Symbolic calculation of the moments of the time of ruin

2004 ◽  
Vol 34 (1) ◽  
pp. 109-120 ◽  
Author(s):  
Steve Drekic ◽  
James E. Stafford ◽  
Gordon E. Willmot
Keyword(s):  
Symbolic Calculation ◽  
Time Of Ruin ◽  
The Time Of Ruin

scholarly journals Ruin problems in Markov-modulated risk models

2017 ◽  
Vol 12 (1) ◽  
pp. 23-48 ◽  
Author(s):  
David C.M. Dickson ◽  
Marjan Qazvini
Keyword(s):  
Continuous Time ◽  
Risk Model ◽  
Numerical Algorithms ◽  
Binomial Model ◽  
Risk Models ◽  
Time Of Ruin ◽  
Compound Binomial ◽  
Compound Binomial Model ◽  
Markov Modulated ◽  
The Time Of Ruin

AbstractChen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

SOME ADVANCES ON THE ERLANG(n) DUAL RISK MODEL

2014 ◽  
Vol 45 (1) ◽  
pp. 127-150 ◽  
Author(s):  
Eugenio V. Rodríguez-Martínez ◽  
Rui M. R. Cardoso ◽  
Alfredo D. Egídio dos Reis
Keyword(s):  
Risk Model ◽  
Waiting Times ◽  
Dual Model ◽  
Time Of Ruin ◽  
Constant Rate ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Renewal Risk ◽  
The Individual

AbstractThe dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

scholarly journals Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

2008 ◽  
Vol 38 (02) ◽  
pp. 399-422 ◽  
Author(s):  
Eric C.K. Cheung ◽  
Steve Drekic
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Size Distribution ◽  
Discrete Time ◽  
Continuous Time ◽  
Risk Model ◽  
Time Model ◽  
Jump Size ◽  
Time Of Ruin ◽  
The Time Of Ruin

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

scholarly journals Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 110 ◽  
Author(s):  
Sooie-Hoe Loke ◽  
Enrique Thomann
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Ruin Probability ◽  
Risk Model ◽  
Constant Force ◽  
Time Of Ruin ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
Force Of Interest ◽  
The Time Of Ruin

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.

scholarly journals On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (02) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to thenth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of thekth moment of the time of ruin in the classical risk model.

scholarly journals Asymptotics for the time of ruin in the war of attrition

2017 ◽  
Vol 49 (2) ◽  
pp. 388-410 ◽  
Author(s):  
Philip A. Ernst ◽  
Ilie Grigorescu
Keyword(s):  
Distribution Function ◽  
Explicit Formula ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Fluid Limit ◽  
War Of Attrition ◽  
Time Of Ruin ◽  
Standard Normal ◽  
The Difference ◽  
The Time Of Ruin

AbstractWe consider two players, starting withmandnunits, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probabilityp(m,n) that the first player wins. Whenm~Nx0,n~Ny0, we prove the fluid limit asN→ ∞. Whenx0=y0,z→p(N,N+z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τNis established as (T- τN) ~N-βW1/β, β = ¼,T=x0+y0. Modulo a constant,W~ χ21(z02/T2).

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