Generating Lévy area of Brownian motion for adaptive and high-order SDE simulation

Jelincic Andraz, University of Bath

It is well known that, when numerically simulating solutions to SDEs, achieving a strong convergence rate better than $O(\sqrt{h})$ (where $h$ is the step-size) requires the use of an iterated integral of Brownian motion $\int W \! \otimes dW$ which is non-Gaussian and difficult to simulate in dimensions $d>2$. We consider an efficient weak approximation of this iterated integral due to Foster (2020), which matches all second and fourth cross moments of the true distribution, as well as all odd moments. We show that this approximation far exceeds state-of-the-art performance across several empirical metrics. We apply Foster's approximation to a numerical simulation of the log-Heston model, which leads to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC) (Jelinčič et. al., 2023). In the second part of the talk we turn to adaptive time-stepping for SDE simulation. We introduce a means of generating an entire Brownian path and its time-integrals such as $\int_s^t W_r dr$, as a deterministic function of a single PRNG seed (Jelinčič, Foster, Kidger, 2024). Our intended use cases are (a) adaptive time stepping of SDE solvers, for which (due to step rejection) Brownian increments may be generated non-chronologically, and (b) higher-order strong SDE solvers, for which Brownian time-integrals are required as inputs. As a worked example, we use our construction to run an adaptive third order underdamped (kinetic) Langevin solver applied to an MCMC problem, where our approach outperforms NUTS whilst using only a tenth of its function evaluations. Using an adaptive solver for strong simulation of a high-volatility CIR model we achieve a twice higher strong order of convergence compared to a constant step solver. We provide a JAX-based implementation of our construction in the popular Diffrax library (\href{}{}).

Area: CS42 - Rough paths and data science (Christian Bayer, Paul Hager and Sebastian Riedel)

Keywords: Lévy area, Brownian motion, adaptive ODE solvers, stochastic analysis, rough path theory, numerical approximation.

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