Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

Atkinson, Michael P.; Singham, Dashi I.

doi:10.1016/j.spl.2015.02.006

Cited by 4 publications

(2 citation statements)

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“…In other words, the non ruin probability conditional to V T = y is equal to the probability that a Brownian bridge starting form 0 and going to y γ + T with length equal to T stays under the line t → x+t γ . Using the results of [13,4], we obtain This implies that the De Vylder-Goovaert conjecture holds true in the heavy trac approximation.…”

Section: Comparison Of the Ruin Probabilitiesmentioning

confidence: 60%

The De Vylder–Goovaerts conjecture holds within the diffusion limit

Ankirchner¹,

Blanchet-Scalliet

Kazi-Tani

2019

J. Appl. Probab.

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The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

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Section: Comparison Of the Ruin Probabilitiesmentioning

confidence: 60%

The De Vylder–Goovaerts conjecture holds within the diffusion limit

Ankirchner¹,

Blanchet-Scalliet

Kazi-Tani

2019

J. Appl. Probab.

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“…where W (t) is a standard Brownian motion. Atkinson and Singham [3] derive a similar relationship for the probability that a Brownian bridge that starts at location −β < x 0 < λ at time 0, ends at location −(αT + β) < x T < (γT + λ) at time T , and having variance parameter σ 2 stays between two asymmetric linear boundaries, −(αt + β) and γt + λ. If we denoteB(t) as the Brownian bridge with the properties described in the previous sentence, then we have the following probability of interest…”

Section: (A2)mentioning

confidence: 83%

Boundary Crossing Probabilities for the Cumulative Sample Mean

Singham¹,

Atkinson²

2017

Prob. Eng. Inf. Sci.

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We develop a new measure of reliability for the mean behavior of a process by calculating the probability the cumulative sample mean will stay within a given distance from the true mean over a period of time. This probability is derived using boundary-crossing properties of Brownian bridges. We derive finite sample results for independent and identically distributed normal data, limiting results for data meeting a functional central limit theorem, and draw parallels to standard normal confidence intervals. We deliver numerical results for i.i.d., dependent, and queueing processes.

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