ROBUST FILTERING AND DETECTION OF AN INSURANCE MODEL

2007 ◽  
Vol 07 (01) ◽  
pp. 91-102 ◽  
Author(s):  
LAKHDAR AGGOUN
Keyword(s):  
Differential Equations ◽  
Stochastic Models ◽  
Point Processes ◽  
Risk Theory ◽  
Robust Filtering ◽  
Stochastic Integration ◽  
Competing Models ◽  
Rate Processes ◽  
The Difference ◽  
Insurance Model

Risk theory deals with stochastic models in insurance business. Usually, in such models claims are described by point processes and the amounts claimed by policy holders are sequences of random variables. The profit or loss, of the company is the difference between premiums income and the claims. We assume that we have a certain number of competing models, describing the claims and premiums rate processes. We are interested in ranking the candidate models based on their likelihood of being most appropriate for describing these processes. We compute robust dynamics for our estimates. In these new dynamics stochastic integration disappear and stochastic differential equations become ordinary differential equations.

scholarly journals Modeling and Analysis of BDS-2 and BDS-3 Combined Precise Time and Frequency Transfer Considering Stochastic Models of Inter-System Bias

2021 ◽  
Vol 13 (4) ◽  
pp. 793
Author(s):  
Guoqiang Jiao ◽  
Shuli Song ◽  
Qinming Chen ◽  
Chao Huang ◽  
Ke Su ◽  
...  
Keyword(s):  
Stochastic Models ◽  
Satellite System ◽  
Modeling And Analysis ◽  
System Bias ◽  
Frequency Transfer ◽  
Global Navigation ◽  
The Difference ◽  
The Stability ◽  
Global Navigation Satellite ◽  
Precise Time

BeiDou global navigation satellite system (BDS) began to provide positioning, navigation, and timing (PNT) services to global users officially on 31 July, 2020. BDS constellations consist of regional (BDS-2) and global navigation satellites (BDS-3). Due to the difference of modulations and characteristics for the BDS-2 and BDS-3 default civil service signals (B1I/B3I) and the increase of new signals (B1C/B2a) for BDS-3, a systemically bias exists in the receiver-end when receiving and processing BDS-2 and BDS-3 signals, which leads to the inter-system bias (ISB) between BDS-2 and BDS-3 on the receiver side. To fully utilize BDS, the BDS-2 and BDS-3 combined precise time and frequency transfer are investigated considering the effect of the ISB. Four kinds of ISB stochastic models are presented, which are ignoring ISB (ISBNO), estimating ISB as random constant (ISBCV), random walk process (ISBRW), and white noise process (ISBWN). The results demonstrate that the datum of receiver clock offsets can be unified and the ISB deduced datum confusion can be avoided by estimating the ISB. The ISBCV and ISBRW models are superior to ISBWN. For the BDS-2 and BDS-3 combined precise time and frequency transfer using ISBNO, ISBCV, ISBRW, and ISBWN, the stability of clock differences of old signals can be enhanced by 20.18%, 23.89%, 23.96%, and 11.46% over BDS-2-only, respectively. For new signals, the enhancements are −50.77%, 20.22%, 17.53%, and −3.69%, respectively. Moreover, ISBCV and ISBRW models have the better frequency transfer stability. Consequently, we recommended the optimal ISBCV or suboptimal ISBRW model for BDS-2 and BDS-3 combined precise time and frequency transfer when processing the old as well as the new signals.

On modelling physical systems with stochastic models: diffusion versus Lévy processes

2008 ◽  
Vol 366 (1875) ◽  
pp. 2455-2474 ◽  
Author(s):  
Cécile Penland ◽  
Brian D Ewald
Keyword(s):  
Differential Equations ◽  
Stochastic Models ◽  
Lévy Processes ◽  
Numerical Models ◽  
Gaussian White Noise ◽  
Levy Processes ◽  
Physical Systems ◽  
Random Forcing ◽  
Weather And Climate ◽  
Made In

Stochastic descriptions of multiscale interactions are more and more frequently found in numerical models of weather and climate. These descriptions are often made in terms of differential equations with random forcing components. In this article, we review the basic properties of stochastic differential equations driven by classical Gaussian white noise and compare with systems described by stable Lévy processes. We also discuss aspects of numerically generating these processes.

scholarly journals Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

2021 ◽  
Vol 58 (2) ◽  
pp. 469-483
Author(s):  
Jesper Møller ◽  
Eliza O’Reilly
Keyword(s):  
Finite Number ◽  
Point Process ◽  
Point Processes ◽  
Parametric Models ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
The Difference ◽  
Weaker Conditions ◽  
Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

scholarly journals Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics

2018 ◽  
Vol 6 (1) ◽  
pp. 41-64 ◽  
Author(s):  
Aslak Tveito ◽  
Mary M. Maleckar ◽  
Glenn T. Lines
Keyword(s):  
Differential Equations ◽  
Markov Model ◽  
Stochastic Models ◽  
Markov Models ◽  
Single Channel ◽  
Probability Density Functions ◽  
Density Functions ◽  
Mathematical Framework ◽  
Wild Type ◽  
Channel Dynamics

AbstractSingle channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

Elaboration of stochastic models to comprehensive evaluation of occupational risks in complex dynamic systems

2021 ◽  
Vol 1 (104) ◽  
pp. 31-41
Author(s):  
A.P. Bochkovskyi
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Stochastic Models ◽  
Comprehensive Evaluation ◽  
System Of Differential Equations ◽  
Partial Derivatives ◽  
Occupational Risks ◽  
The Impact ◽  
Negative Factors ◽  
Over Time

Purpose: Elaborate stochastic models to comprehensive evaluation of occupational risks in “man - machine - environment” systems taking into account the random and dynamic nature of the impact on the employee of negative factors over time. Design/methodology/approach: Within study, the methods of probability theory and the theory of Markov processes - to find the limit distribution of the random process of dynamic impact on the employee of negative factors over time and obtain main rates against which the level of occupational risks within the "man - machine - environment" systems can be comprehensively evaluated; Erlang phases method, Laplace transform, difference equations theory, method of mathematical induction - to elaborate a method of analytical solution of the appropriate limit task for a system of differential equations in partial derivatives and appropriate limit conditions were used. Findings: A system of differential equations in partial derivatives and relevant limit conditions is derived, which allowed to identify the following main rates for comprehensive evaluation of occupational risks in systems "man - machine - environment": probability of excess the limit of the employee's accumulation of negative impact of the harmful production factor; probability of the employee’s injury of varying severity in a random time. An method to the solution the limit task for a system of differential equations, which allows to provide a lower bounds of the probability of a certain occupational danger occurrence was elaborated. Research limitations/implications: The elaborated approach to injury risk evaluation is designed to predict cases of non-severe injuries. At the same time, this approach allows to consider more severe cases too, but in this case the task will be more difficult. Practical implications: The use of the elaborated models allows to apply a systematic approach to the evaluation of occupational risks in enterprises and to increase the objectivity of the evaluation results by taking into account the real characteristics of the impact of negative factors on the employee over time. Originality/value: For the first time, a special subclass of Markov processes - Markov drift processes was proposed and substantiated for use to comprehensive evaluation of occupational risks in “man - machine - environment” systems.

scholarly journals Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

2019 ◽  
Vol 29 (6) ◽  
pp. 1297-1315 ◽  
Author(s):  
Filip Tronarp ◽  
Hans Kersting ◽  
Simo Särkkä ◽  
Philipp Hennig
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Gradient Field ◽  
Space Representation ◽  
Bayesian Filtering ◽  
State Space Representation ◽  
The Difference ◽  
New Perspective ◽  
Non Gaussian ◽  
Filtering Problem

Abstract We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a nonlinear Bayesian filtering problem and all widely used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the nonlinear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

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