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I've read Merton's article "On the Mathematics and Economics Assumptions of Continuous-Time Models" (Reprinted in Continuous-time Finance, Chapter 3), where Merton proved that the price of an asset follows an Itô process (no "rare events", or only those of type I/II) or a jump-diffusion process (with type III "rare events") under certain assumptions when taking the continuous-time limit.

However, in Merton's article he only considered the case of one asset. Are there any generalizations of this result to $n\geqslant 2$ assets? (I assume such generalizations exist, but I can't find any)


Here is a link of the article mentioned above: https://www.dropbox.com/scl/fi/of0rmatcimlrrqn2aucxu/On-the-Mathematics-and-Economics-Assumptions-of-Continuous-time-Models.pdf?rlkey=t498s5p91f8fo1x93smflc3m1&dl=0


Edit: There is a well-known result of prices in the binomial model (Cox-Ross-Rubinstein model) approaching a diffusion process in continuous time. However, the result in Merton's article is much stronger, since it does not specify a functional form.

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  • $\begingroup$ I just glanced quickly but more than one asset would be ( I think ) ito with correlated brownian motions which for $n=2$ has been discussed on this list. So, maybe if you do a search here ( or google instead ) along those lines and combine it with the terms in Merton's title, something might appear ? $\endgroup$
    – mark leeds
    Apr 2 at 5:16
  • $\begingroup$ I know that n-dimensional Ito processes/jump-diffusion processes can be used to model asset prices. However, what I am interested is the mathematical/economic justification for doing so, which is what Merton's article aims to provide. (Actually, I find it quite weird that there is justification like this in the one-dimesional case, but I can't find any paper/books/online material doing this in the n-dimensional case, although n-dimensional Ito/jump-diffusion processes pop up everywhere in finance.) @markleeds $\endgroup$ Apr 2 at 6:12
  • $\begingroup$ Hi Steve: I understand but I don't know where you will find that. Good luck. $\endgroup$
    – mark leeds
    Apr 3 at 1:41

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