A (mathbb{R})-valued stochastic process (left(X_{t}ight)_{t geqslant 0}) such that (mathbb{E}left(X_{t}^{2}ight)
Question:
A \(\mathbb{R}\)-valued stochastic process \(\left(X_{t}ight)_{t \geqslant 0}\) such that \(\mathbb{E}\left(X_{t}^{2}ight)<\infty\) is stationary (in the wide sense) if \(m(t)=\mathbb{E} X_{t} \equiv\) const. and \(C(s, t)=\mathbb{E}\left(X_{S} X_{t}ight)=g(t-s), 0 \leqslant s \leqslant t<\infty\) for some even function \(g: \mathbb{R} ightarrow \mathbb{R}\). Which of the following processes is stationary?
a) \(W_{t}=B_{t}^{2}-t\);
b) \(X_{t}=e^{-\alpha t / 2} B_{e^{\alpha t}}\);
c) \(Y_{t}=B_{t+h}-B_{t}\);
d) \(Z_{t}=B_{e^{t}}\).
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Related Book For
Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
Question Posted: