# 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. – Kevin Dec 31 '19 at 16:40
• Thanks. The definition of a Wiener process requires being independent and Gaussian. Is a non-continuous process which jumps also Gaussian? – buckner Dec 31 '19 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. – Kevin Dec 31 '19 at 17:07
• Thanks, but the question is whether a non-continuous process which jumps is Gaussian, i.e. drawn from the normal distribution. – buckner Dec 31 '19 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. – Kevin Dec 31 '19 at 18:35