Stochastic Calculus Beyond Brownian Motion: Jumps and Itô Formulae

# How to measure a non-normal stochastic process?


# How to measure a non-normal stochastic process?












If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable function $f$, $$\mathrm df(X_t) = f'(X_t)\mathrm dX_t + \frac 1 2 f''(X_t)\mathrm d[X_t] ,$$ where $$[X_t]=\lim_{\|\Pi\|\rightarrow0}\sum_{i=1}^{n}\left(X_{t_i}-X_{t_{i-1}}\right)^2,$$ $0=t_0 < \cdots < t_n=t$, and $\|\Pi\|=\max_{1\le k\le n}|t_k-t_{k-1}|$, is the quadratic variation. We are allowed to ignore all higher power variations because all cumulants higher than the second vanish for the normal distribution. But the additivity of cumulants under convolution leads us naturally to consider a stochastic process based on an arbitrary distribution subject only to the requirements that it be infinitely divisible and possess a moment-generating function. So if we let $A_t$ be the constant-linear-drift+martingale process having such a distribution, and we define the higher power variations $$[A_t]_n=\lim_{\|\Pi\|\rightarrow0}\sum_{i=1}^{n}\left(A_{t_i}-A_{t_{i-1}}\right)^n,$$ then these all exist and are equal to the corresponding cumulants of the distribution of $A_t$. This complicates Itô's lemma, which, again if I understand right, must now be written: $$\mathrm df(X_t) = f'(X_t)\mathrm dX_t + \sum_{k=2}^{\infty}\frac 1{n!} f^{(n)}(X_t)\mathrm d[X_t]_n ,$$ but this is not so bad, because all these higher power variations converge in probability and scale linearly with time, except that now we need $f$ to be infinitely differentiable. So, starting off from here, what are the most important consequences to stochastic calculus of refusing to assume that a process can be adapted to a measure based on a normal distribution?

## Answer by lehalle (score 4)

https://quant.stackexchange.com/a/3707

The standard method to manage your kind of problem (i.e. dealing with stochastic processes that are note presented or built thanks to a Brownian motion) is to use a measure change.

The power of Brownian motion is that you have a lot of representation theorems (Doob-Meyer theorem, Wold theorem, etc) that allows to (thanks to a change of measure or a localization method), provides you a way to define a space in which your process (or the residual of your process once you removed easy to deal with components) behaves like a martingale. So you can use Itô on it.

The only effects that can change the Itô formula should be:

- jumps (but you have an Itô semi-martingale formula as far as the Jumps are of Lévy-type, see for instance Jacod-Shiryaev or Cont-Tankov)

- fractional Brownian motion (but here again you have an Itô-like formula, see form instance Stochastic Calculus for Fractional Brownian Motion, I. Theory, by Duncan)

For others you should first think about the proper cleaning of the process and change of measure before doing something fancy I think.