A recursive formula for the nth survival function and the nth first passage time distribution for jump and diffusion processes. Applications to the pricing of nth-to-default CDS.

We derive some rather general, but combinatorially challenging, formulae to compute the survival function and the first passage time distribution of the n ^th coordinate of a many-body stochastic process in the presence of a killing barrier. First we will study the case of two coordinates and then we will generalize the results to three or more coordinates. Even if the results are difficult to implement, we will provide examples of their use applying them to a physical system, the single file diffusion, and to the financial problem of pricing a n ^th -to-default credit default swap (n ^th -CDS). MSC Classification: 60J70 , 60J25 , 60J27 , 60K35 , 82C31 , 91G20