Delay differential equations driven by Lévy processes: Stationarity and Feller properties

Abstract

We consider a stochastic delay differential equation driven by a general Lévy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.

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

“…For unconstrained systems, one can consider ordinary delay differential equations with an addition to the dynamics in the form of white noise or even a state dependent noise. There is a sizeable literature on such stochastic delay differential equations (SDDE) especially when the associated noiseless system has a globally attracting equilibrium [2,7,11,15,19,20,22,23,30,34,35,36]. To obtain the analogue of such SDDE models with non-negativity constraints, it is not simply a matter of adding a noise term to the ordinary differential equation dynamics, as this will typically not lead to a solution respecting the state constraint (even if the deterministic model was naturally constrained).…”

“…Our proofs in that section are an adaptation of methods developed recently by Hairer, Mattingly and Scheutzow [12] for proving uniqueness of stationary distributions for stochastic delay differential equations without reflection. An important new aspect of the results in [12] is that they enable one to obtain uniqueness of stationary distributions for stochastic delay differential equations when the dispersion coefficient depends on the history of the process over the delay period, in contrast to prior results on uniqueness of stationary distributions for stochastic delay differential equations which often restricted to cases where the dispersion coefficient depended only on the current state X(t) of the process [7,17,30,34,36], with a notable exception being [15]. See Section 4 for further discussion of this point.…”

“…A common method for showing the existence of a stationary distribution for a Markov process is to exhibit a limit point of a sequence of Krylov-Bogulyubov measures [2,7,15,30]. In light of that, given x o ∈ C d I and T > 0, we define the probability measure…”

“…Most previous work on proving uniqueness of stationary distributions relied on showing the mutual equivalence of distributions of X t at some time t > 0 for all starting states, and then applying either Doob's theorem (see Theorem 4.2.1 in [7]) as in [30], or the techniques of Döblin (see, e.g., [24]) as in [17,34]. However, these arguments cannot be easily extended to situations where σ depends on past states, because of the potential for reconstruction of the initial condition from the quadratic variation process (see [30,36]). In [12], this potential difficulty is avoided by use of a different ergodic argument (see Theorem 1.1 of [12]).…”

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

“…given θ . Markovian systems are more easily real-time implementable and the infinitesimal generator of a Markov process can directly be used to construct its probability distribution by solving the Chapman-Kolmogorov forward equation [20][21][22][23].…”