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   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. Analytic regularity of strong solutions for the complexified stochastic nonlinear Poisson-Boltzmann equation

   https://doi.org/10.1016/j.camwa.2025.12.003  

   Choi, Brian; Xu, Jie; Norton, Trevor; Kon, Mark; Castrillón-Candás, Julio E (February 2026, Computers & Mathematics with Applications)

   The nonlinear Poisson-Boltzmann equation (nPBE) is a fundamental partial differential equation (PDE) in electrostatics, widely used in computational biology and chemistry to model potential fields in solvents or plasmas. In this paper, we consider the problem of quantifying the statistical uncertainty of the stochastic nPBE solution under random variations in its coefficients. We establish the existence and uniqueness of solutions of the complexified nPBE using a contraction mapping argument, as conventional convex optimization techniques for the real-valued nPBE do not naturally extend to the complex setting. Using the existence and uniqueness result, we demonstrate that the solutions admit analytic extensions over a well-defined region in the complex hyperplane The analyticity makes the computation for statistics of real-valued quantities of interest amenable to numerical techniques such as sparse grids. Sparse grids are applied to uniformly approximate analytic functions with algebraic to sub-exponential error with respect to the number of knots, thus allowing for efficient approximations of high-dimensional integrals. Our numerical experiments confirm the predicted error behavior. 

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

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   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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5. 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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