Existence of the stationary regime of a Non-Markovian Stochastic Differential Equation

TL;DRAbstract

In this paper, we obtain some existence results of stationary solutions to a class of SDEs driven by continuous Gaussian processes with stationary increments. We propose a constructive approach based on the study of some sequences of empirical measures of Euler schemes of these SDEs. In our main result, we obtain the functional convergence of this sequence to a stationary solution to the SDE. We also obtain some specific properties of the stationary solution. In particular, we show that, in contrast to Markovian SDEs, the initial random value of a stationary solution and the driving Gaussian process are always dependent. This emphasizes the fact that the concept of invariant distribution is definitely different to the Markovian case.

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