Abstract
The paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). The coupling techniques traditionally consist of two components: one is local mixing, the other is recurrence. Often in the articles they do not split. Yet, they are quite different in their nature, and this paper separates them, concentrating only on the former. Most of the techniques discussed here work in dimension $d\ge 1$, although, in $d=1$ there is one additional option to use intersections of trajectories, which requires nothing but the strong Markov property and nondegeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\dots $, or by arranging special stopping time sequences and to use the local Markov–Dobrushin (MD) condition, which is one of the most efficient versions of local mixing. Further applications may be based on one or another version of the MD condition; respectively, this paper is devoted to various methods of verifying one or another form of it.
Highlights
The stochastic differential equation (SDE) in Rd t tXt = x + b(Xs )ds + σ (Xs)dWs, t ≥ 0, (1)is considered
Under this condition the process Xn – that is, our solution Xt considered at integer times t = 0, 1, . . . – is a Markov chain (MC), which is, strong Markov
The advantage of the total variation distance for Markov processes is that once a convergence rate is established, say, μn − μ T V ≤ ψ(n) → 0, n → ∞, where μt is the marginal distribution of Xt, μ is any probability measure (ergodic limit for), this rate of convergence can be nearly verbatim transferred to the continuous time: μt − μ T V ≤ ψ([t]) → 0, t → ∞, where [t] is the integer part of t
Summary
General approaches to coupling for SDEs require a (usually positive) recurrence and some form of local mixing For the latter, beside intersections applicable only in the case d = 1, the following tools can be used. Lower and upper bounds of the transition density (requires Hölder’s continuity, at least, for the diffusion coefficient, as well as an “elliptic” or “hypoelliptic” nondegeneracy); here the petite sets condition, popular in the discrete time theory of ergodic Markov chains, along with recurrence properties may be used. In the absence of lower and upper bounds of the density, under the nondegeneracy condition and for general measurable coefficients of SDE, Harnack inequalities in parabolic or elliptic versions may be applied; a petite sets condition apparently may be proved; they are less efficient than the MD condition because the latter guarantees better estimates of the convergence rate. Neither recurrence – the necessary second ingredient in studying convergence and mixing rates – nor coupling itself (except for the basic Lemma 1 added for the reader’s convenience) are not the goals of this paper
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
