Almost Sure Convergence for Nested Simulations

Almost Sure Convergence for Nested Simulations


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The Solvency II framework provides a standard formula for the calculation of the risk capital (SCR = Solvency Capital Requirement) in one year, but also gives life insurance companies the freedom to develop an own internal model. By using such an internal model, insurance companies aim to reduce their risk capital, as specific business strategies and internal diversification effects can be better taken into account by an internal model instead of the predefined standard formula. One way of determining future risk capital on the basis of an internal model is to use a nested Monte Carlo simulation. From a theoretical point of view, in nested stochastics we analyze the asymptotic behavior and optimal convergence rates of Monte Carlo estimators for E[G(E[V|Z])] ; whereby G(.) represents a non-linear function and V and Z are elements of L2(Ώ; F; P). Thereby outer scenarios are used for modelling Z (e.g. for an insurance framework this represents risk factor modelling) and (conditional) inner scenarios for V , for the approximation of the previously introduced conditional expected value. Essentially, we will show almost sure (a.s.) convergence results for nested Monte-Carlo estimators. For specific setups we will discuss almost sure asymptotic rates of convergence based on the law of iterated logarithm. Finally, these results will be compared with the existing results for the L2-convergence and for the convergence in distribution. We particularly compare to the work of Gordy & Juneja (2010) and Andradóttir & Glynn (2016), who laid the foundation for the L2- and central limit theory of nested simulation.

[1] Andradóttir, S. and Glynn, P.W. (2016), “Computing Bayesian Means Using Simulation.” ACM Transactions on Modeling and Computer Simulation, vol. 26(2).

[2] Gordy, M.B. and Juneja, S. (2010), “Nested Simulation in Portfolio Risk Measurement.” Management Science, vol. 56(10).

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