Are there any ways to transform the product in $$y=\left(\sum_ {k=1}^N a_k \exp\left(it \mu_k-\frac{\sigma^2_k t^2}{2}\right)\right) \times \left(\sum_ {k=1}^M b_k \exp\left(it \nu_k-\frac{\Sigma^2_k t^2}{2}\right)\right) $$ such that there results an equation in the form $$y=A \exp\left(itB -\frac{C^2 t^2}{2}\right).$$
Thereby, $a_k$, $\sigma_k$ $\mu_k$ $b_k$ $\nu_k$ $\Sigma_k$ are real parameters.
The objective is to find $A$, $B$ and $C$ such that the above two equations are equal.
All I know is about Cauchy products but it seems that I cannot do a lot with it. If you have any suggestions, I would be grateful.
As Tobias says, it depends on your constants.
$$\begin{array} .y &= \left(\sum_ {k=1}^N a_k \exp\left(it \mu_k-\frac{\sigma^2_k t^2}{2}\right)\right) \times \left(\sum_ {l=1}^M b_l \exp\left(it \nu_l-\frac{\Sigma^2_l t^2}{2}\right)\right)\\ &=\sum_ {k=1}^N \sum_ {l=1}^M a_kb_l \exp\left(it \mu_k-\frac{\sigma^2_k t^2}{2}\right) \exp\left(it \nu_l-\frac{\Sigma^2_l t^2}{2}\right)\\ &=\sum_ {k=1}^N \sum_ {l=1}^M a_kb_l \exp\left(it(\mu_k+\nu_l)-\frac{(\sigma^2_k + \Sigma^2_l) t^2}{2}\right)\\ \end{array} $$
Unless all but one of your $a_kb_l$s is zero, or if your $\mu_k$s, $\nu_l$s, $\sigma_k$s, and $\Sigma_l$s are all constant, or something similar, you won't be able to combine all of your exponentials.