# 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 $$[X_t]=\lim_{\Delta\Pi\rightarrow0}_{\Pi|[0,t]}\sum_{i=1}^{\#\Pi}\left(X_{\pi_i}-X_{\pi_{i-1}}\right)^2$$ 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 $$[A_t]_n=\lim_{\Delta\Pi\rightarrow0}_{\Pi|[0,t]}\sum_{i=1}^{\#\Pi}\left(A_{\pi_i}-A_{\pi_{i-1}}\right)^n,$$ 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?

-

A change in measure can be thought of as a shift or translation of the cumulant-generating function so that it passes through the origin with the desired slope -- this slope will be the mean of the new measure. The new measure $\mathbb Q$ is then related to the old measure $\mathbb P$ by $$\frac{\mathrm d\mathbb Q}{\mathrm d\mathbb P}=\frac {e^{cA_t}}{M_{\mathbb P}(c)}$$ where $M_{\mathbb P}(c)$ is the moment-generating function of $A_t$ under the old measure evaluated at the point $c$ that was found. –  JL344 Jul 1 '12 at 11:03