# What is Heston's equation?

This paper mentions the elliptic Heston operator:

$Av:= -\frac y2(v_{xx}+2\rho\sigma v_{xy} + \sigma^2v_{yy}) - (c_0 - q - \frac y2)v_x + \kappa(\theta -y)v_y + c_0v$.

Then boundary value problem are discussed:

$Au=f \text{ on } \Omega \\ u = g \text{ on } \partial\Omega$

I would like to know how people use such Dirichlet conditions in mathematical finance.

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Expanding a bit on chrisaycock's answer, and noting in particular from the abstract

In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

we can see that this would be used to price those few rare cases of perpetual options.

The only traded examples I know of are perpetual convertible preferred securities, for example from Wells Fargo's offerings. Such securities are lightly traded by the market players and therefore not always analyzed using the full machinery of a stochastic vol model, even if they should be in principle.

In practice, these "perps" are so bond-like that it is often more useful to think of them as fixed-income instruments. The main concern with them is that the issuer will stop paying the dividends or change capital structure, so it is a bit ridiculous to spend one's time on a fancy stochastic vol model when all the interesting stochastic events have to do with unrelated variables such as alterations in capital structure.

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If I understood you correctly such models have little use and mathematicians just play with them. To do it clear, is there any sense to do research of PDE and SPDE in connection with mathematical finance? –  nikita2 Oct 28 '12 at 10:08