Feynman-Kac formula

The set of Feynman Kac is a result of probability theory , for that. B. is used in financial mathematics . He combines the theory of probability with the theory of partial differential equations . The name goes back to Richard Feynman and Mark Kac .

Statement of the sentence

Let us first be a process adapted to the filtration and the solution of the stochastic differential equation ${\ displaystyle X_ {t}}$ ${\ displaystyle (F_ {t}) _ {t}}$

{\ displaystyle dX_ {t} = \ sigma (t, X_ {t}) dW_ {t} + \ mu (t, X_ {t}) dt}

.

${\ displaystyle (X_ {t}) _ {t}}$ is therefore an Itō process . Be further

{\ displaystyle h: \ mathbb {R} \ rightarrow \ mathbb {R}}

a limited, Borel-measurable function and the conditional expectation of its value in the information . Then the partial (non-stochastic!) Differential equation is fulfilled ${\ displaystyle g (t, x) = \ mathbb {E} (h (X_ {T}) \ mid X_ {t} = x)}$ ${\ displaystyle t}$ ${\ displaystyle X_ {T}}$ ${\ displaystyle g}$

{\ displaystyle g_ {t} (t, x) + g_ {x} (t, x) \ mu (t, x) + {\ frac {1} {2}} g_ {xx} (t, x) \ sigma (t, x) ^ {2} = 0}

with the boundary condition . ${\ displaystyle g (T, x) = h (x)}$

The proof uses the martingale property of conditional expectation and the fact that an Itō process (given in ) is martingale if and only if its drift term disappears. ${\ displaystyle g}$

example

For example, the payout of a financial instrument (such as a call option ) could be based on the value of (such as a share). Then describes the pricing process of this instrument. is the derivative of the price of the underlying asset, in the case of an option therefore their Delta . is theta in the case of a call option. ${\ displaystyle h}$ ${\ displaystyle X_ {t}}$ ${\ displaystyle g}$ ${\ displaystyle g_ {x}}$ ${\ displaystyle {\ frac {g_ {x}} {g}}}$ ${\ displaystyle {\ frac {g_ {t}} {g}}}$

literature

Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. 6th edition, Springer, Berlin 2003, ISBN 978-3-540-04758-2 .
John Michael Steele: Stochastic Calculus and Financial Applications. Springer, New York 2001, ISBN 0-387-95016-8 .