S.M. Ross, 1996. Stochastic Processes . John Wiley & Sons, Inc., New York, USA, second edition. P.V. Shevchenko, 2010. Calculation of aggregate loss distributions. The Journal of Operational Risk , 5 (2), 3–40. P. Whittle, 2000. Probability via Expectation . New York: Springer, 4th edition. D.J. Wilkinson, 2012. Stochastic Modelling for Systems Biology . CRC Press, Boca Raton FL, USA, second edition. Michael Wiper, David Rios Insua, and Fabrizio Ruggeri, 2001. Mixtures of Gamma distributions with applications. Journal of Computational and Graphical Statistics , 10 (3), 440–454. Minimisation for the Moment bound. The Moment bound is min- imised over the control variable k = 1 , 2 , . . . . It is convenient to have an upper bound for k , because it is efficient to compute S k t for a set of k val- ues, rather than one k at a time, as shown in (3). We find an approximate upper bound for k as follows. First, we compute the first two moments of S$_{t}$ exactly using (1) and (2). Then we approximate the distribution of S$_{t}$ using a Gamma distribution matched to these two moments, for which α$_{s}$ = µ 2 t σ 2 t β$_{s}$ = µ$_{t}$ σ 2 t . APPENDIX 31