Itō's lemma

The lemma of Itō (also Itō formula), named after the Japanese mathematician Kiyoshi Itō is a key statement in the stochastic analysis. In its simplest form, it is an integral representation of stochastic processes, the functions of a Wiener process. It thus corresponds to the chain rule or substitution rule of classical differential and integral calculus.

Version of Wiener processes

Be a ( standard ) Wiener process and a twice continuously differentiable function. Then we have

Here, the first integral is to be understood as ITO integral and the second integral as an ordinary Riemann integral (through the continuous paths of the integrand ).

For the process defined by for this representation is in differential notation

Version for Ito processes

A stochastic process is called Itō process, if

For two stochastic processes, applies (for more details, see stochastic integration). In differential notation:

Is a component in the first and once in the second twice continuously differentiable function, so the process is defined by an Ito process, and it is

Here, and denote the partial derivatives of the function according to the first and second variables. The second representation follows from the first by substituting and combining the - and -terms.

Version for semimartingales

Be a -valued semimartingale and be. Then again a semimartingale and it is

Here, the left-hand limit and the associated jump process is. With the quadratic covariation of continuous fractions of components and inscribed. If a continuous semimartingale, the last sum vanishes in the formula and it is.

Examples

• For true.

• Using the lemma we can prove easily that the geometric Brownian motion