In our recent joint paper with Monique Jeanblanc, we have studied a model of a financial market in which the dividend rates of two risky assets change their initial values to other constant ones at the times at which certain unobservable external events occur. The asset price dynamics were described by geometric Brownian motions with random drift rates switching at exponential random times that are independent of each other and the constantly correlated driving Brownian motions. We have obtained closed form expressions for the rational values of European contingent claims through the filtering estimates of occurrence of the switching times and their conditional probability density derived given the filtration generated by the underlying asset price processes.

Building on the results described above, we consider the model in which two underlying assets are driven by dependent (compound) Poisson processes belonging to exponential families. We obtain closed form expressions for the prices in the case in which the parameters of the asset price dynamics change one constants to other at the times of occurrence of unobservable external events and derive stochastic differential equations for the filtering estimates. We also discuss the solution to the problem of pricing of perpetual American options in a one-dimensional continuous diffusion model for the asset price with switching dividend rates under partial information.