PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

RISKINESS LEVERAGE MODELS 59APPENDIX ASOME MATHEMATICAL ASIDES(x) is the step function: zero for negative argument and 1 forpositive. It is also referred to as the index function.±(x) is the Dirac delta function. It can be heuristically thoughtof as the density function of a normal distribution with mean zeroand standard deviation arbitrarily small compared to anythingelse in the problem. This makes it essentially zero everywhereexcept at zero but it still integrates to 1.The index function can also be thought of as the cumulativedistribution function of the same normal distribution, and it isin this sense that the delta function can be thought of as thederivative of the index function. All the usual calculus rules aboutderivatives apply without modification.Always, we are implicitly taking the limit as the standard deviationof this distribution goes to zero. This whole usage canbe justified in the theory of linear functionals, but the author hasno idea where.These notions lead to some fundamental properties of the deltafunction. For any continuous function f(x)Zf(a)= f(x)±(x ¡ a)dx,(A.1)and for c>bZ cf(x)±(x ¡ a)dx = (c ¡ a)(a ¡ b)f(a):bIf h(a)=0thenZf(x)±(h(x))dx = f(a)jh 0 (a)j :(A.2)(A.3)