1 Homework 2
to be handed in: March 2, 2016
Note: always check the web for the latest adaptations in blue in LN and BN.
Exercise 1
Let E be a separable Banach space, i.e. E is a complete, normed vector space over the field R or C, and it has a countably dense subset. Equip E with the Borel-σ-algebra, and denote it by E . Let (Ω, F , P) be a probability space.
In LN Definition 1.1.2 we have defined what we mean by an (E, E ) valued stochastic process X with independent increments. Implicitly, this requires that the difference Xt− Xs be measurable, i.e. Xt− Xs : (Ω, F ) → (E, E ).
Clearly, addition and substraction should be defined, whence, the vector space require- ment. It should be shown that measurability is preserved under addition (and substraction).
So: let X, Y : (Ω, F ) → (E, E ). Prove that X + Y : (Ω, F ) → (E, E ).
Hint: study BN section 3. You may use the results mentioned there.
Exercise 2
Prove Corollary 1.3.4. This means that you have to show that there exists a stochastic process with the properties (i,ii,ii,iv) of Definition 1.1.3.
Excercise 3 LN Exercise 1.5.
Exercise 4 LN Exercise 1.6.
