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?


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).


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