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August 29, 2010 2:18 AM   Subscribe

Where can I find a proof for the strong order of convergence of the Euler-Maruyama or Milstein schemes used to model stochastic differential equations?
posted by moorooka to Technology (4 answers total)
Curious, you taking a Continuous Time Finance class? These SDEs seem to arise almost exclusively in such contexts (AFAIK, if any other industry uses such processes I'd like to know, interesting topic …) but in industry we typically accept convergence if the baseline conditions of Lipschitz and linear growth are met (ha! trusting souls, I know!). Seems like you see Euler-Maruyama used mostly for path independent European options, while Milstein is often applied to path dependent European options, so I'd be curious what you're up to, if it isn't proprietary, that is.

But to try to answer your query: I haven't been through this material in a while but there are a couple of papers I'm aware of which might help:

Higham, D, 2000, 'An Algorithmic Introduction to Numerical Simulation of SDE', SIAM Review, Vol 43, No 3

Higham, D., J., Mao, X., 2005, 'Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process', Journal of Computational Finance, Vol 8, No 3

We used Seydel, Tools for Computational Finance in my CTF class, I seem to recall a proof of Euler-Maruyama convergence was presented. That being said, I don't believe he addressed Milstein convergence, but its been a while since I've read it (don't even know where my copy is, to be honest), so I'd suggest a trip to a University library before purchasing.

Hope this helps!
posted by Mutant at 4:03 AM on August 29, 2010 [1 favorite]

yes I'm just a student taking a computational finance course and proving the order of convergence for these schemes and also the Balance Method is part of an assignment. I can code up a loglog plot that shows the order of convergence as a slope, but I've got to present an analytical proof, which is something I'm terrible at. I thought that I'd be able to find these proofs easily by googling but I just can't find them!
posted by moorooka at 3:03 PM on August 29, 2010

Ah excellent topic to study !

Ok I doubt you'll find those proofs just by googling, and your professor knows this for sure (which is why they were assigned).

As you probably know you'll see lots of discussions where they present the baseline conditions, then work it through part way, ending with some type of leading phrase (which is incredibly frustrating, I can empathise) implying the proof is obvious.

What I'd do: when I was working through this material I had similar assignments and I'd find as many of these presentations as I could. Then I'd sequester myself alone in a room, emerging a couple of days later with my proof. I couldn't do it any other way, realise some folks are sharper but there you go.

Still, try Seydel in the first instance, and Higham's papers will probably help as well. Best of luck!
posted by Mutant at 4:11 AM on August 30, 2010

I've got to present an analytical proof, which is something I'm terrible at.

I don't know anything about stochastic differential equations, but if you are inexperienced with proofs in general I could post some tips later if you're interested.
posted by Jpfed at 6:52 AM on August 30, 2010

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