3 A SELECTION OF UPPER BOUNDS Our interest is in a bound for the probability Pr( S$_{t}$ ≥ s ) for some specified ELT and time period t ; we assume, as is natural, that Pr( S$_{t}$ ≤ 0) = 0. We pose the question: is Pr( S$_{t}$ ≥ s ) small enough to be tolerable for specified s and t ? We are aware of four useful upper bounds on Pr( S$_{t}$ ≥ s ), explored here in terms of increasing complexity. The following material is covered in standard textbook such as Grimmett and Stirzaker (2001), and in more specialised books such as Ross (1996) and Whittle (2000). To avoid clutter, we will drop the ‘ t ’ subscript on S$_{t}$ and N$_{t}$ . The Markov inequality. The Markov inequality states that if Pr( S ≤ 0) = 0 then where µ := E( S ). As S is a compound process, the second equality following because N is Poisson. The second expectation is simply We do not expect this inequality to be very tight, because it imposes no conditions on the integrability of S $^{2}$, but it is so fast to compute that it is always worth a try for a large s . Pr( S ≥ s ) ≤ µ s (Mar) µ = E( N ) E( Y ) = λt E( Y ) , (1) E( Y ) = m ∑ i =1 λ$_{i}$ λ E( X$_{i}$ ) . 5