Comparación de acciones y precios de MySQL

¿Has intentado simplemente agregar una condición where dentro de la subconsulta?

SELECT a.name , a.partnumber , a.supplier , a.cost 
FROM table1 a 
JOIN (
    SELECT name , partnumber , MIN(cost) cost
    FROM table1 
    where stock > 0
    GROUP BY name , partnumber 
    ) b ON a.name = b.name AND a.cost = b.cost AND a.partnumber = b.partnumber