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We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA…
In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with…
This paper mainly investigates the strong convergence and stability of the truncated Euler-Maruyama (EM) method for stochastic differential delay equations with variable delay whose coefficients can be growing super-linearly. By…
This work addresses uncertainty quantification of electromagnetic devices determined by the eddy current problem. The multilevel Monte Carlo (MLMC) method is used for the treatment of uncertain parameters while the devices are discretized…
We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's…
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…
This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for L\'evy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain…
In this work, we will show strong convergence of the Multilevel Monte-Carlo (MLMC) algorithm with split-step backward Euler (SSBE) and backward (drift-implicit) Euler (BE) schemes for nonlinear jump-diffusion stochastic differential…
In this paper, we describe an explicit extension formula in sensitivity analysis regarding the Malliavin weight for jump-diffusion mean-field stochastic differential equations whose local Lipschitz drift coefficients are influenced by the…
In the stochastic gradient descent (SGD) for sequential simulations such as the neural stochastic differential equations, the Multilevel Monte Carlo (MLMC) method is known to offer better theoretical computational complexity compared to the…
The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y_n is expensive to simulate, e.g., the solution of a…
To construct positivity-preserving numerical methods, a vast majority of existing works employ transformation techniques such as the Lamperti transformation or logarithmic transformation. However, using these techniques often leads to the…
We study Monte Carlo estimation of the expected value of sample information (EVSI) which measures the expected benefit of gaining additional information for decision making under uncertainty. EVSI is defined as a nested expectation in which…
This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of SDEs with a drift which is not globally Lipschitz. It is proved that if the timestep is bounded appropriately, then over a finite time interval the…
Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean--Kawasaki equation, a…
In this paper we consider the Euler-Maruyama scheme for a class ofstochastic delay differential equations driven by a fractional Brownian motion with index $H\in(0,1)$. We establish the consistency of the scheme and study the rate of…
Sample-based Bayesian inference provides a route to uncertainty quantification in the geosciences, and inverse problems in general, though is very computationally demanding in the naive form that requires simulating an accurate computer…
Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in…
This work develops novel error expansions with computable leading order terms for the global weak error in the tau-leap discretization of pure jump processes arising in kinetic Monte Carlo models. Accurate computable a posteriori error…
In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. We explore the capabilities of a recovery technique based on an enhanced…