In English, a verb that expresses a state can also express the entrance into a state. This is called inchoative aspect. The simple past is sometimes inchoative. For example, the present-tense verb in the sentence "He understands his friend" is stative, while the past-tense verb in the sentence "Suddenly he understood what she said" is inchoative, because it means "He began to understand"
I'm not sure if a sentence similar to that in Japanese would work. For example, 突然信じた sounds wrong, but I'm not sure. There are some other parts I'm not too sure about.
Dowty gives some tests to decide whether an English verb is stative. They are as follows: They cannot be complements of "force":
I forced John to run. *I forced John to know the answer.
This one is tricky. 信じさせる is a valid word, but I'm not sure if that's because there is an inchoative aspect to it or something like Japanese perceiving stative verbs differently.
They do not occur as imperatives, except when used in an inchoative manner.
Run! *Know the answer! Know thyself! (inchoative, not stative; archaic)
Since expressions like 信じて and そばに居て are valid, one would be inclined to think Japanese stative verbs would also have an inchoative aspect, but I'm not so sure if it can be used similarly in a way like 突然信じた. I'm also curious if verbs like 信じられる and 信じがたい have the inchoative aspect like "can believe" and "hard to believe" do.