Exercise 1 (1.5 pts) LN Exercise 1.9

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1 Homework 3

to be handed in: March 9, 2016

Note: always check the web for the latest adaptations in blue in LN and BN.

This time we will give you the option of selecting exercises according to your interest. You should select exercises worth in total at most 10 points. If you hand in exercises worth more than 10 points, then this will therefore not be beneficial to your grade.

Exercise 1 (1.5 pts) LN Exercise 1.9.

Excercise 6 (3 pts) LN Exercise 1.24.

This exercise deals with the validity of the Galmarino test. This is a test for checking whether a random time is a stopping time τ for a canonical process on a path space (Ω = E^T, F = E^T, P).

Basically, the test tells us that a random time τ is a stopping time if and only if for any trajectory ω, it holds that if τ (ω) ≤ t and ω⁰ ∈ Ω is equal to ω on [0, t], then also τ (ω⁰) ≤ t.

This intuitive characterisation clearly requires the underlying probability space to be the

‘path space’. But as we have seen, we can always view a stochastic process to live on a ‘path space’.