Volume 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes. The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness "dx". The calculation for the volume of a sphere with center 0 and radius "r" is as follows. The surface area of the circular disk is formula_10. The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is or where y or z can be taken to represent the radius of a disk at a particular x value. Using y
Feynman–Kac formula term contains, in parentheses, the above partial differential equation and is therefore zero. What remains is Integrating this equation from "t" to "T", one concludes that Upon taking expectations, conditioned on "X" = "x", and observing that the right side is an Itô integral, which has expectation zero, it follows that The desired result is obtained by observing that and finally In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks. Feynman–Kac formula The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial