What positive feedback cycle can you identify in the formation and action of pepsin?

It Activates Itself (Pepsinogen=>Pepsin, Cleavage By Pepsin).

pepsin is a digestive enzyme the body produces when we eat, it is a positive feedback loop in the fact that once produced, pepsin acts to break down food in the stomach aka digestion, mainly proteins,and as long as there is food in the stomach, the body will continue to produce the pepsin; pepsin breaks down food, but presence of food produces more pepsin so on and so forth until there is no more food for the pepsin to work on, thus higher levels of the end product shuts off the production of the product

In the stomach, chief cells release pepsinogen. This zymogen is activated by hydrochloric acid , which is released from parietal cells in the stomach lining. The hormone gastrin and the vagus nerve trigger the release of both pepsinogen and HCl from the stomach lining when food is ingested. HCl creates an acidic environment which allows pepsinogen to unfold and cleave itself in an autocatalytic fashion, thereby generating pepsin (the active form). Pepsin cleaves the 44 amino acids from pepsinogen to create more pepsin. Pepsin will digest up to 20% of ingested carbon bonds by cleaving preferentially after the N-terminal of aromatic amino acids such as phenylalanine and tyrosine. It will not cleave at bonds containing valine, alanine or glycine. Peptides may be further digested by other proteases (in the duodenum) and eventually absorbed by the body.

Pepsin is stored as pepsinogen so it will only be released when needed, and does not digest the body's own proteins in the stomach's lining.

Pepsin functions best in acidic environments, particularly those in a pH of 1.

I agree with the facts that have been stated by the others, but to address your question more directly, it is a positive feedback (also called "feed-forward") mechanism because by definition it responds to an environmental change, and consequently acts to MAINTAIN a state (of digestion).  This is different from a negative feedback system that responds to change, but acts to REDUCE the presence of some mediator.

With pepsin, the change is presence of food in the stomach, and pepsin is released to digest it.  Pepsin also activates other pepsin enzymes though, and thus strengthens its own response by amplifying the level of pepsin-mediated digestion.  In this way, pepsin acts to maintain the state of digestion by further increasing pepsin levels.

If it was a negative feedback system, digestion would lead to pepsin activity, but would actually INHIBIT further release and activation of pepsin.