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1. Stochastic Differential Games: A Sampling Approach via FBSDEs

   https://doi.org/10.1007/s13235-018-0268-4  

   Exarchos, Ioannis; Theodorou, Evangelos; Tsiotras, Panagiotis (June 2018, Dynamic Games and Applications)

   The aim of this work is to present a sampling-based algorithm designed to solve various classes of stochastic differential games. The foundation of the proposed approach lies in the formulation of the game solution in terms of a decoupled pair of forward and backward stochastic differential equations (FBSDEs). In light of the nonlinear version of the Feynman–Kac lemma, probabilistic representations of solutions to the nonlinear Hamilton–Jacobi–Isaacs equations that arise for each class are obtained. These representations are in form of decoupled systems of FBSDEs, which may be solved numerically. 

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2. Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

   https://doi.org/10.1115/1.4046093  

   Kharazmi, Ehsan; Zayernouri, Mohsen (December 2019, Journal of Verification, Validation and Uncertainty Quantification)

   null (Ed.)

   Abstract Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure. 

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3. Global solution for superlinear stochastic heat equation on $R^d$ under Osgood-type conditions

   https://doi.org/10.1088/1361-6544/add3b2  

   Chen, Le; Foondun, Mohammud; Huang, Jingyu; Salins, Michael (May 2025, Nonlinearity)

   We study thestochastic heat equationon ${\mathbb{R}}^d$subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists random field solution which do not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations. 

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4. Dynamic modelling of sparse longitudinal data and functional snippets with stochastic differential equations

   https://doi.org/10.1093/jrsssb/qkae116  

   Zhou, Yidong; Müller, Hans-Georg (December 2024, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Sparse functional/longitudinal data have attracted widespread interest due to the prevalence of such data in social and life sciences. A prominent scenario where such data are routinely encountered are accelerated longitudinal studies, where subjects are enrolled in the study at a random time and are only tracked for a short amount of time relative to the domain of interest. The statistical analysis of such functional snippets is challenging since information for far-off-diagonal regions of the covariance structure is missing. Our main methodological contribution is to address this challenge by bypassing covariance estimation and instead modelling the underlying process as the solution of a data-adaptive stochastic differential equation. Taking advantage of the interface between Gaussian functional data and stochastic differential equations makes it possible to efficiently reconstruct the target process by estimating its dynamic distribution. The proposed approach allows one to consistently recover forward sample paths from functional snippets at the subject level. We establish the existence and uniqueness of the solution to the proposed data-driven stochastic differential equation and derive rates of convergence for the corresponding estimators. The finite sample performance is demonstrated with simulation studies and functional snippets arising from a growth study and spinal bone mineral density data. 

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5. Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

   https://doi.org/10.1051/m2an/2021010  

   Uy, Wayne Isaac; Peherstorfer, Benjamin (May 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions. 

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