Obtaining the dividends–penalty identities by interpretation

The dividends–penalty identity is a relation between three functions: the discounted penalty function without dividends, the discounted penalty function if a barrier dividend strategy is applied, and the expected discounted dividends until ruin. The classical model of risk theory is modified in that the deterministic premiums are replaced by a compound Poisson process with exponential jumps. In this model, the dividends–penalty identity is new and can be derived by interpretation. Then the dividends–penalty identity in the classical model is obtained as a limit.