of the newly developed method includes SDE model fitting on weekly closing prices of Honda Motor Company stock data between 02 July 2018 and 25 March 2019.

scholarly journals Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140679 ◽  
Author(s):  
Eike H. Müller ◽  
Rob Scheichl ◽  
Tony Shardlow
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Operator Splitting ◽  
Random Variables ◽  
Weak Approximation ◽  
Multilevel Monte Carlo ◽  
Discrete Random Variables ◽  
Modified Equations

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

scholarly journals Diffusion Processes Satisfying a Conservation Law Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
J. Bakosi ◽  
J. R. Ristorcelli
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conservation Law ◽  
Evolution Equations ◽  
Diffusion Processes ◽  
Random Variables ◽  
Planck Equation ◽  
Equation Model ◽  
First Four Moments ◽  
And Diffusion

We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

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.

scholarly journals Multiplexing: B

2021 ◽  
pp. 59-69
Author(s):  
Jean Walrand
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Function ◽  
Central Limit ◽  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wifi Networks ◽  
The Central Limit Theorem ◽  
Communication Links

AbstractChapter 10.1007/978-3-030-49995-2_3 used the Central Limit Theorem to determine the number of users that can safely share a common cable or link. We saw that this result is also fundamental to calculate confidence intervals. In this section, we prove this theorem. A key tool is the characteristic function that provides a simple way to study sums of independent random variables.Section 4.1 introduces the characteristic function and calculates it for a Gaussian random variable. Section 4.2 uses that function to prove the Central Limit Theorem. Section 4.3 uses the characteristic function to calculate the moments of a Gaussian random variable. The sum of squares of Gaussian random variables is a common model of noise in communication links. Section 4.4 proves a remarkable property of such a sum. Section 4.5 shows how to use characteristic functions to approximate binomial and geometric random variables. The error function arises in the calculation of the probability of errors in transmission systems and also in decisions based on random observations. Section 4.6 derives useful approximations of that function. Section 4.7 concludes the chapter with a discussion of an adaptive multiple access protocol similar to one used in WiFi networks.

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