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  • Picard, Philippe  (34)
  • Mathematics  (34)
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  • Mathematics  (34)
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  • 1
    Online Resource
    Online Resource
    Probabilité de ruine éventuelle dans un modèle de risque à temps discret
    Cambridge University Press (CUP) ; 2003
    In:  Journal of Applied Probability Vol. 40, No. 03 ( 2003-09), p. 543-556
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 40, No. 03 ( 2003-09), p. 543-556
    Abstract: We continue the study of the discrete-time risk model introduced by Picard et al . (2003). The cumulative loss process ( S t ) t ∊ℕ has independent and stationary increments, the increments per unit of time having nonnegative integer values with distribution { a i , i ∊ ℕ and mean ā . The premium receipt process ( c k ) k ∊ℕ is deterministic, nonnegative and nonuniform; in addition, we assume it to be regular in order for there to exist a constant c & gt; ā such that the deviation is bounded as the time t varies. We are interested in whether or not ruin occurs within a finite time. If T is the time of ruin, we obtain P( T = ∞) as the limit of P( T & gt; t ) as t → ∞, firstly in the particular case where c = 1/ d for some positive d ∊ ℕ, and then in the general case for positive c under the condition that a 0 & gt; ½.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200019550
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2003
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 2
    Online Resource
    Online Resource
    Applications of martingale theory to some epidemic models
    Cambridge University Press (CUP) ; 1980
    In:  Journal of Applied Probability Vol. 17, No. 03 ( 1980-09), p. 583-599
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 17, No. 03 ( 1980-09), p. 583-599
    Abstract: The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here X t and Y t respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200033702
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1980
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 3
    Online Resource
    Online Resource
    Applications of martingale theory to some epidemic models, II
    Cambridge University Press (CUP) ; 1984
    In:  Journal of Applied Probability Vol. 21, No. 04 ( 1984-12), p. 677-684
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 21, No. 04 ( 1984-12), p. 677-684
    Abstract: We consider Weiss's and Downton's models with parametersπ, αand β depending on i number of susceptibles and j number of carriers. A martingale argument is performed when πand α /β only depend on i or, in Weiss's case, when α /β is the product of a function of i by a function of j. In these cases the martingale approach proves very valuable and gives explicit results quite easily. In particular it shows that well-known relations between moments and integrals along a trajectory are still true for any stopping time and for more general models than the classic ones.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200037396
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1984
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 4
    Online Resource
    Online Resource
    On the algebraic structure in Markovian processes of death and epidemic types
    Cambridge University Press (CUP) ; 1999
    In:  Advances in Applied Probability Vol. 31, No. 03 ( 1999-09), p. 742-757
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 31, No. 03 ( 1999-09), p. 742-757
    Abstract: This paper is concerned with the standard bivariate death process as well as with some Markovian modifications and extensions of the process that are of interest especially in epidemic modeling. A new and powerful approach is developed that allows us to obtain the exact distribution of the population state at any point in time, and to highlight the actual nature of the solution. Firstly, using a martingale technique, a central system of relations with two indices for the temporal state distribution will be derived. A remarkable property is that for all the models under consideration, these relations exhibit a similar algebraic structure. Then, this structure will be exploited by having recourse to a theory of Abel-Gontcharoff pseudopolynomials with two indices. This theory generalizes the univariate case examined in a preceding paper and is briefly introduced in the Appendix.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S0001867800009393
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1999
    detail.hit.zdb_id: 1474602-5
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  • 5
    Online Resource
    Online Resource
    Abelian-type expansions and non-linear death processes(II)
    Cambridge University Press (CUP) ; 1996
    In:  Advances in Applied Probability Vol. 28, No. 03 ( 1996-09), p. 877-894
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 28, No. 03 ( 1996-09), p. 877-894
    Abstract: This paper is concerned with the study of death processes with time-homogeneous non-linear death rates. An explicit formula is obtained for the joint distribution of the state X T and the variable ∫ T 0 g ( X t ), where g is any given real function and T corresponds to some appropriate stopping time. This is achieved by constructing a family of martingales and then by using a particular family of Abel–Gontcharoff pseudopolynomials (the theory of which has been introduced in a companion paper) and related Abelian-type expansions. Moreover, the distribution of the first crossing level of such a death process through a general upper boundary is also evaluated in terms of pseudopolynomials of that kind. The flexibility of the methods developed makes easy the extension to multidimensional death processes.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S000186780004653X
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1996
    detail.hit.zdb_id: 1474602-5
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  • 6
    Online Resource
    Online Resource
    Abel-Gontcharoff pseudopolynomials and the exact final outcome of SIR epidemic models (III)
    Cambridge University Press (CUP) ; 1999
    In:  Advances in Applied Probability Vol. 31, No. 02 ( 1999-06), p. 532-550
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 31, No. 02 ( 1999-06), p. 532-550
    Abstract: The paper is concerned with the final state and severity of a number of SIR epidemic models in finite populations. Two different classes of models are considered, namely the classical SIR Markovian models and the collective models introduced recently by the authors. First, by applying a simple martingale argument, it is shown that in both cases, there exists a common algebraic structure underlying the exact law of the final state and severity. Then, a unified approach to these statistics is developed by exploiting the theory of Abel-Gontcharoff pseudopolynomials (presented in a preceding paper).
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S0001867800009228
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1999
    detail.hit.zdb_id: 1474602-5
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  • 7
    Online Resource
    Online Resource
    On the algebraic structure in Markovian processes of death and epidemic types
    Cambridge University Press (CUP) ; 1999
    In:  Advances in Applied Probability Vol. 31, No. 3 ( 1999-09), p. 742-757
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 31, No. 3 ( 1999-09), p. 742-757
    Abstract: This paper is concerned with the standard bivariate death process as well as with some Markovian modifications and extensions of the process that are of interest especially in epidemic modeling. A new and powerful approach is developed that allows us to obtain the exact distribution of the population state at any point in time, and to highlight the actual nature of the solution. Firstly, using a martingale technique, a central system of relations with two indices for the temporal state distribution will be derived. A remarkable property is that for all the models under consideration, these relations exhibit a similar algebraic structure. Then, this structure will be exploited by having recourse to a theory of Abel-Gontcharoff pseudopolynomials with two indices. This theory generalizes the univariate case examined in a preceding paper and is briefly introduced in the Appendix.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1239/aap/1029955202
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1999
    detail.hit.zdb_id: 1474602-5
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  • 8
    Online Resource
    Online Resource
    Abelian-type expansions and non-linear death processes(II)
    Cambridge University Press (CUP) ; 1996
    In:  Advances in Applied Probability Vol. 28, No. 3 ( 1996-09), p. 877-894
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 28, No. 3 ( 1996-09), p. 877-894
    Abstract: This paper is concerned with the study of death processes with time-homogeneous non-linear death rates. An explicit formula is obtained for the joint distribution of the state X T and the variable ∫ T 0 g ( X t ), where g is any given real function and T corresponds to some appropriate stopping time. This is achieved by constructing a family of martingales and then by using a particular family of Abel–Gontcharoff pseudopolynomials (the theory of which has been introduced in a companion paper) and related Abelian-type expansions. Moreover, the distribution of the first crossing level of such a death process through a general upper boundary is also evaluated in terms of pseudopolynomials of that kind. The flexibility of the methods developed makes easy the extension to multidimensional death processes.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1428185
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1996
    detail.hit.zdb_id: 1474602-5
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  • 9
    Online Resource
    Online Resource
    On branching models with alarm triggerings
    Cambridge University Press (CUP) ; 2020
    In:  Journal of Applied Probability Vol. 57, No. 3 ( 2020-09), p. 734-759
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 57, No. 3 ( 2020-09), p. 734-759
    Abstract: We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/jpr.2020.24
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2020
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 10
    Online Resource
    Online Resource
    A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes
    Cambridge University Press (CUP) ; 1990
    In:  Advances in Applied Probability Vol. 22, No. 1 ( 1990-03), p. 25-48
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 22, No. 1 ( 1990-03), p. 25-48
    Abstract: This paper provides a global treatment of the final size distribution of Reed–Frost epidemic processes. Exact and asymptotic results are derived for both single and multipopulation situations. The key tool is a non-standard family of polynomials, introduced initially by Gontcharoff (1937) for one variable, revisited and extended here for several variables. The attractiveness of these polynomials will be enhanced in forthcoming works in the epidemic context as well as in other fields.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1427595
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1990
    detail.hit.zdb_id: 1474602-5
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