# Probability distributions as solutions to differential equations

As far as what I can tell, the popularity of the Black-Scholes-Merton model partly stems from the fact that it formulates the value of a derivative in a differential form in which the solution has a known distribution. From this perspective, the dynamic hedging argument was simply a modeling assumption which allowed its authors to pose the problem in a way which already had a known solution. Anyway, I was hoping to make use of such transformations to identify solutions to other similarly posed problems. However, I have not been able to find a comprehensive reference on the topic of transformations of differential equations to probability distributions.

My intuition is that such a resource might be helpful in researching solutions to other types derivatives, such as path dependent options.

For example, the current Wiki entries on the Inverse-chi-squared distribution and Normal-inverse-gamma distribution lack the differential equations to which these distributions are solutions. Moreover, historical Wiki entries actually contained such solutions, such as this redacted entry on Inverse-chi-squared distribution:

$\left\{2x^{2}f_{}'(x)+f_{}(x)(-\nu +\nu x+2x)=0,f_{}(1)={\frac {(2e)^{{-\nu /2}}v^{{\nu /2}}}{\Gamma \left({\frac {\nu }{2}}\right)}\sim\frac{1}{\chi^2[\nu]}}\right\}$

Were these solutions redacted because they were wrong? If not, why?

Moreover, does comprehensive resource exist which connects known distributions as solutions to various differential equations?

• "[...] the dynamic hedging argument was simply a modeling assumption which allowed its authors to pose the problem in a way which already had a known solution." I think this is inaccurate, the authors started from a purely financial argument (i.e. dynamic hedging of an option), which can be represented mathematically as a PDE, and through transformations of the original PDE ended up finding the heat equation. Apr 28 '18 at 9:54
• Maybe you might want to look at Feynman-Kac theorem, which establishes a connection between the (parabolic) PDE and the distribution of the underlying asset. Apr 28 '18 at 9:56