Proving a stochastic process is Gaussian and continuous 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).

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.
