1
$\begingroup$

Let $\mathcal{X}\triangleq (\Omega,\mathfrak{F}_t,\mathbb{P})$ be a filtered probability space.
Is it possible for a stochastic process $X_t$ defined on $\mathcal{X}$ with increments satisfying

  • $X_t-X_s$ are normal
  • $ Var(X_t - X_s) \underset{t\to s}{\to} 0 , $

for every $t\geq s$ to have sample paths to not have $\mathbb{P}$-a.s. continuous sample paths?

$\endgroup$
2
$\begingroup$

It is possible, the following is an example. Let $X=\{X_t, t\in \mathbb{R}\}$ be stationary Gaussian process with zero mean $\mathsf{E}[X_t]=0$ and the following covariance function, $$ B(h)=\mathsf{E}[X_{t+h}X_t]=\int_0^\infty\frac{\cos(h\lambda)}{(1+\lambda)[\log(1+\lambda)]^{3/2}}\,d\lambda.\qquad h,t\in \mathbb{R}.$$ Then $\lim\limits_{t\to s}\mathsf{Var}(X_t-X_s)=0$, and $X$ has discontinuous sample path with probability one from a result in following book: J. Alder, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Proceses, IMS Lecture Notes-Monograph Series, Vol.12. pp.15 (1.20).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.