Continuity mistake: After Kellijian aims to choke his wife, they kiss. Right afterwards, the position of her left arm on his body is not continuous between shots.
Continuity mistake: When Jessica talks to Lt. Bergkamp, she takes her left hand to her ear to improve her hearing, but from the opposite shot, her hand is lowered.
Continuity mistake: Before Joan leaves the restaurant, she has a glass of wine on her right, but from Jessica's POV, the glass has moved to Joan's left side. Also, Jessica's glass is in front of her, but moves to the right side of the table after Joan leaves.
Continuity mistake: When Bud and Andy arrive at Cagliostro's table, Jessica stares with her glasses on, while Joan turns her head around. An instant cut later, Jessica's glasses are off and Joan is turning her head again.
Continuity mistake: When Jessica is having dinner with Joan, Jessica's right arm is either extended on the table next to the cutlery or not, depending on the shot.
Continuity mistake: When Joan leaves the restaurant, her glass is full. Next time we see it, it's empty.
Answer: In a flashback scene, it was shown that he had a violent temper, so it stands to reason that he finally killed her in a fit of rage.