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We study the weak approximation error of a skew diffusion with bounded measurable drift and H\"older diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first…
Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. These objectives motivate the desire for efficient safety-theoretic reasoning that can be embedded in core decision-making tasks such…
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 develop via Girsanov's transformation a perturbation argument to investigate weak convergence of Euler-Maruyama (EM) scheme for path-dependent SDEs with H\"older continuous drifts. This approach is available to other…
The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of…
Many practical problems involve estimating low dimensional statistical quantities with high-dimensional models and datasets. Several approaches address these estimation tasks based on the theory of influence functions, such as…
The combination of continuum Many-Body Quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a…
In this paper we study a type of stochastic McKean-Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler-Maruyama approximation existence of its weak solutions is proved. And then we observe pathwise uniqueness of its weak…
Indirect imaging problems in biomedical optics generally require repeated evaluation of forward models of radiative transport, for which Monte Carlo is accurate yet computationally costly. We develop a novel approach to reduce this…
Many modern computational approaches to classical problems in quantitative finance are formulated as empirical loss minimization (ERM), allowing direct applications of classical results from statistical machine learning. These methods,…
Latent variable models are increasingly used in economics for high-dimensional categorical data like text and surveys. We demonstrate the effectiveness of Hamiltonian Monte Carlo (HMC) with parallelized automatic differentiation for…
We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose…
Hierarchical modeling provides a framework for modeling the complex interactions typical of problems in applied statistics. By capturing these relationships, however, hierarchical models also introduce distinctive pathologies that quickly…
We discuss numerical approximation methods for Random Time Change equations which possess a deterministic drift part and jump with state-dependent rates. It is first established that solutions to such equations are versions of certain…
In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability…
We propose a new least-squares Monte Carlo algorithm for the approximation of conditional expectations in the presence of stochastic derivative weights. The algorithm can serve as a building block for solving dynamic programming equations,…
This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian…
We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently…
The advantages of sequential Monte Carlo (SMC) are exploited to develop parameter estimation and model selection methods for GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) style models. It provides an alternative method…
The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_sds+\int_0^t\sigma (X^+_s)^pdW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive…