Solving Random Ordinary and Partial Differential Equations Through the Probability Density Function: Theory and Computing with Applications

2018 ◽  
pp. 261-282 ◽  
Author(s):  
J. Calatayud ◽  
J.-C. Cortés ◽  
M. Jornet ◽  
A. Navarro-Quiles
Keyword(s):  
Partial Differential Equations ◽  
Probability Density Function ◽  
Differential Equations ◽  
Probability Density ◽  
Density Function ◽  
Function Theory ◽  
Density Function Theory ◽  
Partial Differential

MULTIVARIATE PARTIAL DIFFERENTIAL EQUATION DESCRIBING THE EVOLUTION OF A GAUSSIAN PROCESS

2009 ◽  
Vol 09 (04) ◽  
pp. 493-518
Author(s):  
MURAD S. TAQQU ◽  
MARK VEILLETTE
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Differential Equations ◽  
Gaussian Process ◽  
Probability Density ◽  
Density Function ◽  
Single Equation ◽  
Marginal Probability ◽  
Partial Differential ◽  
Finite Dimensional

If {X(t), t ≥ 0} is a Gaussian process, the diffusion equation characterizes its marginal probability density function. How about finite-dimensional distributions? For each n ≥ 1, we derive a system of partial differential equations which are satisfied by the probability density function of the vector (X(t1), …, X(tn)). We then show that these differential equations determine uniquely the finite-dimensional distributions of Gaussian processes. We also discuss situations where the system can be replaced by a single equation, which is either one member of the system, or an aggregate equation obtained by summing all the equations in the system.

scholarly journals Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
M.-C. Casabán ◽  
J.-C. Cortés ◽  
J.-V. Romero ◽  
M.-D. Roselló
Keyword(s):  
Stochastic Process ◽  
Probability Density Function ◽  
Differential Equations ◽  
Probability Density ◽  
Density Function ◽  
Initial Value Problems ◽  
Random Variable ◽  
Initial Value ◽  
Variable Transformation ◽  
Comprehensive Study

Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

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 Function theory for a beltrami algebra

1985 ◽  
Vol 8 (2) ◽  
pp. 247-256
Author(s):  
B. A. Case
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic System ◽  
Analytic Functions ◽  
Function Theory ◽  
Real Parameter ◽  
Partial Differential ◽  
Complex Functions ◽  
Parameter Function

Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functionsfassociated with a particualr parameter functiongon a domainDform a Beltrami algebra denoted by the pair(D,g)and a function theory is developed in this algebra. A strong conformality property holds for all functions in a(D,g)algebra. Forg≡|z|=rthe algebra(D,r)is that of the analytic functions.

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