Lawrence is Stefano's Son
Let me explain just how and why Lawrence Alamain is Stefano DiMera's son....
When "Tony" (really Andre DiMera) returned in 2002, he showed John a letter from his mother Daphne DiMera. In this letter, Daphne explained that she had given her child to her sister Philomena and her husband Leopold Alamain. A DNA test showed that John and "Tony" were related by blood, so everyone believed John and "Tony" were brothers. (Well, the DNA matched because John is Stefano's brother and therefore Andre's uncle).
In 2008, we learned that Philomena and Leopold Alamain had actually adopted John, a.k.a. Ryan Brady, the son of Colleen Brady and Santo DiMera, which meant that John couldn't possibly be Daphne's child.
Well, the Alamains only had one other child: Lawrence. Since the Alamains adopted John from Colleen, that means Lawrence is Daphne DiMera's son.
Years ago, when John first learned he was Forrest Alamain, he and Lawrence took a DNA test that matched. Now, since John was a Brady/DiMera child and not an Alamain, he and Lawrence's DNA shouldn't have matched. But it did. And here's my theory on why....
As we all know, Tony DiMera was not Stefano's biological son. Daphne DiMera had an affair with the gardener, producing Tony. So I think it's safe to assume that Daphne, noticing Stefano's evil influence on Tony, wanted to spare her second child (Stefano's biological son) the same fate, so she pretended to have a stillbirth and gave the child to her sister Philomena Alamain and her husband Leopold. Stefano and Daphne's biological son was raised as Lawrence Alamain.
This would make Lawrence the nephew of John Black, explaining why their DNA matched even though John was never a biological Alamain.
Now, think of the drama this could stir up: Stefano would instantly hate Carly for murdering his son. He and Vivian could team up to ruin her life. Also, Carly and Lawrence's son Nicholas is now also a biological DiMera, adding a twentysomething DiMera descendant to the mix.
I doubt the show will ever outright say this about Lawrence, but the facts speak for themselves. Leopold and Philomena only had two children, and since John was Colleen and Santo's son, that means Lawrence is Daphne and Stefano's child.