Where can I read on properties of characteristic functions used in Protter's book?

[justinmir]

I am reading Protter's book on stochastic integration and differential equations.

On a couple of occasions the author used some properties of characteristic functions I am not familiar with. I would appreciate if someone could point me in a right direction.

1. p22 T31 - We are working with Levy process $X$ with independent stationary increments and cadlag paths. We sample $X$ in $s_{j}$, construct linear combinations of $X_{s_{j}}$ and show that $\mathbb{E}\left\{ e^{i\sum u_{j}X_{s_{j}}}\mid\mathcal{G}_{t+}\right\} =\mathbb{E}\left\{ e^{i\sum u_{j}X_{s_{j}}}\mid\mathcal{G}_{t}\right\}$ . From this the author concludes that the sigma algebras $\mathcal{G}_{t}$ and $\mathcal{G}_{t+}$ are identical except probably for events of measure zero.

Could you help me understand why this equality of characteristic functions implies the equality of sigma algebras? Could you point me to a theorem, book or article that will help me fill in the blanks?

2. p30 T39 - We work with two Levy processes $J^{1}$ and $J^{2}$, we show that the characteristic function of their linear combination is equal to the product of characteristic functions $\mathbb{E}\left\{ e^{i(uJ_{t}^{1}+vJ_{t}^{2})}\right\} =\mathbb{E}\left\{ e^{iuJ_{t}^{1}}\right\} \mathbb{E}\left\{ e^{ivJ_{t}^{2}}\right\}$ (subindependence). After that we sample these processes in $s_{j}$, construct linear combinations of their increments and show subindependence of linear combinations of increments. The author then concludes that this is enough to establish independence of $J^{1}$ and $J^{2}$.

Could you please point me in a direction of the missing bit on how we establish independence?

 

[Valkarie]

For both questions, I recommend looking at the book "An Introduction to Measure-theoretic Probability" by B.R. Bhattacharya and E.C. Waymire. This book contains a comprehensive discussion on measure-theoretic probability, including topics such as characteristic functions, sigma algebras, and independence.