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Quasi-Monte Carlo sampling can attain far better accuracy than plain Monte Carlo sampling. However, with plain Monte Carlo sampling it is much easier to estimate the attained accuracy. This article describes methods old and new to quantify…
The multi-level Monte Carlo method proposed by M. Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified…
Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation.…
In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of…
This paper focuses on the numerical scheme for multiple-delay stochastic differential equations with partially H\"older continuous drifts and locally H\"older continuous diffusion coefficients. To handle with the superlinear terms in…
Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady state performance of the…
We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the…
In this paper, we extend the logarithmic Euler-Maruyama scheme for stochastic delay differential equation in one dimension to the part where we propose a scheme for a system of stochastic delay differential equations. We then show that the…
This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion…
We demonstrate the use of a variational method to determine a quantitative lower bound on the rate of convergence of Markov Chain Monte Carlo (MCMC) algorithms as a function of the target density and proposal density. The bound relies on…
We present an efficient algorithm that, given a discrete random variable $X$ and a number $m$, computes a random variable whose support is of size at most $m$ and whose Kolmogorov distance from $X$ is minimal, also for the one-sided…
We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is…
In automated driving, crash mitigation is crucial to ensure passenger safety. Accurate avoidance requires precise knowledge of the object's position and orientation. However, sensor noise and occlusions often result in tracking and…
The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different…
This paper concerns the numerical approximation for the invariant distribution of Markovian switching L\'evy-driven stochastic differential equations. By combining the tamed-adaptive Euler-Maruyama scheme with the Multi-level Monte Carlo…
Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro--differential equations with uncertainties in the coefficient…
Low rank matrix approximations appear in a number of scientific computing applications. We consider the Nystr\"{o}m method for approximating a positive semidefinite matrix $A$. In the case that $A$ is very large or its entries can only be…
In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the…
Stochastic PDE eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic…
The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been…