On the reflected Ornstein–Uhlenbeck process with catastrophes

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Abstract

The reflected Ornstein–Uhlenbeck process in the presence of catastrophes that occur at exponential rate is considered. Explicit expressions for the transition probability density and for related moments are determined. The first visit time to zero state is also investigated. A systematic computational analysis is performed to elucidate the role played by the parameters.

Introduction

Reflected Ornstein–Uhlenbeck (OU) process is widely used in various applied fields, such as neuroscience, mathematical biology, finance and queueing theory. Specifically, the unrestricted OU process is the most frequently proposed stochastic model for the single neuronal activity (see, for instance, [8], [18], [21], [24], [36], [38], [46]). In such context, the membrane potential evolution can be described by focusing the attention on the OU process confined by a lower reflecting boundary that can be interpreted as the neuronal reversal hyperpolarization potential (cf. [23], [39]). The OU process with reflection at the origin also arises as an approximating process for population dynamics (cf. [1], [37]) and for queueing systems (see, for instance, [47], [48]). Furthermore, the diffusion limits of a number of classical birth and death processes asymptotically approach the OU process (cf. [22], [27], [33], [37]). In particular, in [22] the reflected OU process arises as a heavy-traffic approximation for queueing systems characterized by linear arrival and service rates. Recently, the reflected OU process has been applied to the so-called regulated financial market (see, for instance, [4], [5], [6]). In various types of instances, first-passage time densities are invoked to describe events such as firing times in neuronal modeling, extinction in population dynamics and busy period in queueing systems (cf. for instance, [2], [3], [9], [17], [34], [40], [44]). References to other applications in economics, in finance and in queueing systems can be found in [29].

During the last three decades, great attention has been paid to the description of biological, physical and engineering systems subject to various types of catastrophes, a catastrophe being defined as a random event resulting in the extinction of all individuals or customers in the system. The usual framework is that the system evolves according to the dynamics of some continuous-time Markov chain and it is influenced by catastrophes that occur at exponential rate ξ (cf. [7], [10], [13], [14], [31], [32], [35], [41], [42], [43]). These works are concerned with various quantities, such as the transient and the stationary probabilities, the time of extinction and the first occurrence time of effective catastrophe. The results obtained for continuous-time Markov chains have suggested the possibility of deriving corresponding results for diffusion processes bounded by one reflected boundary and subject to catastrophes that occur at exponential rate ξ. Indeed, in [12], [13], [15], [16] some general results for the transient and steady-state probability density functions (pdf’s) of diffusion processes in the presence of catastrophes have been obtained. In particular, in [12] analytical and computational results for transient and steady-state pdf’s for the Wiener process in the presence of catastrophes have been determined. Instead, in [13] a heavy-traffic approximation for the M/M/1 queue in the presence of catastrophes is given that is seen to be equivalent to a Wiener process subject to randomly occurring jumps. Furthermore, in [15] time-non-homogeneous processes in the presence of catastrophes that occur with a time-varying intensity function are considered. Instead, in [16] a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate and followed by exponentially-distributed repair times, is studied. A heavy-traffic approximation of this system is a Wiener process with jumps. Jump-diffusion processes as models for neuronal activity are also considered in [24], [25], [28].

The aim of the present paper is to provide some quantitative informations on the reflected OU process subject to catastrophes that occur at exponential rate ξ reducing the state of the system instantaneously to zero. The paper is organized as follows. In Section 2 we obtain various relations between functions characterizing the process with catastrophes and those of the same process in the absence of catastrophes. In Sections 3 Reflected OU process with catastrophes, 4 First-visit time for OU process with catastrophes we focus on the OU process restricted to [0,+) by a reflecting boundary in zero state under the assumption that random catastrophes occur with exponential rate. In particular, we determine the transition and steady-state pdf’s and their moments (Section 3) and we analyze the problem of the first visit time to zero state (Section 4). Extensive numerical computations with MATHEMATICA have been performed to show the role played by the involved parameters. Moreover, in Section 5 an application of the OU process with catastrophes is considered. Finally, in Appendix A some classical results concerning the reflected OU process are given.

Section snippets

Diffusion process with catastrophes

Let {X(t),t0} be a regular one-dimensional time-homogeneous diffusion process with drift A1(x) and infinitesimal variance A2(x) restricted to the interval I=[0,r2) by a reflecting boundary in zero state (cf. [11]). For this process we denote with r(x,t|x0)=dP{X(t)x|X(0)=x0}/dx the transition pdf, with 0x0<r2. We construct a new stochastic process X(t) defined in I as follows. Starting from the state x0 at the initial time t=0, the process X(t) evolves according to the process X(t) until

Reflected OU process with catastrophes

Let X(t) be an OU process restricted to the interval [0,+) by a reflecting boundary in zero state, characterized by drift and infinitesimal varianceA1(x)=ηx,A2(x)=σ2(η0,σ>0).The boundary x=+ is nonattracting-natural if η<0 and attracting-natural if η>0, whereas x = 0 is a regular boundary (see, for instance, [30] for the classification of the boundaries). Some classical results on the reflected OU process are given in Appendix A.

Furthermore, let X(t) be the reflected OU process, defined in [0,

First-visit time for OU process with catastrophes

In this Section we focus our attention on the random variable Tx0 that describes the first-visit time of X(t) from X(0)=x0>0 to the zero state.

Proposition 4.1

The transition pdf of X(t) in the presence of an absorbing boundary at 0 is:α(x,t|x0)=e-ξtf(x,t|x0)-f(x,t|-x0)(x,x0>0),where f(x,t|x0) is given in (A.2). Furthermore, the FVT pdf of X(t) is:g(0,t|x0)=e-ξtg(0,t|x0)+ξe-ξtErfx0eηtσηe2ηt-1(x0>0),with g(0,t|x0) given in (A.8).

Proof

Eq. (31) can be easily obtained by virtue of (4), (A.7). Furthermore, making

A heavy-traffic approximation to a queueing system

Let {N(t),t0} be a birth–death process with catastrophes, describing a queueing system such that the transitions occur according to the following scheme: (i) nn+1 with rate αn=αn+γ (n=0,1,); (ii) nn-1 with rate βn=βn+γ (n=2,3,); (iii) 10 with rate β1+ξ; (iv)  n0 with rate ξ (n=2,3,). The effect of each catastrophe is to make the queue instantly empty. For all n=0,1, and t>0 the transition probabilities pn(t)=Prob{N(t)=n|N(0)=0} satisfy the following system of forward equations:ddtp0(t)=-

Acknowledgments

This work has been performed under partial support by MIUR (PRIN 2008). We acknowledge the constructive criticism of an anonymous reviewer on an earlier version of this paper.

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