A Poisson process is a real-valued stochastic process (left(N_{t} ight)_{t geqslant 0}) such that (N_{0}=0), (N_{t}-N_{s} sim
Question:
A Poisson process is a real-valued stochastic process \(\left(N_{t}\right)_{t \geqslant 0}\) such that \(N_{0}=0\), \(N_{t}-N_{s} \sim N_{t-s}\) and for \(t_{0}=0
a) Show that the process \(\left(N_{t}\right)_{t \geqslant 0}\) does not satisfy the assumptions of the Kolmogorov-Slutsky-Chentsov theorem, Theorem 10.1.
b) Show that (10.1) holds true for \(n=1, \alpha>0\) and \(\beta=0\). Discuss the role of \(\beta\) for Theorem 10.1.
c) Let \(\lambda=1\). Determine for the process \(X_{t}=N_{t}-t\) the mean value \(m(t)\) and the covariance \(C(s, t)=\mathbb{E}\left(X_{s} X_{t}\right), s, t \geqslant 0\).
Data From Theorem 10.1
Data From (10.1)
Step by Step Answer:
Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher