2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Kevin Dec 31 '19 at 16:40
  • $\begingroup$ Thanks. The definition of a Wiener process requires being independent and Gaussian. Is a non-continuous process which jumps also Gaussian? $\endgroup$ – buckner Dec 31 '19 at 17:02
  • $\begingroup$ 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. $\endgroup$ – Kevin Dec 31 '19 at 17:07
  • $\begingroup$ Thanks, but the question is whether a non-continuous process which jumps is Gaussian, i.e. drawn from the normal distribution. $\endgroup$ – buckner Dec 31 '19 at 17:30
  • 1
    $\begingroup$ 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. $\endgroup$ – Kevin Dec 31 '19 at 18:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.