On the cross-variation of a class of stochastic processes

The present paper deals with the study of the cross-variation of two-dimensional stochastic process defined using the Young integral with respect to a continuous, $\alpha$-self-similar Gaussian process that does not necessarily have stationary increments, with increment exponent some $\beta >0$. We analyze the limit, in probability, of the so-called cross-variation when $\beta$ in $\left(0,2\alpha \right)$, and we finish by providing some examples of known processes that satisfy the required assumptions.