Finding the auto-correlation function when $X(t)$ has independent increment

Question

Let $X(t)$ be a zero mean stochastic process with independent increment, namely for every $t_1<t_2<\ldots<t_k$ the random variables $X_{t_2}-X_{t_1}$, $X_{t_3}-X_{t_2}$, $\ldots,X_{t_k}-X_{t_{k-1}}$ are independent.

I wish to find the autocorrelation of $X(t)$, which is $R_X(t, t+\tau)$

I wonder if there is a way to simplify my expression

$R_X(t_1, t_2) = \mathbb{E}[X(t_1)X(t_2)] = \int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty x_{t_1} x_{t_2} f_{X(t_1),X(t_2)}(x_{t_1}, x_{t_2}) dt_1 dt_2$

Using the independent increment property, we have that

$R_X(t_1, t_2) = \int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty x_{t_1} x_{t_2} f_{X(t_1),X(t_2)}(x_{t_1}, x_{t_2}) dt_1 dt_2 $

$= \int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty x_{t_1} x_{t_2} f_{X(t_1)}(x_{t_1}) f_{X(t_2) - X(t_1)}(x_{t_2} - x_{t_1}) dt_1 dt_2$

Here I was hoping to obtain a more useful expression. Does anyone see how to proceed?

Answer 1

Writing $X(t_{2})=X(t_{1})+(X(t_{2})-X(t_{1}))$,

$$ \begin{align*} \mathbb{E}(X(t_{1})X(t_{2}))& =\mathbb{E}(X(t_{1})^{2})+\mathbb{E}[X(t_{1})(X(t_{2})-X(t_{1}))] \\ & = \mathbb{E}(X(t_{1})^{2})+\mathbb{E}(X(t_{1}))\mathbb{E}(X(t_{2})-X(t_{1})) \\ & = \mathbb{E}(X(t_{1})^{2}). \end{align*} $$