Properties of Kagi and Renko moments for homogeneous diffusion processes (Q2435935)

Let \(X_t\) be the process satisfying \(dX_t=\mu(X_t)dt+\sigma(X_t)dB_t,\) \(t\geq 0,\) where \(B_t\) is the standard Brownian motion, \(\mu\) and \(\sigma>0\) satisfy the Lipschitz condition: for \(x,y\in\mathbb R,\) \[ |\mu(x)-\mu(y)|+|\sigma(x)-\sigma(y)|\leq L|x-y| \] and \[ |\mu(x)|^2+|\sigma(x)|^2\leq C(1+x^2). \] The author obtains the Laplace transforms for the stopping times: \[ \gamma_{\max}=\inf\{t\geq 0:\sup_{s\leq t}X_s-X_t\geq H\}, \] \[ \gamma_{\min}=\inf\{t\geq 0:X_t-\inf_{s\leq t}X_s\geq H\}, \] and \[ \kappa_0=\inf\{t\geq 0:\sup_{s\leq t}X_s-\inf_{s\leq t}X_s\geq H\}. \] Also the distribution functions for the random variables \[ \sup_{s\leq\gamma_{\max}}X_s,\qquad\inf_{s\leq\gamma_{\min}}X_s,\qquad X_{\kappa_0} \] are given. As an application these results are used to get the distribution of the increments of the process \(X_t\) between two adjacent Kagi and Renko instants of time.