منابع مشابه
First Passage Time of Skew Brownian Motion
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...
متن کاملOn the First-passage Time of Integrated Brownian Motion
Let (Bt; t ≥ 0) be a Brownian motion process starting from B0 = ν and define Xν(t) = ∫ t 0 Bs ds. For a≥ 0, set τa,ν := inf{t : Xν(t) = a} (with inf φ=∞). We study the conditional moments of τa,ν given τa,ν <∞. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean E(τa,ν | τa,ν <∞)...
متن کاملFirst passage time for Brownian motion and piecewise linear boundaries
We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the fi...
متن کاملOn Skew Brownian Motion
We consider the stochastic equation X(t) = W(t) + βlX0(t), where W is a standard Wiener process and lX0(⋅) is the local time at zero of the unknown process X. There is a unique solution X (and it is adapted to the fields of W) if |β| ≤ 1, but no solutions exist if |β| > 1. In the former case, setting α = (β + 1)/2, the unique solution X is distributed as a skew Brownian motion with parameter α....
متن کاملDistribution of the time at which a Brownian motion is maximal before its first-passage time
We calculate analytically the probability density P (tm) of the time tm at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density P (M, tm) of the maximum M and tm. In the driftless case, we find that P (tm) has power-law tails: P (tm) ∼ t −3/2 m for large tm and P (...
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ژورنال
عنوان ژورنال: Journal of Applied Probability
سال: 2012
ISSN: 0021-9002,1475-6072
DOI: 10.1239/jap/1346955326