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Generative Modeling of Levy Area for High Order SDE Simulation
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DOI：10.1137/23M161077X.png)
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Abstract
En 中文
It is well known that when numerically simulating solutions to stochastic differential equations (SDEs), achieving a strong convergence rate better than O(root h) (where h is the step-size) usually requires the use of certain iterated integrals of Brownian motion, commonly referred to as its ``Levy areas, However, these stochastic integrals are difficult to simulate due to their non-Gaussian nature. and for a d-dimensional Brownian motion with d > 2, no fast almost-exact sampling algorithm is known. In this paper, we propose LevyGAN, a deep-learning-based model for generating approximate samples of Levy area conditional on a Brownian increment. Due to our ``bridge-flipping operation, the output samples match all joint and conditional odd moments exactly. Our generator employs a tailored graph neural network (GNN)-inspired architecture, which enforces the correct dependency structure between the output distribution and the conditioning variable. Furthermore, we incorporate a mathematically principled characteristic-function-based discriminator. Lastly, we introduce a novel training mechanism, termed ``Chen-training, which circumvents the need for expensive-to-generate training data-sets. This new training procedure is underpinned by our two main theoretical results. For four-dimensional Brownian motion, we show that LevyGAN exhibits state-of-the-art performance across several metrics which measure both the joint and marginal distributions. We conclude with a numerical experiment on the log-Heston model, a popular SDE in mathematical finance, demonstrating that a high-quality synthetic Levy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).
Keywords:
generative modeling
Levy area
adversarial learning
probability theory
stochastic analysis
rough path theory
numerical approximation
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