# Itô’s formula and Wiener process

The Wikipedia page on the formula https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma and some textbooks I have looked at say we must assume that the relevant time-dependent function is over an Itô drift-diffusion process, and that the diffusion process is a Wiener process https://en.wikipedia.org/wiki/Wiener_process.

Is it absolutely essential that the process be a Wiener process?

Regards and New Year best wishes.

• I'm not sure I get your question completely but you can apply Itô's Lemma to any semimartingale (a sum of a local martingale and a bounded variation process), see here en.wikipedia.org/wiki/…. Amongst others, these include non-continuous processes which jump. All Itô processes are semimartingales, so are e.g. Lévy processes. Dec 31, 2019 at 16:40
• Thanks. The definition of a Wiener process requires being independent and Gaussian. Is a non-continuous process which jumps also Gaussian? Dec 31, 2019 at 17:02
• Look at Levy processes for example. They are a straight forward generalisation of the Wiener process and really only rely on independent increments. They can jump infinitely often and still you can apply Ito's Lemma to them. Dec 31, 2019 at 17:07
• Thanks, but the question is whether a non-continuous process which jumps is Gaussian, i.e. drawn from the normal distribution. Dec 31, 2019 at 17:30
• No neither. A Brownian motion is a Gaussian process and a Lévy process. A Brownian bridge is a Gaussian process which is not a Lévy process. A Poisson process is a Lévy process which is not a Gaussian process. The Heston Stochastic volatility model is neither Gaussian nor Levy. So, neither Lévy and Gaussian are contained in the other one. Dec 31, 2019 at 18:35