scholarly journals Interloss Time in Loss System

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Pierpaolo Ferrante
Keyword(s):  
Cauchy Problem ◽  
Probability Density Function ◽  
Differential Equations ◽  
Laplace Transform ◽  
Probability Density ◽  
Density Function ◽  
Loss Rate ◽  
Loss System ◽  
Second Order Differential Equations ◽  
Erlang Loss

We consider the interloss times in the Erlang Loss System. Here we present the explicit form of the probability density function of the time spent between two consecutive losses in the model. This density function solves a Cauchy problem for the second-order differential equations, which was used to evaluate the corresponding laplace transform. Finally the connection between the Erlang's loss rate and the evaluated probability density function is showed.

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 On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

scholarly journals On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (01) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

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 A Directed Continuous Time Random Walk Model with Jump Length Depending on Waiting Time

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Long Shi ◽  
Zuguo Yu ◽  
Zhi Mao ◽  
Aiguo Xiao
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Density Function ◽  
Waiting Time ◽  
Continuous Time ◽  
Dimensional Space ◽  
Continuous Time Random Walk ◽  
Random Walk Model

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density functionP(x,t)of finding the walker at positionxat timetis completely determined by the Laplace transform of the probability density functionφ(t)of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.

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