Properties of Kagi and Renko Moments for Homogeneous Diffusion Processes

M. A. Spiryaev

M. V. Lomonosov Moscow State University

Abstract: For a homogeneous diffusion process $(X_t)_{t\geqslant 0}$, we consider problems related to the distribution of the stopping times
\begin{gather*} \gamma_{\max}=\inf\Bigl\{t\ge 0:\sup_{s\le t}X_s-X_t \ge H\Bigr\},\qquad \gamma_{\min}=\inf \Bigl\{t\ge 0: X_t-\inf_{s\le t}X_s \ge H \Bigr\}, \\ \kappa_0=\inf\Bigl\{t\ge 0:\sup_{s\le t}X_s-\inf_{s\le t}X_s \ge H\Bigr\}. \end{gather*}
The results obtained are used to construct an inductive procedure allowing us to find the distribution of the increments of the process $X$ between two adjacent kagi and renko instants of time.

Keywords: homogeneous diffusion process, Brownian motion, stopping time, kagi instant of time, renko instant of time, Laplace transform.

UDC: 517

Received: 14.03.2011

DOI: 10.4213/mzm9077

 English version: Mathematical Notes, 2012, 91:2, 259–271

Bibliographic databases:

• 
• 
• 
• 
•