Solution: Properties of a counting process (Exam 2021)

a) \(P(dN(t)>1)=0\) is correct because a counting process is defined only to have jumps of size \(1\).

b) The statement "\(N(t)\) is Poisson distributed with mean value \(\Lambda (t) = \int_0^t \lambda(s) ds\)" is incorrect because \(N(t)\) does not need to be Poisson distributed. If \(N(t)\) counts the number of events in a Poisson process the statement would be true, but the statement is not generally correct.

c) \(N(t)\) is a sub-martingale. Since the \(N(t)\) is an increasing function one must also have that \(\mbox{E}[N(t)|{\cal F}_s] \geq N(s)\).

d) \(2N(t)+5\) is not a counting process because a counting process is required to start at zero at time zero.

e) For \(0<t_1<t_2<t_3\) the increments \(N(t_2)-N(t_1)\) and \(N(t_3)-N(t_2)\) do not need to be independent. If \(N(t)\) counts the number of events in a Poisson process it would be true, but the statement is not generally true for counting processes.

f) It is not correct that \(\lambda(t)\in [0,1]\). The \(\lambda(t)\) can not be negative, but it may be larger than one.