scholarly journals Numerical Solutions of the Hattendorff Differential Equation for Multi-state Markov Insurance Models

2021 ◽  
Author(s):  
Nathan Ritchey ◽  
Rajeev Rajaram
Keyword(s):  
Differential Equation ◽  
Time Evolution ◽  
Continuous Time ◽  
Numerical Solutions ◽  
Random Variable ◽  
The State ◽  
Numerical Results ◽  
State Process ◽  
Differential Equa

We provide methodology and numerical results for the Hattendorff differential equa- tion for the continuous time evolution of the variance of L(j)t , the loss at time t random variable for a multi-state process, given that the state at time t is j.

scholarly journals Numerical Solutions of the Hattendorff Differential Equation for Multi-state Markov Insurance Models

2021 ◽  
Author(s):  
Nathan Ritchey ◽  
Rajeev Rajaram
Keyword(s):  
Differential Equation ◽  
Time Evolution ◽  
Continuous Time ◽  
Numerical Solutions ◽  
Random Variable ◽  
The State ◽  
Numerical Results ◽  
State Process ◽  
Differential Equa

We provide methodology and numerical results for the Hattendorff differential equa- tion for the continuous time evolution of the variance of L(j)t , the loss at time t random variable for a multi-state process, given that the state at time t is j.

scholarly journals Hattendorff Differential Equation for Multi-State Markov Insurance Models

Risks ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 169
Author(s):  
Rajeev Rajaram ◽  
Nathan Ritchey
Keyword(s):  
Differential Equation ◽  
Time Evolution ◽  
Discrete Time ◽  
Random Variable ◽  
The State ◽  
Numerical Computations ◽  
Matrix Notation ◽  
Insurance Model

We derive a Hattendorff differential equation and a recursion governing the evolution of continuous and discrete time evolution respectively of the variance of the loss at time t random variable given that the state at time t is j, for a multistate Markov insurance model (denoted by 2σt(j)). We also show using matrix notation that both models can be easily adapted for use in MATLAB for numerical computations.

Filters and parameter estimation for a partially observable system subject to random failure with continuous-range observations

2004 ◽  
Vol 36 (4) ◽  
pp. 1212-1230 ◽  
Author(s):  
Daming Lin ◽  
Viliam Makis
Keyword(s):  
Condition Monitoring ◽  
Continuous Time ◽  
The State ◽  
Parameter Estimates ◽  
State Process ◽  
Observation Process ◽  
Special Cases ◽  
Random Failure ◽  
Prediction Problems ◽  
Continuous Range

We consider a failure-prone system operating in continuous time. Condition monitoring is conducted at discrete time epochs. The state of the system is assumed to evolve as a continuous-time Markov process with a finite state space. The observation process with continuous-range values is stochastically related to the state process, which, except for the failure state, is unobservable. Combining the failure information and the condition monitoring information, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Updated parameter estimates are obtained using the expectation-maximization (EM) algorithm. Some practical prediction problems are discussed and finally an illustrative example is given using a real dataset.

Recursive filters for a partially observable system subject to random failure

2003 ◽  
Vol 35 (1) ◽  
pp. 207-227 ◽  
Author(s):  
Daming Lin ◽  
Viliam Makis
Keyword(s):  
Condition Monitoring ◽  
Continuous Time ◽  
The State ◽  
Parameter Estimates ◽  
State Process ◽  
Observation Process ◽  
Special Cases ◽  
Random Failure ◽  
Partially Observable System ◽  
Prediction Problems

We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.

Filters and parameter estimation for a partially observable system subject to random failure with continuous-range observations

2004 ◽  
Vol 36 (04) ◽  
pp. 1212-1230 ◽  
Author(s):  
Daming Lin ◽  
Viliam Makis
Keyword(s):  
Condition Monitoring ◽  
Continuous Time ◽  
The State ◽  
Parameter Estimates ◽  
State Process ◽  
Observation Process ◽  
Special Cases ◽  
Random Failure ◽  
Prediction Problems ◽  
Continuous Range

We consider a failure-prone system operating in continuous time. Condition monitoring is conducted at discrete time epochs. The state of the system is assumed to evolve as a continuous-time Markov process with a finite state space. The observation process with continuous-range values is stochastically related to the state process, which, except for the failure state, is unobservable. Combining the failure information and the condition monitoring information, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Updated parameter estimates are obtained using the expectation-maximization (EM) algorithm. Some practical prediction problems are discussed and finally an illustrative example is given using a real dataset.

scholarly journals Maximum approximate likelihood estimation of general continuous-time state-space models

2022 ◽  
pp. 1471082X2110657
Author(s):  
Sina Mews ◽  
Roland Langrock ◽  
Marius Ötting ◽  
Houda Yaqine ◽  
Jost Reinecke
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Likelihood Estimation ◽  
The State ◽  
State Space Models ◽  
State Transitions ◽  
Model Parameters ◽  
State Process ◽  
Approximate Likelihood ◽  
Non Gaussian

Continuous-time state-space models (SSMs) are flexible tools for analysing irregularly sampled sequential observations that are driven by an underlying state process. Corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretization of the state process. The corresponding reframing of the SSM as a continuous-time hidden Markov model, with structured state transitions, enables us to apply the associated efficient algorithms for parameter estimation and state decoding. We illustrate the modelling approach in a case study using data from a longitudinal study on delinquent behaviour of adolescents in Germany, revealing temporal persistence in the deviation of an individual's delinquency level from the population mean.

scholarly journals Representation of simulation errors in single step methods using state dependent noise

2021 ◽  
Vol 347 ◽  
pp. 00001
Author(s):  
Edward Boje
Keyword(s):  
Differential Equation ◽  
Continuous Time ◽  
Single Step ◽  
Error Model ◽  
The State ◽  
Rate Of Change ◽  
Local Error ◽  
Step Size ◽  
Discrete State ◽  
Simulation Error

The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.

scholarly journals Genealogical constructions and asymptotics for continuous-time Markov and continuous-state branching processes

2018 ◽  
Vol 50 (A) ◽  
pp. 197-209
Author(s):  
Thomas G. Kurtz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Population Size ◽  
Time Evolution ◽  
Continuous Time ◽  
Branching Processes ◽  
Supercritical Case ◽  
Continuous State ◽  
Population Processes

AbstractGenealogical constructions of population processes provide models which simultaneously record the forward-in-time evolution of the population size (and distribution of locations and types for models that include them) and the backward-in-time genealogies of the individuals in the population at each timet. A genealogical construction for continuous-time Markov branching processes from Kurtz and Rodrigues (2011) is described and exploited to give the normalized limit in the supercritical case. A Seneta‒Heyde norming is identified as a solution of an ordinary differential equation. The analogous results are given for continuous-state branching processes, including proofs of the normalized limits of Grey (1974) in both the supercritical and critical/subcritical cases.

scholarly journals Time triggered stochastic hybrid system with nonlinear continuous dynamics

2021 ◽  
Author(s):  
Zahra Vahdat ◽  
Abhyudai Singh
Keyword(s):  
State Space ◽  
Cell Size ◽  
Time Evolution ◽  
Continuous Time ◽  
Linear Time ◽  
The State ◽  
Closed Form Expression ◽  
Model Parameters ◽  
Continuous Growth ◽  
Piecewise Deterministic Markov Processes

Time triggered stochastic hybrid systems (TTSHS) constitute a class of piecewise-deterministic Markov processes (PDMP), where continuous-time evolution of the state space is interspersed with discrete stochastic events. Whenever a stochastic event occurs, the state space is reset based on a random map. Prior work on this topic has focused on the continuous-time evolution being modeled as a linear time- invariant system, and in this contribution, we generalize these results to consider nonlinear continuous dynamics. Our approach relies on approximating the nonlinear dynamics between two successive events as a linear time-varying system and using this approximation to derive analytical solutions for the state space’s statistical moments. The TTSHS framework is used to model continuous growth in an individual cell’s size and its subsequent division into daughters. It is well known that exponential growth in cell size, together with a size- independent division rate, leads to an unbounded variance in cell size. Motivated by recent experimental findings, we consider nonlinear growth in cell size based on a Michaelis- Menten function and show that this leads to size homeostasis in the sense that the variance in cell size remains bounded. Moreover, we provide a closed-form expression for the variance in cell size as a function of model parameters and validate it by performing exact Monte Carlo simulations. In summary, our work provides an analytical approach for characterizing moments of a nonlinear stochastic dynamical system that can have broad applicability in studying random phenomena in both engineering and biology.

Recursive filters for a partially observable system subject to random failure

2003 ◽  
Vol 35 (01) ◽  
pp. 207-227 ◽  
Author(s):  
Daming Lin ◽  
Viliam Makis
Keyword(s):  
Condition Monitoring ◽  
Continuous Time ◽  
The State ◽  
Parameter Estimates ◽  
State Process ◽  
Observation Process ◽  
Special Cases ◽  
Random Failure ◽  
Partially Observable System ◽  
Prediction Problems

We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.

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