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In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary value problems in mathematical finance?

I came across this Q&A: "Hyperbolic and Elliptic PDEs in Quant Finance" on the Quantitative Finance StackExchange, but I cannot find a clear mathematical formulation of a "perpetual exchange option" (whatever that is)...

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    $\begingroup$ Long shot, but they may be relevant in ergodic control problems, where the long term behaviour of SDE and stationary measures is what matters. $\endgroup$
    – Nate River
    Commented Jan 17 at 19:30
  • $\begingroup$ @NateRiver That sounds like a nice example, do you know of a reference? $\endgroup$
    – ABIM
    Commented Jan 17 at 19:57
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    $\begingroup$ I do not, unfortunately. $\endgroup$
    – Nate River
    Commented Jan 17 at 19:59
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    $\begingroup$ A perpetual exchange option would be an option to trade one asset for another that never expires. For example, you could have the option to trade 1 share of Apple for 1 share of Microsoft at the time of your choosing, until the end of time. $\endgroup$
    – arsmath
    Commented Jan 17 at 21:19
  • $\begingroup$ Ya that's cool. I think I may go for a nice control problem given by some reasonable BSDE? $\endgroup$
    – ABIM
    Commented Jan 17 at 21:36

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For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "mathematical finance" includes insurance mathematics, there is one application.

A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics (2010)

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations.

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