# Where Can I Learn About Characteristic Functions in Protter's Book?

• justinmir
Specifically, Chapter 5.5 covers the relationship between characteristic functions and sigma algebras, and Chapter 6.2 discusses independence and subindependence. These chapters should provide the necessary information to fill in the missing gaps and understand the arguments made by the author in the book by Protter. In summary, I recommend consulting "An Introduction to Measure-theoretic Probability" by B.R. Bhattacharya and E.C. Waymire for further clarification on the concepts discussed in Protter's book on stochastic integration and differential equations.
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?

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.

## 1. What are characteristic functions and why are they important in Protter's book?

Characteristic functions are mathematical tools used to describe the properties of a random variable. They are important in Protter's book because they provide a way to analyze and understand the behavior of stochastic processes.

## 2. Can I find information on characteristic functions in Protter's book?

Yes, Protter's book, "Stochastic Integration and Differential Equations," extensively covers characteristic functions and their applications in stochastic processes. It is a valuable resource for understanding these concepts.

## 3. How do I use characteristic functions to analyze stochastic processes?

Characteristic functions can be used to calculate various statistical properties of stochastic processes, such as mean, variance, and higher moments. They also provide a way to determine the distribution of a random variable.

## 4. Are there any other resources I can use to learn about characteristic functions?

Yes, there are various online resources and textbooks that cover characteristic functions and their applications in stochastic processes. Some recommended resources include "Probability and Stochastic Processes" by Roy D. Yates and David J. Goodman and "Introduction to Stochastic Processes" by Gregory F. Lawler.

## 5. Can I use characteristic functions in other areas of science?

Yes, characteristic functions have applications in various fields, including statistics, physics, and engineering. They are also used in financial mathematics and actuarial science to model and analyze random processes.

Replies
1
Views
436
Replies
3
Views
1K
Replies
36
Views
3K
Replies
12
Views
2K
Replies
2
Views
2K
Replies
9
Views
2K
Replies
9
Views
2K
Replies
5
Views
2K
Replies
12
Views
1K
Replies
4
Views
416