Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation

This work introduces a novel approach that combines the multi-index Monte Carlo (MC) method with importance sampling (IS) to estimate rare event quantities expressed as an expectation of a smooth observable of solutions to a broad class of McKean-Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator, previously introduced in our works (Ben Rached et al., 2022a,b), to the multi-index setting. We formulate a new multi-index DLMC estimator and conduct a comprehensive cost-error analysis, leading to improved complexity results. To address rare events, an importance sampling scheme is applied using stochastic optimal control of the single level DLMC estimator. This combination of IS and multi-index DLMC not only reduces computational complexity by two orders but also significantly decreases the associated constant compared to vanilla MC. The effectiveness of the proposed multi-index DLMC estimator is demonstrated using the Kuramoto model from statistical physics. The results confirm a reduced complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single level DLMC estimator (Ben Rached et al., 2022a) to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$ for the considered example, while ensuring accurate estimation of rare event quantities within the prescribed relative error tolerance $\mathrm{TOL}_\mathrm{r}$.Comment: Extension to works 2207.06926 and 2208.0322

Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting a boundary

In this paper, we study the non-linear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests

Antithetic multilevel sampling method for nonlinear functionals of measure

Let $\mu\in \mathcal{P}_2(\mathbb R^d)$, where $\mathcal{P}_2(\mathbb R^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $\Phi:\mathcal P_2(\mathbb R^d)\rightarrow \mathbb R$. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $\Phi(\mu)$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws $\mu^N = \frac1N \sum_{i=1}^{N}\delta_{X_i}$, where a) $(X_i)_{i=1}^N$ is a sequence of i.i.d samples from $\mu$ or b) $(X_i)_{i=1}^N$ is a system of interacting particles (diffusions) corresponding to a McKean-Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform \textit{strong propagation of chaos}, as well as an $L^2$ estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures

Iterative multilevel particle approximation for McKean–Vlasov SDEs

The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean \cite{mckean1966class}. The interacting diffusions pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs, but provide a great challenge for Monte Carlo simulations. This is due to the nonlinear dependence of the bias on the statistical error arising through the approximation of the law of the process. This and the fact that particles/diffusions are not independent render classical variance reduction techniques not directly applicable and consequently make simulations of interacting diffusions prohibitive. In this article, we provide an alternative iterative particle representation, inspired by the fixed point argument by Sznitman \cite{sznitman1991topics}. This new representation has the same mean field limit as the classical SIPS. However, unlike classical SIPS, it also allows decomposing the statistical error and the approximation bias. We develop a general framework to study integrability and regularity properties of the iterated particle system. Moreover, we establish its weak convergence to the McKean-Vlasov SDEs (MVSDEs). One of the immediate advantages of iterative particle system is that it can be combined with the Multilevel Monte Carlo (MLMC) approach for the simulation of MVSDEs. We proved that the MLMC approach reduces the computational complexity of calculating expectations by an order of magnitude. Another perspective on this work is that we analyse the error of nested Multilevel Monte Carlo estimators, which is of independent interest. Furthermore, we work with state dependent functionals, unlike scalar outputs which are common in literature on MLMC. The error analysis is carried out in uniform, and what seems to be new, weighted norms

Milstein schemes for delay McKean equations and interacting particle systems

In this paper, we derive fully implementable first order time-stepping schemes for point delay McKean stochastic differential equations (McKean SDEs), possibly with a drift term exhibiting super-linear growth in the state component. Specifically, we propose different tamed Milstein schemes for a time-discretised interacting particle system associated with the McKean equation and prove strong convergence of order 1 and moment stability, making use of techniques from calculus on the space of probability measures with finite second order moments. In addition, we introduce a truncated tamed Milstein scheme based on an antithetic multi-level Monte Carlo approach, which leads to optimal complexity estimators for expected functionals without the need to simulate L\'evy areas.Comment: 33 pages, 4 figure

Double-Loop Importance Sampling for McKean--Vlasov Stochastic Differential Equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with solutions to the $d$-dimensional McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting $P$-particle system, which is a set of $P$ coupled $d$-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a $P \times d$-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in [dos Reis et al., 2023], generating a $d$-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ with a significantly reduced constant to achieve a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given $\mathrm{TOL}_{\mathrm{r}}$ compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics