Risk Theory

Abstract

Suppose the claims arrive according to a Poisson process {N(t)} with rate λ and interarrival intervals {X ₁, ⋯, X _n , ⋯ }. The consecutive claim amounts {Y ₁, ⋯, Y _n , ⋯ } are identically and independently distributed with continuous distribution F(y) with mean μ > 0, and independent of the arrival times. The premier rate c satisfies c ∕ λ > μ, and $c = \lambda \mu (1 + \theta )$. This guarantees the profitability in the long run. The risk process R _t is defined as

$${R}_{t} = ct -{\sum }_{i=1}^{N(t)}{Y }_{ i}.$$

For initial capital reserve U ₀ = u, we define the surplus process as $U_t = u + R_t$ and the time of ruin as

$$T = \inf \{t > 0: U_t = u + R_t < 0\}.$$