Abstract. The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable ζ, there exists a timevarying barrier b such that P{Bs > b(s), 0 ≤ s ≤ t} = P{ζ > t}. We study a “smoothed” version of this problem and ask whether there is a “barrier” b such that E[exp(−λ ∫ t 0 ψ(Bs − b(s)) ds)] = P{ζ > t}, where λ is a killing rate parameter and ψ : R → [0...

Abstract: This paper deals with the design of Optimal Step-Stress Partially Accelerated Degradation Test (PADT) using Wiener Process for modelling degradation paths. In partial step-stress loading, the unit is tested at usual stress for some time and then stress is accelerated. This helps in preventing over-stressing the test specimens. Failure occurs when the performance characteristic crosses...

1. Estimates for the proof 2 1.1. Estimating the error in terms of ξk and ζk 2 1.2. ∑N−1 k=1 ETN [|ξk|] = O(λ−1/2) 3 1.3. ETN [ζ 2 k ] = O(λ −1/2) 3 1.4. ETc [sup0≤t≤T Et] = O(λ−1/2) 4 2. Proof of Lemma 4.2 5 3. Proof of Lemma 4.3 6 4. Calculation of the Moments of the jumping time and First Passage time using the Fokker-Planck equation 7 4.1. The Moments of the Jumping Time τ k 7 4.2. The mome...

Previously counting embeddings of planar graphs [5] used P-Q trees and was restricted to biconnected graphs. Although the P-Q tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give formulas that count the number of embeddings of general planar graphs (not necessarily connected or bic...

Evaluation of default correlation is an important task in credit risk analysis. In many practical situations, it concerns the joint defaults of several correlated firms, the task that is reducible to a first passage time (FPT) problem. This task represents a great challenge for jump-diffusion processes (JDP), where except for very basic cases, there are no analytical solutions for such problems...

Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper, we first generalize results of Pitman and Yor (2001) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use this result to derive the first passage time distribut...

0009-2614/$ see front matter 2010 Elsevier B.V. A doi:10.1016/j.cplett.2010.08.019 * Fax: +972 2 651 3742. E-mail address: [email protected] The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing number of applications in physics and chemistry. A partial differential equation for the residence time moments is derived, as a generalization of the ...

In this paper we study the two-dimensional joint distribution of the first passage time of a constant level by spectrally negative generalized Ornstein–Uhlenbeck processes and their primitive stopped at this first passage time. By using martingales techniques, we show an explicit expression of the Laplace transform of the distribution in terms of new special functions. Finally, we give an appli...

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several p...

There is an extensive literature on the complexity of planning, but explicit bounds on time and space complexity are very rare. On the other hand, problems like the constraint satisfaction problem (CSP) have been thoroughly analysed in this respect. We provide a number of upperand lower-bound results (the latter based on various complexitytheoretic assumptions such as the Exponential Time Hypot...