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Can some one kindly go over some of the applications and use of Lie groups in finance? The math is very rigorous and I don't fully understand it or the potential it could have.

Let me share some examples with you, I'm sure there are many more. There even is a financial math graduate program in Europe where there is required course work that teaches Lie groups.

http://www.sciencedirect.com/science/article/pii/S0096300308009156

http://math.sut.ac.th/school/faculty/sergey/filespublic/2006/SrihirunMeleshkoSchultz2006_I.pdf

http://www.phy.cuhk.edu.hk/~cflo/Finance/papers/publications/lie_QF.pdf

Bob Jansen
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pyCthon
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    Could you please explain what lie groups are and why you think it's applicable to finance (suggest areas of application})? This would make a much better question and it would be much more useful for other users of the site. – SRKX Apr 20 '12 at 06:21
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    Please provide a link to a finance article that makes explicit use of lie group theory – Alexey Kalmykov Apr 20 '12 at 10:22
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    Where did you see this? I've never heard of lie groups applied to finance. – chrisaycock Apr 20 '12 at 12:49
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    Well, we do use $\mathbb{R}^n$ all the time. – Brian B Apr 20 '12 at 15:49
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    I've even seen some applications of Lie algebras to option pricing (in the academic literature, needless to say). I'll try to compose an answer over the weekend. – olaker Apr 20 '12 at 17:11
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    It is a strange question! As far as Lie groups are needed to define infinitesimal generators, you need them as soon as you speak about markovian processes. Of course it is the case in quantitative finance. But we are not talking about an application of Lie groups to finance here. It is like asking if the Zorn lemma has applications in finance. – lehalle Apr 21 '12 at 02:43
  • I like this i've seem to generate some controversy. – pyCthon Apr 21 '12 at 18:34
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    I've updated the question with some papers – pyCthon Apr 21 '12 at 18:38
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    Which graduate program is it? – Bob Jansen Apr 22 '12 at 19:36
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    I like the second one: Boonlert Srihirun, Sergey V. Meleshko, Eckart Schulz, On the definition of an admitted Lie group for stochastic differential equations, Communications in Nonlinear Science and Numerical Simulation, Volume 12, Issue 8, December 2007, Pages 1379-1389. – lehalle Apr 30 '12 at 21:27
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    It underlines the relationship between Lie groups and infinitesimal generators. Any tool that can be of use on Lie groups could be on SDE. – lehalle Apr 30 '12 at 21:39

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I didn't read the papers you linked but I can understand that lie groups may be used much as there are used in quantum field theories to build up gauge theories for interaction of particles. The purpose is to have a model that is invariant according to a given transformation group. This introduce interaction terms in the equations.

I can imagine that the same technique can be used in financial stochastic processes to force the introduction of new terms in the equations.

microcosme
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