On ltration immersions and credit events

We study a model where two assets are paying dividends with rates changing from one fixed value to another when any credit event occurs. The credit events are associated with the first times when the asset values fall to some given constant levels. The behavior of asset values is described by exponential diffusion processes with random drift rates and independent driving Brownian motions. We obtain closed form expressions for the ex-dividend prices of certain barrier-type contingent claims with structure similar to first- and second-to-default options in credit risk theory.