Let (X n : n ≥ 0) be a sequence of iid rv's with mean zero and finite variance. We describe an efficient state-dependent importance sampling algorithm for estimating the tail of S n = X 1 + ... + X n in a large deviations framework as n ր ∞. Our algorithm can be shown to be strongly efficient basically throughout the whole large deviations region as n ր ∞ (in particular, for probabilities of the form P (S n > κn) as κ > 0). The techniques combine results of the theory of large deviations for sums of regularly varying distributions and the basic ideas can be applied to other rare-event simulation problems involving both light and heavy-tailed features.
Our focus is on efficient estimation of tail probabilities of sums of correlated lognormals. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose three different procedures that can be rigorously shown to be asymptotically optimal as the tail probability of interest decreases to zero. The first algorithm is based on importance sampling and is as easy to implement as crude Monte Carlo. The second algorithm is based on an elegant conditional Monte Carlo strategy which involves polar coordinates and the third one is an importance sampling algorithm that can be shown to be strongly efficient.
We consider a model of an irreducible network in which each node is subjected to a random demand, where the demands are jointly normally distributed. Each node has a given supply that it uses to try to meet its demand; if it cannot, the node distributes its unserved demand equally to its neighbors, which in turn do the same. The equilibrium is determined by solving a linear program (LP) to minimize the sum of the unserved demands across the nodes in the network. One possible application of the model might be the distribution of electricity in an electric power grid. This paper considers estimating the probability that the optimal objective function value of the LP exceeds a large threshold, which is a rare event. We develop a conditional Monte Carlo algorithm for estimating this probability, and we provide simulation results indicating that our method can significantly improve statistical efficiency.
We discuss rare event simulation techniques based on state-dependent importance sampling. Classical examples and counter-examples are shown to illustrate the reach and limitations of the state-independent approach. State-dependent techniques are helpful to deal with these limitations. These techniques can be applied to both light and heavy tailed systems and often are based on subsolutions to an associated Isaacs equation and on Lyapunov bounds.
We explore a bottom-up approach to revisit the problem of cash flow modeling in insurance business, and propose a methodology to efficiently simulate the related tail quantities, namely the fixed-time and the finitehorizon ruin probabilities. Our model builds upon the micro-level contract structure issued by the insurer, and aims to capture the bankruptcy risk exhibited by the aggregation of policyholders. This distinguishes from traditional risk theory that uses random-walk-type model, and also enhances risk evaluation in actuarial pricing practice by incorporating the dynamic arrivals of policyholders in emerging cost analysis. The simulation methodology relies on our model's connection to infinite-server queues with non-homogeneous cost under heavy traffic. We will construct a sequential importance sampler with provable efficiency, along with large deviations asymptotics.
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