Problem: Properties of a counting process (Exam 2021)

Eksempel

Exam June 2021, problem 2.

Let \( \{ N(T); t \geq 0 \} \) be a counting process with intensity process \(\lambda(t)\), i.e.

\[ \lambda(t) dt = P(dN(t) = 1 \mid \mathcal{F}_{t-})\]

where \( \{\mathcal{F}_{t-}\} \) is the history containing all information about up to time and including time \(t\).

Problem

Adopting the standard notation used in the course, specify which of the following
statements you from the above information can know for sure is true.

a) \(P(dN(t) > 1) = 0\)

b) \(N(t)\) is Poisson distributed with mean value \(\Lambda(t) = \int_0^t\lambda(s) ds \)

c) \(N(t)\) is a sub-martingale

d) \(2N(t) + 5\) is a counting process

e) For all \(0<t_1<t_2<t_3\), \(N(t_2)-N(t_1)\) and \(N(t_3)-N(t_2)\) are independent

f) \(\lambda(t) \in [0, 1] \) for all \(t\geq 0\)