What type of immunity exists even in the absence of a stimulus?

The correct answer is: adaptive immunity.

Explanation:

Adaptive immunity (also called acquired immunity) is one of the two parts of the immune system.

Unlike the innate immunity that responds to any pathogens, the adaptive immunity is highly specific and provides long-lasting protection because once the cells from adaptive immunity contact the antigen (any molecule that triggers an immune response) they become ready to fight them again later in life with a much more efficient approach - this is what is known as immunological memory and is the reason vaccines work.

Acquired immunity does not need a stimulus to exist, unlike innate immunity (the later only acts when there are pathogens involved).

The acquired immune system is mostly integrated by lymphocytes.