Names tend to stay the same in any language. Thus, Jessica would stay Jessica in French. Although, French speaking people would say it with a different accent.
*This is true for existing persons for whom a name has already been given. Your name belongs to you as it was given when and where you were born. Of course, you may choose to be identified however you want.
There are many names that have been adopted into English or from English into another language, such as John/Juan. However, there is no form of Jessica in French. This name was invented by Shakespeare in the 16th century and became common only as recently as the 20th century!
The French version of that first name is spelled the same, Jessica.
Jessica is spelled the same in French.
Jessa French's birth name is Jessica Erin French.
Jessica Simonsson's birth name is Jessica Cederholm.
Jessica Soss's birth name is Jessica Baron.
Jessica Aldean's birth name is Jessica Ussery.
Jessica Bangkok's birth name is Jessica Batuhan.
Jessica Boehrs's birth name is Jessica Boehrs.
Jessica Oyelowo's birth name is Jessica Watson.
Jessica Mandeville's birth name is Jessica Novotny.
Jessica Donohoe's birth name is Jessica Webster.
Jessica Charlebois's birth name is Jessica Cr.