Calculation of aggregate loss distributions

Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed

On Maximal Inequalities for some Jump Processes

We present a solution to the considered in [5] and [22] optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.Jump process, stochastic differential equation, maximum process, optimal stopping problem, compound Poisson process, Ito’s formula, integro-differential free-boundary problem, normal reflection, continuous and smooth fit, maximality principle, maximal inequalities

Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon

We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.Discounted optimal stopping problem, finite horizon, geometric Brownian motion, maximum process, parabolic free-boundary problem, smooth fit, normal reflection, a nonlinear Volterra integral equation of the second kind, boundary surface, a change-of-variable formula with local time on surfaces, American lookback option problem