Exit problem of a two-dimensional risk process from a cone: exact and large deviations results

1 The Cramer-Lundberg, renewal and perturbed reserves models: X t = x + p t − S(t) + Y (t) (1) S(t) = N(t) i=1 Z i where x denotes the initial reserve of an insurance company , p is the premium rate per unit of time, the i.i.d. random variables Z i with distribution function B(x) are the " claims " , and N t is a counting process 1). Alternatively , we have a SDE with constant coefficients dX t = pdt − dS(t) + dY (t). A more realistic model, including an interest rate r, is the GOU dX t = (p + rX t)dt − dS(t) + dY (t) 1 Quite similar to the classical risk model is the M/G/1 workload model used in queueing theory, where only upward jumps are allowed (the stationary distribution of the M/G/1 workload coincides with that of the ruin problem).