Related papers: Multilevel Monte Carlo for L\'evy-driven SDEs: Cen…
A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping…
As the size of engineered systems grows, problems in reliability theory can become computationally challenging, often due to the combinatorial growth in the cut sets. In this paper we demonstrate how Multilevel Monte Carlo (MLMC) - a…
In this paper, we estimate the variance of two coupled paths derived with the Multilevel Monte Carlo method combined with the Euler Maruyama discretization scheme for the simulation of McKean-Vlasov stochastic differential equations with…
We describe general multilevel Monte Carlo methods that estimate the price of an Asian option monitored at $m$ fixed dates. Our approach yields unbiased estimators with standard deviation $O(\epsilon)$ in $O(m + (1/\epsilon)^{2})$ expected…
Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high…
Monte Carlo experiments produce samples in order to estimate features of a given distribution. However, simultaneous estimation of means and quantiles has received little attention, despite being common practice. In this setting we…
We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or…
This paper is devoted to proving a (Lindeberg-Feller type ) central limit theorem for the multilevel Monte Carlo estimator associated with the Euler discretization scheme for the stochastic Volterra equations with fractional kernels…
We consider a class of finite time horizon nonlinear stochastic optimal control problem, where the control acts additively on the dynamics and the control cost is quadratic. This framework is flexible and has found applications in many…
We develop algorithms for computing expectations of the laws of models associated to stochastic differential equations (SDEs) driven by pure L\'evy processes. We consider filtering such processes and well as pricing of path dependent…
We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and…
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…
We present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for…
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of…
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential…
The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variables can be replaced by similar approximate random variables…
We study the Euler scheme for a stochastic differential equation driven by a Levy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(X^n-X), where X is the true solution and X^n is its Euler…
We present a multilevel stochastic gradient descent method for the optimal control of systems governed by partial differential equations under uncertain input data. The gradient descent method used to find the optimal control leverages a…
Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While L\'evy processes provide a natural mathematical framework for…
We generalize the multilevel Monte Carlo (MLMC) method of Giles to the simulation of systems of particles that interact via a mean field. When the number of particles is large, these systems are described by a McKean-Vlasov process - a…