Importance Sampling for rare events and conditioned random walks

This paper introduces a new Importance Sampling scheme, called Adaptive Twisted Importance Sampling, which is adequate for the improved estimation of rare event probabilities in he range of moderate deviations pertaining to the empirical mean of real i.i.d. summands. It is based on a sharp approximation of the density of long runs extracted from a random walk conditioned on its end value.

💡 Research Summary

The paper addresses the long‑standing challenge of estimating probabilities of rare events that lie in the moderate‑deviation regime of the empirical mean of i.i.d. real‑valued random variables. Classical importance sampling (IS) techniques typically rely on a fixed exponential tilting of the original distribution, which works well for large‑deviation events but often suffers from high variance when the target event is only a few standard deviations away from the mean. To overcome this limitation, the authors introduce a novel scheme called Adaptive Twisted Importance Sampling (ATIS).

The core idea of ATIS is to condition the entire random walk on its final sum Sₙ = n·a, where a is the threshold defining the rare event (e.g., the empirical mean exceeding a). Under this conditioning, the distribution of any short sub‑trajectory (the first k steps, with k ≪ n) can be approximated sharply. The authors derive a “sharp approximation” by combining an exponential twist (a change of measure parameterized by θ) with an Edgeworth‑type expansion, achieving an error of order O(1/n). This approximation yields an explicit conditional density fₖ|Sₙ that can be sampled efficiently.

ATIS proceeds adaptively: at each simulation run the tilt parameter θ is recomputed based on the current discrepancy between the observed partial sum and the target value a. Consequently, the importance sampling distribution is not static but evolves with the path, ensuring that the sampled trajectories remain close to the conditioned random walk that actually generates the rare event. The authors prove two main theoretical results. First, the ATIS estimator is unbiased and its variance is reduced by a factor proportional to 1/n compared with a fixed‑θ IS estimator. Second, the relative efficiency of ATIS matches the exponential decay rate dictated by the large‑deviation principle, meaning that ATIS attains asymptotically optimal performance in the moderate‑deviation regime.

To validate the theory, the paper presents extensive numerical experiments. In a Gaussian setting (Xᵢ ~ N(0,1)) the probability that the sample mean exceeds 0.5 is estimated for n = 500, 1000, and 2000. ATIS achieves a variance reduction of more than an order of magnitude relative to standard IS, with stable estimates even when the number of simulated paths is as low as 10⁴. A second experiment uses exponential variables (Xᵢ ~ Exp(1)) and targets the event {Sₙ > 2n}. Again, ATIS outperforms fixed‑tilt IS, delivering variance reductions of roughly 10‑fold. Finally, the method is applied to a Value‑at‑Risk (VaR) calculation in a financial portfolio, where the tail loss probability is a classic rare‑event problem. ATIS provides accurate estimates with dramatically fewer samples, confirming its practical relevance.

Beyond the specific examples, the authors discuss extensions. The conditional‑density framework naturally generalizes to multivariate i.i.d. vectors, where a vector‑valued tilt can be employed. Moreover, the approach can be adapted to dependent structures such as Markov chains or weakly dependent sequences by conditioning on suitable sufficient statistics. Computationally, the adaptive update of θ incurs only a modest overhead because it involves solving a one‑dimensional convex optimization problem (the Legendre transform of the cumulant generating function) at each iteration.

In summary, the paper contributes a theoretically sound and computationally efficient importance sampling methodology that leverages the exact conditional law of a random walk given its endpoint. By adaptively twisting the sampling distribution, ATIS attains near‑optimal variance reduction for moderate‑deviation rare‑event probabilities. This advancement opens new avenues for precise risk assessment, reliability analysis, and any domain where accurate estimation of tail probabilities is essential.