Exit identities for diffusion processes observed at Poisson arrival times

Abstract De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.

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The M/G/1 processor sharing queue as the almost sure limit of feedback queues

Journal of Applied Probability ◽

10.2307/3214834 ◽

1990 ◽

Vol 27 (4) ◽

pp. 913-918 ◽

Cited By ~ 10

Author(s):

J. A. C. Resing ◽

G. Hooghiemstra ◽

M. S. Keane

Keyword(s):

Almost Sure Convergence ◽

Processor Sharing ◽

Regenerative Processes ◽

Arrival Times ◽

Sojourn Times ◽

Feedback Model ◽

Poisson Arrival ◽

Arrival Processes ◽

Feedback Queue ◽

Feedback Queues

In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.

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The M/G/1 processor sharing queue as the almost sure limit of feedback queues

Journal of Applied Probability ◽

10.1017/s0021900200028084 ◽

1990 ◽

Vol 27 (04) ◽

pp. 913-918 ◽

Cited By ~ 1

Author(s):

J. A. C. Resing ◽

G. Hooghiemstra ◽

M. S. Keane

Keyword(s):

Almost Sure Convergence ◽

Processor Sharing ◽

Regenerative Processes ◽

Arrival Times ◽

Sojourn Times ◽

Feedback Model ◽

Poisson Arrival ◽

Arrival Processes ◽

Feedback Queue ◽

Feedback Queues

In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.

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Exit identities for Lévy processes observed at Poisson arrival times

Bernoulli ◽

10.3150/15-bej695 ◽

2016 ◽

Vol 22 (3) ◽

pp. 1364-1382 ◽

Cited By ~ 47

Author(s):

Hansjörg Albrecher ◽

Jevgenijs Ivanovs ◽

Xiaowen Zhou

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

Arrival Times ◽

Poisson Arrival

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On Periodic Dividends for the Classical Risk Model with Debit Interest

Mathematical Problems in Engineering ◽

10.1155/2020/6395717 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Hua Dong ◽

Xianghua Zhao

Keyword(s):

Differential Equations ◽

Risk Model ◽

Classical Risk Model ◽

Arrival Times ◽

Numerical Examples ◽

Dividend Payments ◽

Poisson Arrival ◽

Optimal Dividend ◽

Debit Interest ◽

The Classical Risk Model

A periodic dividend problem is studied in this paper. We assume that dividend payments are made at a sequence of Poisson arrival times, and ruin is continuously monitored. First of all, three integro-differential equations for the expected discounted dividends are obtained. Then, we investigate the explicit expressions for the expected discounted dividends, and the optimal dividend barrier is given for exponential claims. A similar study on a generalized Gerber–Shiu function involving the absolute time is also performed. To demonstrate the existing results, we give some numerical examples.

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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