The stationary distributions of state-dependent diffusions reflected at one and two sides

Abstract

Consider a one-dimensional diffusion process which has state-dependent drift and deviation and is reflected at the origin, which is called a one-side reflected diffusion or simply reflected diffusion. We are particularly interested in the case that its drift and deviation are discontinuous. We define this reflected diffusion as the solution of a stochastic integral equation, and find conditions for its positive recurrence, We then derive its stationary distribution under these conditions. As a related problem, we also consider the case that it is reflected at two sides, which is called a two-sides reflected diffusion. Its existence, positive recurrence and stationary distribution are similarly studied. In the literature, these problems are studied through a state-dependent diffusion on the whole line particularly when the drift and deviation are discontinuous. However, the reflected process itself is not defined in such a study. Thus, the stationary distribution has not been fully studied for a general state-dependent reflected diffusion. We aim to fills this insufficiency and to make the stationary distributions of reflected diffusions widely available in application.

Figures (1)

• Figure 1: Test function $g$

Theorems & Definitions (16)

• Lemma 2.1: Theorem 5.15 of KaraShre1998
• Lemma 2.2
• Remark 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['lem:Z-X-existence']}
• Theorem 3.1
• Remark 3.1
• proof
• Lemma 4.1
• Theorem 4.1