Assume $\xi_t$ is a stochastically continuous random process on $\left.[0,\,\infty\right.)$.
Prove $$\varphi(t_1,\,t_2,\,\lambda)=Ee^{i\lambda\left(\xi_{t_2}-\xi_{t_1}\right)}$$ is continuous in $\left(t_1,\,t_2\right)\in \left[0,\,1\right]^2$ where $\lambda\in \mathbb{R}$.
We have $$ \begin{aligned} &\left|\varphi\left(t_1,\,t_2,\,\lambda\right)-\varphi\left(t_1^\prime,\,t_2^\prime,\,\lambda\right)\right|\\[.4cm] =&\left|Ee^{i\lambda\left(\xi_{t_2}-\xi_{t_1}\right)}-Ee^{i\lambda\left(\xi_{t_2^\prime}-\xi_{t_1^\prime}\right)}\right|\\[.4cm] =&\left|E\left[e^{i\lambda\left(\xi_{t_2^\prime}-\xi_{t_1^\prime}\right)}\left(e^{i\lambda\left[\left(\xi_{t_2}-\xi_{t_2^\prime}\right)-\left(\xi_{t_1}-\xi_{t_1^\prime}\right)\right]}-1\right)\right]\right|\\[.4cm] \leq&\lambda E\left|\xi_{t_2}-\xi_{t_2^\prime}\right|+\lambda E\left|\xi_{t_1}-\xi_{t_1^\prime}\right|. \end{aligned} $$ Since we only have $\xi_{t^\prime}\to_P\xi_{t}\;(t^\prime\to t)$, it does not guarantee the last formula can be as small as possible.
So I want to know how to prove it.
