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Asymptotic expansions for the first hitting times of bessel processes

Hamana Yuji, Kaikura Ryo, Shinozaki Kosuke

Opuscula Mathematica

2021

Vol. 41, no. 4

509--537

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

The area of a spectrally positive stable process stopped at zero

Letemplier J., Simon T.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

27--37

A multiplicative identity in law for the area of a spectrally positive Lévy α-stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.

Probabilistic solution to a two-dimensional LQG homing problem

Makasu C., Lefebvre M.

Control and Cybernetics

2012

Vol. 41, no. 1

5-12

Let xt be an arbitrary one-dimensional diffusion process and yt be a one-dimensional controlled diffusion process starting from y0 = y ∈ (a, b). The process is controlled until yt crosses either y = a or y = b for the first time. Our aim is to find the control ut that minimizes an expected cost functional with both quadratic control and boundary crossing costs. An explicit form for the optimal control is obtained under certain conditions.