Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

Abstract

AbstractA time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity.