Ruin Probabilities

Asmussen, Søren; Albrecher, Hansjörg

doi:10.1142/9789814282536

Cited by 464 publications

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“…However, increasing N t and N x brings us closer to the 'target'. In Table 1 we show the precomputing time needed 11.9935 to compute the three-dimensional array p * 1 (x i ; t j , x k ) and the time needed to perform each iteration (3).…”

Section: Compute the Approximation Of The First Passage Time Densitŷmentioning

confidence: 99%

“…A wellstudied class of processes for which this approach works well consists of Lévy processes with phase-type distributed jumps (see [1], [3], [4, p. 81], and [14]). Phase-type distributions are defined as the first passage time for a continuous-time, finite-state Markov chain, they form a dense class in the set of all distributions on R + , and, most importantly, if a Lévy process has phase-type jumps, its characteristic exponent is a rational function (though the converse is not true; see [19]).…”

Section: Introductionmentioning

confidence: 99%

“…The second problem is that the degree of the polynomial equation q + ψ(λ) = 0 would necessarily grow to ∞, which will make solving this equation very complicated. Also, see [3] and [19] for an interesting example of a distribution with rational transform which would require an infinite-degree phase-type representation.…”

Section: Introductionmentioning

confidence: 99%

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On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Hurd

Kuznetsov

2009

J. Appl. Probab.

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In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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Section: Compute the Approximation Of The First Passage Time Densitŷmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Hurd

Kuznetsov

2009

J. Appl. Probab.

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“…Due to a sample path duality (see e.g. [7,Ch.III.2]), risk processes of that type have a counterpart in workload models of queueing theory, and a similar semi-Markovian structure was considered in [1] in a queueing context. In [5] it was proposed to assume that (A k , B k ) are i.i.d.…”

Section: Introductionmentioning

confidence: 99%

On Simple Ruin Expressions in Dependent Sparre Andersen Risk Models

Albrecher¹,

Boxma²,

Ivanovs³

2014

J. Appl. Probab.

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In this note we provide a simple alternative derivation of an explicit formula of Kwan and Yang [14] for the probability of ruin in a risk model with a certain dependence between general claim inter-occurrence times and subsequent claim sizes of conditionally exponential type. The approach puts the type of formula in a general context, illustrating the potential for similar simple ruin probability expressions in more general risk models with dependence.

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“…This approach saves us from the burden of rederiving the integral equations that would arise, and we can readily use the already existing numerically stable and efficient methods that have been proposed for the steady-state solution of MRMFQs. The connection between workload-dependent MAP/PH/1 queues and MRMFQs is obtained using sample path arguments as in [9,11,12], but the derivation of this connection in the case of complete rejection policy is one of the main contributions of this paper. Fluid queues in which a fluid acts as the input to and output of the buffer have been used extensively in various stochastic modeling contexts since their introduction in the early studies of [13].…”

Section: Introductionmentioning

confidence: 99%

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

Yazıcı

Akar

2013

Performance Evaluation

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a b s t r a c tWe propose a numerical algorithm for finding the steady-state queue occupancy distribution for a workload-dependent MAP/PH/1 queue in which the arrival process and the service rate depend continuously on the instantaneous workload in the system. Both infinite and finite queue capacity scenarios are considered, including partial rejection and complete rejection policies for the latter. Using discretization, this system is approximately described by a multi-regime Markov fluid queue for which numerical algorithms are available. The computational complexity of the proposed method is linear in the number of regimes used for discretization. We provide numerical examples to validate the proposed approach.

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Ruin Probabilities

Cited by 464 publications

References 0 publications

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

On Simple Ruin Expressions in Dependent Sparre Andersen Risk Models

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

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