Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

Tzaninis, Spyridon M.

doi:10.15559/21-vmsta192

Cited by 3 publications

(5 citation statements)

References 27 publications

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“…For applications of Theorem 4.5 to a characterization of equivalent martingale measures for CMRPs, and to the pricing of actuarial risks (premium calculation principles) in an insurance market possessing the property of no free lunch with vanishing risk we refer to [27]. Finally, for applications of Theorem 4.5 to the ruin problem for CMRPs we refer to [26].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

“…Note also that the above proposition extends Lemma 8.4 of [22] to CMRPs. Applications of Proposition 4.15 to the ruin problem for CMRPs are presented in [26].…”

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confidence: 99%

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

Tzaninis¹,

Macheras²

2020

Preprint

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“…According to (42) and (43), we check that the net profit condition is satisfied: ES 2 = 1 + 1 + 0.9 + 1 = 3.9 < 4 = κN. The probability-generating function of S 2 = X 1 + X 2 is G S 2 (s) = s 2 e 1.9(s−1) and the equation G S 2 (s) = s 4 has one nonzero solution inside the unit circle: α = −0.2928.…”

Section: Examplementioning

confidence: 99%

“…[28,24]), and the research methods (cf. [42]) are the reasons which make the recent literature voluminous. Next to the mentioned references, see [11,36,4,12,10,34,26,25,32,8] as the recent ones on the subject, too.…”

Section: Introductionmentioning

confidence: 99%

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Gervė,

Grigutis

2024

Modern Stochastics: Theory and Applications

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In this paper, the distribution function \[ \varphi (u)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }{\sum \limits_{i=1}^{n}}({X_{i}}-\kappa )\lt u\Bigg),\] and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

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“…Appendix A collects some long proofs. For applications of Theorem 4.5 to a characterization of equivalent martingale measures for CMRPs, and to the pricing of actuarial risks (premium calculation principles) in an insurance market possessing the property of no free lunch with vanishing risk we refer to Tzaninis and Macheras (2020), while for applications of Theorem 4.5 to the ruin problem for CMRPs we refer to Tzaninis (2022) and Tzaninis and Macheras (2020).…”

Section: Introductionmentioning

confidence: 99%

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

Tzaninis¹,

Macheras²

2023

ALEA

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Generalizing earlier work of Delbaen & Haezendonck 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 one through a change of measures.

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Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

Cited by 3 publications

References 27 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

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

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

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