1. Introduction
In mathematical finance, especially in the field of interest rate theory, the Cox–Ingersoll–Ross (CIR for short) model explains the evolution of interest rates. The CIR model is a type of one-factor model (short-rate model), as it describes interest rate movements as driven by only one source of market risk. The model was introduced by [
1] as an extension of the Vasicek’s interest rate model, and it has the following stochastic differential equation (SDE for short):
where
is a Wiener process (modeling the random market risk factor) and
and
are positive constants. The parameter
a is the mean level or long-term interest rate constant, the parameter
b is the speed of the mean reversion and corresponds to the speed of adjustment to the mean
a, and
regulates the volatility. The drift factor,
, is the same as in the Vasicek model; see [
2]. It ensures the mean reversion of the interest rate towards the long-run value
a, with the speed of adjustment governed by the strictly positive parameter
b. The stochastic volatility term
has a standard deviation that is proportional to the square root of the current rate. This implies that as the interest rate increases, its standard deviation increases, and as it falls and approaches zero, the stochastic volatility term also approaches 0.
In the following section, we mainly study the Equation (
1) as the CIR model or CIR process. The same process is used in the Heston model, see [
3], to model stochastic volatility. The SDE (
1) has no explicit solution in general, even though its mean and variance can be calculated explicitly, and the probability transition density can be easily determined by using the time–space transformation. This CIR process
can be defined as a sum of squared Ornstein–Uhlenbeck process or be constructed using a BESQ process of dimension
; see [
4]. Refs [
5,
6] proved that the CIR process is an affine process and the semigroup of every stochastic continuous affine process is a Feller semigroup; hence, the CIR process is a regular Feller process on the interval
. The CIR process
is an ergodic process, and it possesses a stationary distribution. Furthermore, the CIR process is positive recurrent and nonexplosive on the interval
.
Some diffusion processes in time-dependent domains have always been the focus of scholars’ attention in the field of probability, and some various sample path properties involving diffusion processes in the time domain are constantly being discovered. The time-dependent domain problem that this article focuses on is actually a domain problem with deterministic moving boundaries, also known as noncylindrical domains. This type of time-dependent domain problem originates from both random environment problems and classic PDE problems with various boundary conditions; see [
7]. In [
8], the authors provided motivation for studying this issue of diffusion processes in time-dependent domains, through the theoretical explanation of a partial differential equation, and they focused on the heat equation in the time-dependent domain with Neumann rather than Dirichlet boundary conditions, that is, Brownian motion reflected on rather than killed at the boundary of a time-dependent domain. In [
9], the most fundamental question of recurrence versus transience for normally reflected Brownian motion with time-dependent domains has been carefully studied, and the authors provided some sharp criterions for the recurrence versus transience of normally reflected Brownian motion in terms of the growth rate of the boundary. In [
10] the author provided precise conditions for the recurrence versus transience of one-dimensional Brownian motion with a locally bounded drift, which belongs to the time-dependent domain with a normal reflection at the time-dependent boundary, and the precise conditions provided by the author naturally depend on the growth rates of the boundary and the drift terms of the diffusion processes.
Considering that the CIR model in time-dependent domains has important practical significance and value in the financial field, due to the fact that the evolution of interest rates is often limited to a regional scope, it often changes with the policies of interest rate makers or government management departments. In addition, this CIR model in time-dependent domains has theoretical significance in the field of mathematics and also promotes the research of the properties of transience versus recurrence for stochastic processes.
Table 1 below gives the related progress in this field of transience versus recurrence for stochastic processes in time-dependent domains through the aid of the expression of the generator corresponding to the one-dimensional diffusion process. For more topics on the aspect of transience versus recurrence for stochastic processes, please refer to [
11,
12,
13,
14]. It should be pointed out that, in addition to transience versus recurrence for the conservative random walk, scaling limits for the conservative random walk have also been studied in the work of [
11]. However, we did not address scaling limits for stochastic processes in this article.
Here, we need to emphasize that in [
8,
9,
10], they not only deal with one-dimensional situations, but also with multidimensional situations. For more detailed conclusions, please refer to the literature above for interested readers. In this paper, we only deal with the one-dimensional situations for technical reasons, but we deal with situations where
is not a constant. At present, in this paper, we only deal with the case where
is linear, and of course, we can also consider the nonlinear case (which is not the CIR model). This problem will also be considered in a future work.
When , the CIR process hits zero repeatedly but after each hit becomes positive again; this behavior of hitting zero will also occur even if . Therefore, we do not intend to handle this simple situation; we will only consider . At this point, the CIR process will have an upward positive constant slope a, and the evolution of the CIR process will still have a mean reversion property when . However, when we started considering , we saw that the CIR process will have a completely positive slope, which will encourage the CIR process to continuously move upwards and hit our constantly changing time-dependent upper boundary. If there are no time-dependent boundary restrictions, this will cause the CIR process to explode, thus possessing the property of transience. How is it possible to conduct the CIR process so as not to explode? In other words, how is it possible to transfer from transience to recurrence for the CIR process when ? A natural idea is to add a boundary to the explosion diffusion process, just like the boundary of a time-dependent domain we mentioned above, and when this diffusion process hits the boundary, it will reflect back to our time-dependent domain. This is the main research topic of this paper, which is a fundamental problem in the field of probability, that is, recurrence versus transience, for this normally reflected CIR process with time-dependent domains.
In addition, in the transience case, we also investigate the last passage time, which plays an important and increasing role in financial modelling. The theory of the last passage time is a very important topic in the field of mathematical finance. In this paper, we only provide the probability distribution of the last passage time through the scale function, without exploring its application in the financial modelling field. See [
15], as well as [
16], for the applications the last passage time to hazard processes and models of default risk.
Let us briefly explain the analytical method we used to prove recurrence versus transience for this normally reflected CIR process with time-dependent domains. The first major tool is the well-known Feynman–Kac formula of diffusion process, which provides the stochastic representation for the solution to the boundary value problem. It is worth noting here that the common Feynman–Kac formula is a boundary value problem with a Dirichlet condition or Cauchy condition; however, we still need the Feynman–Kac formula for the boundary value problem with a Neumann boundary condition here, as we need to handle the normally reflected CIR process with time-dependent domains. The second tool we use is the criticality theory of second-order elliptic operators; in particular, the maximum principle or comparison theorem is frequently used in our proofs. It is worth mentioning that some comparison theorems are not clearly found in the literature, and we provide detailed proofs of them in the Appendix. Regarding the criticality theory, we refer the reader to [
17] for more details. Due to the need to obtain precise conditions for coefficients in the CIR process, the selection of certain parameters is also crucial in our proof process.
This paper is structured as follows. In
Section 2, we give some basic notations used throughout this paper and provide some auxiliary results about the moment generating function of the first hitting time. In
Section 3, we prove the results of two recurrent properties, recurrence and positive recurrence, and provide the precise conditions that the coefficients of the CIR process should meet for recurrence and positive recurrence in terms of the growth rates of the boundary, the drift terms, and the diffusion terms of the CIR processes in time-dependent domains. In
Section 4, we prove the result of the transient for the CIR process in time-dependent domains and also provide the precise conditions that the coefficients of the CIR process should meet.
Section 5 concludes, and in
Appendix A, we provide some comparison theorems of second-order ordinary differential equations with nonconstant coefficients. in
Appendix B, the exact solution of a second-order ordinary differential equation with nonconstant coefficients is given by transforming it into one-dimensional Riccati equation.