Three Essays on Stopping

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

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First Passage Time Distributions for Brownian Motion with Drift and a Local Limit Theorem

10.1007/978-3-030-78939-8_16 ◽

2021 ◽

pp. 191-198

Author(s):

Rabi Bhattacharya ◽

Edward C. Waymire

Keyword(s):

Brownian Motion ◽

Limit Theorem ◽

First Passage Time ◽

Local Limit Theorem ◽

Local Limit ◽

Passage Time ◽

First Passage ◽

Brownian Motion With Drift

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AN IMPROVED APPROACH TO EVALUATE DEFAULT PROBABILITIES AND DEFAULT CORRELATIONS WITH CONSISTENCY

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024916500369 ◽

2016 ◽

Vol 19 (05) ◽

pp. 1650036 ◽

Cited By ~ 1

Author(s):

WEIPING LI ◽

TIM KREHBIEL

Keyword(s):

Risk Management ◽

Closed Form ◽

First Passage Time ◽

Passage Time ◽

Default Probability ◽

Default Correlation ◽

First Passage ◽

Management Implications ◽

Asset Correlation ◽

Closed Form Formula

We provide (i) a simplified analytic closed form formula for evaluating joint default probability, (ii) an improved method to resolve the inconsistency between the univariate process underlying firm-specific default probability and the correlated bivariate process of the first-passage-time default correlation model, (iii) illustration of risk management implications from misspecification of the default state space. Our closed form formula provides a natural extension of previous structural first-passage-time models and shows the sensitivities of default correlation numerically with respect to the underlying asset correlation, obligor credit quality and time horizon. We emphasize the disparate credit risk management implications of our result in contrast to commonly used risk measurement methods.

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Asymptotics for First-Passage Times on Delaunay Triangulations

Combinatorics Probability Computing ◽

10.1017/s0963548310000477 ◽

2011 ◽

Vol 20 (3) ◽

pp. 435-453 ◽

Cited By ~ 2

Author(s):

LEANDRO P. R. PIMENTEL

Keyword(s):

First Passage Time ◽

Passage Time ◽

Upper Bounds ◽

Sufficient Condition ◽

First Passage Percolation ◽

First Passage ◽

Delaunay Triangulations ◽

Negative Constant ◽

Additivity Theory ◽

Passage Times

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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