Could someone help me to figure out solutions of following problems?:
Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that the process has stationary increments, meaning that for all $t_1 \geq s_1, t_2 \geq s_2, ... , t_n \geq s_n$, the distribution of the vector $(X_{t_1} - X_{s_1} , ... ,X_{t_n} - X_{s_n})$ only depends on the time points through the differences $t_1 - s_1, ... , t_n - s_n$.
$(a)$
Show that for all $s, t \geq 0$
$EX_sX_t =1/2 (v(s) + v(t) - v(|t - s|))$
where the function $v$ is given by $v(t) = EX^2_t$
$(b)$
In addition to stationarity of the increments we now assume that $X$ is $H$-self similar for some parameter $H > 0$. Recall that this means that for every $a > 0$, the process $(X_{at})_t$ has the same finite dimensional distributions as $(a^HX_t)_t$.
Show that the variance function $v(t) = EX^2_t$ must be of the form $v(t) = Ct^{2H}$ for some constant $C>0$.
$(c)$
In view of the $(a),(b)$ we now assume that $X$ is a zero-mean Gaussian process with covariance function $EX_sX_t = 1/2(s^{2H} + t^{2H} - |t - s|^{2H})$ for some $H > 0$.
Show that we must have $H \leq 1$. (Hint: you may use that by Cauchy-Schwarz, the (semi-)metric $d(s, t) = \sqrt{ E(X_s - X_t)^2}$ on $[0, 1)$ satisfies the triangle inequality).
$(d)$
Show that for $H = 1$, we have $X_t = tZ$ a.s., for a standard normal random variable $Z$ not depending on $t$.
$(e)$
Show that for every value of the parameter $H \in (0, 1]$, the process $X$ has a continuous modification.
For now I have some ideas about $(a)$ and $(d)$ but have no clue about $(b), (c), (e)$
For $(a)$ my reasoning would be as follows:
$EX_sX_t = E[(X_s-X_t)+X_t][(X_t - X_s)+X_s]$ $=-E(X_s-X_t)^2+E(X_s-X_t)X_s+EX_t(X_t-X_s)+EX_tX_s$ $=-E(X_s-X_t)^2+EX_s^2-EX_tX_s+EX_t^2-EX_tX_s+EX_tX_s$
so we get
$EX_sX_t=1/2 (v(s) + v(t) - E(X_s-X_t)^2)$
but how to show that $E(X_s-X_t)^2 = v(|t - s|)$?
I would say that if $s \leq t$ then $E(X_s-X_t)^2 = E(X_{t-s}-X_0)^2= E(X_{t-s})^2=v(t - s)$ and by symmetry in the case of $s \geq t$ we get $E(X_s-X_t)^2 = v(|t - s|)$. Is this correct?
For $(d)$ I think in this way:
If $H=1$ then $EX_sX_t = 1/2(s^{2H} + t^{2H} - |t - s|^{2H})$ simplyfies to $1/2(s^2 + t^2 - |t - s|^2)$ and this is equal to $st$
on the other hand
$EtZsZ=stEZ^2=st$ in the case of $Z$ being standard Gaussian. Because we are talking about Gaussian proccess it is described by mean $(=0)$ and covariance (st) we are done. Is this the right way?
As I wrote above I have no reasonable ideas about $(b), (c), (e)$.