Is it a problem to invest in a mutual fund (based in the U.S.) that invests in, among other things, food corporations (problem of owning chametz on Pesach)? Is there a problem in the fact that we cannot know all of the things that this fund invests in (all over the world)? It is being recommended to us by a broker from a reputable investment firm.

Hashem Be With You, I am not a lawyer and am not familiar with the exact definition of "mutual funds", but in general, many poskim have allowed buying stocks or partnership in companies where there is chametz, just one must sell his share in the chametz before Pesach with his local rabbi (see many sources brought in Minchat Yitzchak, III, 1). With Love of Israel, Rav Ari Shvat