On Existence and Uniqueness of Stationary Distributions for Stochastic Delay Differential Equations with Positivity Constraints

Abstract

Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering. There is interest in understanding what effect noise has on the behavior of such models. Here we consider a multidimensional stochastic delay differential equation with normal reflection as a noisy analogue of a deterministic system with delayed feedback and non-negativity constraints. We obtain sufficient conditions for existence and uniqueness of stationary distributions for such equations. The results are applied to an example from Internet rate control and a simple biochemical reaction system. AMS 2000 subject classifications: 34K50, 37H10, 60H10, 60J25, 93E15.

Stability in distribution of Markov-modulated stochastic differential delay equations with reflection

“…Hence, the action of L(t ) is termed reflection at the boundary of the orthant n + . Further, the regulator L(t ) satisfies the following properties (see also Harrison [4] ): We will refer to Definition 2.1.1 of Kinnally and Williams [6] for the definition of solution to equation (4). For the existence and uniqueness of the solution we shall impose the following hypothesis:…”

“…In particular, due to properties (L1) and (L2), (X (t ), L(t )) is a solution to the Skorokhod problem for S(X ). Using the form of solution to Skorokhod problem, there exist Lipschitz functions φ with Lip-coefficient k φ > 0 and ψ with Lip-coefficient k ψ > 0 (under the sup-norm, see Appendix A of [6] ) such that…”

“…In Ref. [6] , Kinnally and Williams studied the existence and uniqueness of stationary distribution of the segment process X t corresponding to the solution X (t ) of equation (1). The stability in distribution for Markov-modulated SDEs has been studied in many papers, for example: in Ref.…”

“…We would like to point out that the method used in this paper is different from that of Ref. [6] , but is similar to that of Ref. [14] , where the stability in distribution of SDDEs with Markovian switching was investigated.…”

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

“…This can be obtained in the following customary way: Set Z(t) = x(0) + It is known (see e.g. [8,13]) that we have an explicit formula for the regulator term Y in terms of Z, the so-called reflector term: for each i = 1, . .…”