Feynman–Kac formula

The set of Feynman - Kac is a result of probability theory, which finds application in financial mathematics for example. It combines the probability theory with the theory of partial differential equations. The name goes back to Richard Feynman and Mark Kac.

Statement of the theorem

Be first one to the filtration adapted process and solution of the stochastic differential equation

Is therefore an Itō process. further, let

A bounded Borel - measurable function and in the Information desk in conditional expectation of its value. Then the partial ( nicht-stochastische! ) satisfies the differential equation

With the constraint.

The proof uses the martingale of the conditional expectation and the fact that an Ito process ( given in ) if and martingale if its drift term vanishes.

Example

For example, could the payment of a financial instrument (such as a call option ), based on the value of (about one share). Then describes the price process of this instrument. is the derivative of the price of the underlying instrument, in the case of an option, therefore, is its delta. in the case of a call option theta.