An ergodic-type theorem for generalized Ornstein–Uhlenbeck processes

Анотація

Abstract. Let be an ℝ d -valued càdlàg random process, and let F be an increasing from zero ℝ-valued random process whose values at all times are measurable w.r.t. some σ-algebra (the class of all such processes is denoted by ). Conditions guaranteing that for every bounded continuous function are found. In the most general theorem they are formulated in terms of F and . Further we consider the case when satisfies an equation of the kind where is an -measurable random variable, is a continuous process of class , A is a matrix-valued random process -measurable in , and S is an ℝ d -valued semimartingale with conditionally on independent increments and initial value 0. In this case, sufficient conditions for ( ) are stated in terms of F and the constituents of the equation. Besides, the characteristic function of an n -dimensional distribution of is found.

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