scholarly journals A Review on Phase-type Distributions and their Use in Risk Theory

2005 ◽  
Vol 35 (01) ◽  
pp. 145-161 ◽  
Author(s):  
Mogens Bladt
Keyword(s):  
Stochastic Modeling ◽  
Ruin Probability ◽  
Risk Theory ◽  
Jump Processes ◽  
Positive Real ◽  
Markov Jump ◽  
Premium Rate ◽  
Markov Jump Processes ◽  
Phase Type ◽  
Phase Type Distributions

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

scholarly journals A Review on Phase-type Distributions and their Use in Risk Theory

2005 ◽  
Vol 35 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Mogens Bladt
Keyword(s):  
Stochastic Modeling ◽  
Ruin Probability ◽  
Risk Theory ◽  
Jump Processes ◽  
Positive Real ◽  
Markov Jump ◽  
Premium Rate ◽  
Markov Jump Processes ◽  
Phase Type ◽  
Phase Type Distributions

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

scholarly journals Heat Release by Controlled Continuous-Time Markov Jump Processes

2013 ◽  
Vol 150 (1) ◽  
pp. 181-203 ◽  
Author(s):  
Paolo Muratore-Ginanneschi ◽  
Carlos Mejía-Monasterio ◽  
Luca Peliti
Keyword(s):  
Heat Release ◽  
Continuous Time ◽  
Jump Processes ◽  
Markov Jump ◽  
Markov Jump Processes

scholarly journals Optimal Filtering of Markov Jump Processes Given Observations with State-Dependent Noises: Exact Solution and Stable Numerical Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 506
Author(s):  
Andrey Borisov ◽  
Igor Sokolov
Keyword(s):  
Exact Solution ◽  
Nonlinear Filtering ◽  
Innovation Process ◽  
Jump Processes ◽  
State Transitions ◽  
Observation System ◽  
Optimal Filtering ◽  
Markov Jump ◽  
Approximation Accuracy ◽  
Markov Jump Processes

The paper is devoted to the optimal state filtering of the finite-state Markov jump processes, given indirect continuous-time observations corrupted by Wiener noise. The crucial feature is that the observation noise intensity is a function of the estimated state, which breaks forthright filtering approaches based on the passage to the innovation process and Girsanov’s measure change. We propose an equivalent observation transform, which allows usage of the classical nonlinear filtering framework. We obtain the optimal estimate as a solution to the discrete–continuous stochastic differential system with both continuous and counting processes on the right-hand side. For effective computer realization, we present a new class of numerical algorithms based on the exact solution to the optimal filtering given the time-discretized observation. The proposed estimate approximations are stable, i.e., have non-negative components and satisfy the normalization condition. We prove the assertions characterizing the approximation accuracy depending on the observation system parameters, time discretization step, the maximal number of allowed state transitions, and the applied scheme of numerical integration.

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