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My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them.
To be more precise, ( if I understand correctly what my professor told me), for some classes of SDEs, we can approach the solution of those SDEs by scaled discrete random walks.
For example,
  • Donsker's theorem gives us a way to construct a Brownian motion by scaled simple symmetric random walks.
  • There is some literature showing that we can approach the Bessel process by some nearest neighbourhood random walks (with suitable scaling and transition probability)

(To be honest, I'm not even sure if I had wrapped that idea correctly).
However, when I asked for a reference, he gave me "Continuous Martingale" by Professor Revuz and Professor Marc Yor which I don't think really related to that subject.
So, I have some references requests that I hope can be helped by the community
Request

  • Would you please give me some references on that matter? And if those references are well-written for newcomers, it would be great.
  • Or at least, some references on "Bessel random walk"?

Thank you.

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2 Answers 2

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Bessel process is a bit different, because of the singularity in the equation. For convergence of Bessel random walks you can have a look at Lamperti, J. (1962). A new class of probability limit theorems. J. Math. Mech. 11, 749– In this paper Lamperti proves convergence to the Bessel process for a class of random walks, which includes nearest-neighbour random walks. You can find a number of references about Lamperti Markov chains in our book at https://arxiv.org/abs/1612.01592 A new revision of this preprint will also include functional convergence as well, but it is not published yet, still revising it.

Some further references about convergence of Markov chains to diffusions are

  1. Ethier and Kurtz have convergence theorems in Chapter 4 of their book Markov processes: characterisation and convergence.
  2. Chapter 11 of Stroock and Vardhan Multidimensional diffusion processes have convegence of Markov chainsto diffusions.
  3. If I remember correctly, Gikhman and Skorokhod have convergence of Markov chains to diffusion in the third volume of Theory of Stochastic Processes.
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  • $\begingroup$ Thank you, I have a strong feeling that this is exactly what I've been searching. Thank you immensely. $\endgroup$
    – Paresseux Nguyen
    Commented Dec 8, 2020 at 14:49
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See Jacod, Shiryaev, "LImit theorems for stochastic processes", Chapter IX.

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  • $\begingroup$ Would you expanding your answer? Why is this a good reference, what is in it? Thanks. $\endgroup$
    – András Bátkai
    Commented Dec 3, 2020 at 8:32
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    $\begingroup$ @AndrásBátkai it contains various limit theorems for stochastic processes :). $\endgroup$
    – Kostya_I
    Commented Dec 3, 2020 at 9:00
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    $\begingroup$ It's a long book, so it would take quite long to write a review. Chapter IX has a section that specifically considers convergence of jump processes to diffusions, i. e., more or less precisely what the OP is asking for. $\endgroup$
    – Kostya_I
    Commented Dec 3, 2020 at 9:03
  • $\begingroup$ Thank you for your response, it'll take some time for me to read through that chapter. At first you, that chapter is as you said, covers more or less what I'm looking for. I'm still looking for other answers. $\endgroup$
    – Paresseux Nguyen
    Commented Dec 3, 2020 at 11:22

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