Some Characterizations Of Mixed Renewal Processes

Lyberopoulos, Demetrios P.; Macheras, N. D.

doi:10.48550/arxiv.1205.4441

Cited by 7 publications

(18 citation statements)

References 12 publications

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“…Ad (ii): Since W is P -and Q-conditionally i.i.d. and Q Θ " P Θ , it follows by [15], Lemma 3.7, that there exists a P Θ -null set…”

Section: Throughout What Follows We Denote Again By Kpθq and Kpθq The...mentioning

confidence: 95%

“…Assertion (i) is equivalent to the fact that pN, Xq is a risk process and that N is a P -MRP(KpΘq). But according to [16], Lemma 2.3, and [15], Proposition 3.8, we equivalently obtain two P Θ -null sets L P,1 , L P,2 P BpDq such that the pair pN, Xq is a P θ -risk process for any θ R L P,1 , and for any θ R L P,2 the process N is a P θ -RPpKpθqq.…”

Section: Throughout What Follows We Denote Again By Kpθq and Kpθq The...mentioning

confidence: 99%

“…The following definition has been introduced in [15], Definition 3.1, see also [17], Definition 3.2(b). Definition 3.1.…”

Section: Compound Mixed Renewal Processes and Progressively Equivalen...mentioning

confidence: 99%

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A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

Tzaninis¹,

Macheras²

2020

Preprint

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Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P , we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.

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“…Ad (ii): Since W is P -and Q-conditionally i.i.d. and Q Θ " P Θ , it follows by [15], Lemma 3.7, that there exists a P Θ -null set…”

Section: Throughout What Follows We Denote Again By Kpθq and Kpθq The...mentioning

confidence: 95%

Section: Throughout What Follows We Denote Again By Kpθq and Kpθq The...mentioning

confidence: 99%

See 1 more Smart Citation

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

Tzaninis¹,

Macheras²

2020

Preprint

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“…Because cpΘq " ppQ, Θq, there exists a P Θ -null set L Q,1 P BpDq such that cpθq " ppQ θ q for any θ R L Q,1 by Remark 5.2(a); hence Q θ P M ruin,ℓ S,Exppρpθqq for any θ R Ă M ˚:" M ˚Y L Q,1 . Condition (9) follows by (7) along with [18], Remark 3.4(b), while (10) is a consequence of (RP M θ ) and (9).…”

Section: ˆR˚ℓmentioning

confidence: 98%

“…The following definition has been introduced in [10], Definition 3.1, see also [13], Definition 3.2 (b). Definition 3.1.…”

Section: )mentioning

confidence: 99%

A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory

Tzaninis¹,

Macheras²

2020

Preprint

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If a given aggregate process S is a compound mixed renewal process under a probability measure P , we provide a characterization of all probability measures Q on the domain of P such that Q and P are progressively equivalent and S is converted into a compound mixed Poisson process under Q. This result extends earlier works of Delbaen & Haezendonck [2], Embrechts & Meister [5], Lyberopoulos & Macheras [11], and of the authors [14]. Implications to the ruin problem and to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.

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Some characterizations for Markov processes as mixed renewal processes

Macheras

Tzaninis

2018

Mathematica Slovaca

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In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from [6] (enlarging Huang's [3] original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.MSC 2010: Primary 60G55 ; secondary 60K05, 28A50, 60A10, 60G05, 60J27, 91B30.

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Some Characterizations Of Mixed Renewal Processes

Cited by 7 publications

References 12 publications

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory

Some characterizations for Markov processes as mixed renewal processes

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