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I am having difficulties understanding when and how to prove if a given stochastic process is Gaussian and continuous. In my assignments and exams I am given a process, e.g. $$ X_t = (1-t)\int_0^t \frac{1}{1-s}dB_s $$ or $$ Y_t = tB_t - \sigma \int_0^t B_s ds$$ Here, I am asked if the process is Gaussian and/or continuous. The professor could not explain the reasoning very well, he said if a process can be written as a discrete time Gaussian vector, then in the limit it would be a discrete Gaussian process.

My approach would be to divide the time grid into a partition $ \pi = \lbrace t_1= 0, t_2,\; \dots \; ,t_m =t \rbrace$ and discretise the process as a vector:

$$(X_{t_1},X_{t_2},... X_{t_m} ) = \left((1-t_1)\int_0^{t_1} \frac{1}{1-s}dB_s \;, \; \dots \;,\; (1-t_m)\int_0^{t_m} \frac{1}{1-s}dB_s \right) \\ = \lim_{n \to +\infty} \left( (1-t_1)\sum_{k=0}^{n-1} \frac{1}{1-t_{\frac{k}{n}}}\left[B(t_{\frac{k+1}{n}})-B(t_{\frac{k}{n}})\right] , \;\dots \;, \;(1-t_m)\sum_{k=0}^{n-1} \frac{1}{1-t_{\frac{mk}{n}}}\left[B(t_{\frac{mk+1}{n}})-B(t_{\frac{mk}{n}})\right] \right) $$ Is this approach correct for Gaussianity? How would you go about showing continuity? (He simply says in his notes "by construction it is continuous" but this is not satisfactory for me).

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The distribution of a stochastic process is expressed through the finite-dimensional distributions, i.e., the collection of distributions of all vectors of the form $\left(X_{t_1},X_{t_2},...,X_{t_k}\right)$ for all possible $k = 1,2,...$ and $t_1 < ... < t_k$. A continuous-time stochastic process is Gaussian if all of its finite-dimensional distributions are Gaussian -- so, yes, it is sufficient to look at discrete points in time, but your argument has to work for arbitrarily many such points.

For continuity, you should fix a sample path $\omega$ and argue that the mapping $t \mapsto X_t\left(\omega\right)$ is continuous for a.e. $\omega$. The Brownian motion itself has this property, so you can use it as a building block for other processes.

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    $\begingroup$ So my discretisation and approach was correct for the Gaussianity in my example? Also, could you recommend a book or website where I can find more explanation and examples of these two properties ? Thanks. $\endgroup$ Aug 31 '20 at 14:58
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    $\begingroup$ For Gaussianity, you first have to take a finite-dimensional vector from the process; then, yes, you can approximate it using a sequence of vectors of discretized increments, and study its convergence in distribution. For a reference on Gaussian processes in general, I believe Karatzas & Shreve, "Brownian motion and stochastic calculus" is pretty standard. $\endgroup$ Sep 2 '20 at 13:02

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