I am reading Buehler et al. (2022) "Learning to Trade II: Deep Hedging" and the slide on p. 44 states
Fun fact: in discrete time, we can change also the volatility of a process by changing measure.
I am familiar with continuous-time stochastic processes and the change of measure there, which (through Girsanov's theorem) would only change the drift part but not the diffusion part of the process.
Now, according to the quote above, I am assuming that a discrete time version of Girsanov's theorem exists with the additional property of changing the diffusion part in a difference equation. However, I haven't been able to find papers or textbooks describing this. I would be grateful if somebody could point me to such an article (preferably with examples) or could outline how this could be derived.
The authors of those slides don't mention the Girsanov theorem and I don't think we need it to see the fun fact that we can change the volatility of a discrete process by changing the measure. Take $X_i\sim N(0,1)$ i.i.d. and consider the martingale $$ M_n=\sum_{i=1}^nX_i\,. $$ Its volatility is one. To simplify things I assume $n\le N<\infty\,.$ The probability space on which $M_n$ is defined is $\Omega=\mathbb R^N$ equipped with the $N$-fold product of the standard normal PDF $$ p(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,. $$ An obvious equivalent measure that changes $M$'s volatlity to $\sigma$ is the $N$-fold product of $$ q(x)=\frac{1}{\sqrt{2\pi\sigma}}e^{-x^2/(2\sigma)}\,. $$