Hallo, is the sum of two stochastically continuous processes again a stochastically continuous process? why? Thank you very much, Paolo.
Yes. A stochastically continuous process is a mapping $t \mapsto X_t \in L^0$ into the space of random variables that is continuous with $L^0$ given the topology of convergence in probability. Since $L^0$ is a topological vector space for this topology  in particular, addition is continuous  it is clear that $t \mapsto (X_t, Y_t) \mapsto X_t + Y_t$ is continuous as a composition of continuous mappings. 

