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This paper develops a two-stage stochastic model to investigate the evolution of random fields on the unit sphere S 2 in R 3. The model is defined by a time-fractional stochastic diffusion equation on S 2 governed by a diffusion operator with a time-fractional derivative defined in the Riemann–Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on S 2 as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on S 2. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree L ≥ 1. The rate of convergence of the truncation errors as a function of L and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical approximation to stochastic differential equations with positive solutions. Inspired by the logarithmic truncated Euler–Maruyama method developed by Yi et al. (2021) and Lei et al. (2023), we construct a novel numerical method, called the positivity preserving logarithmic transformed truncated Milstein method, for the efficient solution of the underlying equation. Under certain assumptions, we prove the exponential integrability of the numerical solutions. On this basis, we further obtain the strong convergence and strong convergence rate of order 1 for the presented method under additional conditions. Especially, this method is applied to the stochastic tumor model with Allee effect. Several numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently. [ABSTRACT FROM AUTHOR]

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β ∈ (− 1 , 2 ]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min { 1 , 1 + β / 2 − ϵ } , which is | β | / 2 order higher than the order of min { 1 , 1 + β } in the existing works when β ∈ (− 1 , 0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min { 3 / 2 , 1 + β / 2 − ϵ } -order strong convergence in time when β ∈ [ − 1 / 4 , 2 ] , so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings. [ABSTRACT FROM AUTHOR]

We consider the time discretization of fractional stochastic wave equation with Gaussian noise, which is negatively correlated. Major obstacles to design and analyze time discretization of stochastic wave equation come from the approximation of stochastic convolution with respect to fractional Brownian motion. Firstly, we discuss the smoothing properties of stochastic convolution by using integration by parts and covariance function of fractional Brownian motion. Then the regularity estimates of the mild solution of fractional stochastic wave equation are obtained. Next, we design the time discretization of stochastic convolution by integration by parts. Combining stochastic trigonometric method and approximation of stochastic convolution, the time discretization of stochastic wave equation is achieved. We derive the error estimates of the time discretization. Under certain assumptions, the strong convergence rate of the numerical scheme proposed in this paper can reach 1 2 + H. Finally, the convergence rate and computational efficiency of the numerical scheme are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. In Cheng et al. (2013) a dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within the framework of generalized multiscale finite element method (GMsFEM) for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical results are presented to demonstrate the efficiency of the proposed method in solving time-dependent PDEs with multiscale and random features. [ABSTRACT FROM AUTHOR]

Abstract This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis. Highlights • The model covers many nonlinear stochastic (fractional) integro-differential models. • Well-posedness of the models is investigated under the non-Lipschitz condition. • The modified EM approximation of the models admits strong first-order convergence. [ABSTRACT FROM AUTHOR]

The Basel Committee of Banking Supervision has recently set out the revised standards for minimum capital requirements for market risk. The Committee has focused, among other things, on the two key areas of moving from Value-at-Risk (VaR) to Expected Shortfall (ES) and considering a comprehensive incorporation of the risk of market illiquidity by extending the risk measurement horizon. The estimation of the ES for several trading desks and taking into account different liquidity horizons is computationally very involved. We present a novel numerical method to compute the VaR and ES of a given portfolio within the stochastic holding period framework. Two approaches are considered, the delta–gamma approximation, for modelling the change in value of the portfolio as a quadratic approximation of the change in value of the risk factors, and some of the state-of-the-art stochastic processes for driving the dynamics of the log-value change of the portfolio like the Merton jump–diffusion model and the Kou model. Central to this procedure is the application of the SWIFT method developed for option pricing, that appears to be a very efficient and robust Fourier inversion method for risk management purposes. [ABSTRACT FROM AUTHOR]

This paper is concerned with almost sure convergence of θ -EM scheme for neutral stochastic differential delay equations under local Lipschitz conditions with θ ∈ 1 2 , 1 . It is revealed that the strong convergence rate under local Lipschitz condition is 1/2, while almost sure convergence rate is less than 1/2, provided the local Lipschitz constant, valid on balls of radius R , grows not faster than log R . [ABSTRACT FROM AUTHOR]

Recently, Mao (2015) developed a new explicit method, called the truncated Euler–Maruyama (EM) method, for the nonlinear SDE and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. In his another follow-up paper (Mao, 2016), he discussed the rates of L q -convergence of the truncated EM method for q ≥ 2 and showed that the order of L q -convergence can be arbitrarily close to q ∕ 2 under some additional conditions. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to establish the convergence rate without these restrictions. The other aim is to study the stability of the truncated EM method. The advantages of our new results will be highlighted by the comparisons with the results in Mao (2015, 2016) as well as others on the tamed EM and implicit methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a numerical investigation of a strongly convex and smooth optimization problem subject to a convection–diffusion equation with uncertain terms. Our approach is based on stochastic approximation where true gradient is replaced by a stochastic ones with suitable momentum term to minimize the objective functional containing random terms. A full error analysis including Monte Carlo, finite element, and stochastic momentum gradient iteration errors is done. Numerical examples are presented to illustrate the performance of the proposed stochastic approximations in the PDE-constrained optimization setting. [ABSTRACT FROM AUTHOR]

We establish a parametric adaptive importance sampling variance reduction method for general multivariate probability laws. Employing the principle of bypass distributions makes it possible to develop adaptive algorithms without relying on particular properties of the target and proposal laws, both of which in the proposed framework are as general as the uniform law on the unit hypercube, without changing the sampling distribution at each iteration. We establish the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity. Although implementation of the proposed methodology requires a small amount of initial work, it has the potential to yield substantial improvements in estimator efficiency in various general problem settings. To illustrate the applicability and effectiveness, we provide numerical results throughout, in which we apply exponential and normal bypass distributions, as well as demonstrate that well-known adaptive importance sampling formulations in the literature can be easily rewritten in the proposed framework. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast stochastic approximation-based subgradient extragradient algorithm with variance reduction for solving the stochastic variational inequality, where the Lipschitz constant is not necessarily known. Each iteration of our algorithm requires only one projection and one oracle call, and so reducing the computation time. By combining the iterative variance reduction procedure and the stochastic approximation approach, we discuss the asymptotic convergence, the optimal oracle complexity and the sublinear convergence rate in terms of the mean natural residual function. We also obtain the linear convergence rate with finite computational budget under the assumption of the strongly Minty variational inequality and the bounded projection error bound condition, respectively. Finally, several numerical experiments illustrate the efficiency and competitiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We investigate the numerical approximation of the stochastic Allen–Cahn equation on a bounded domain D under Dirichlet boundary conditions and with multiplicative noise. The considered numerical method combines the two-step backward differentiation formula (BDF2) for the temporal discretization in conjunction with an abstract Galerkin scheme for the spatial approximation. In dependence on the regularity of the exact solution we derive a rate of convergence for the BDF2-Maruyama method with respect to the root-mean-square error in discrete analogues of the spaces L ∞ ([ 0 , T ] ; L 2 (D)) and L 2 ([ 0 , T ] ; H 0 1 (D)). Our error analysis is based on the variational approach for stochastic evolution equations. Finally, several numerical experiments illustrate our theoretical results, where a finite element method is used as an example for a Galerkin scheme. [ABSTRACT FROM AUTHOR]

Classical approximation results for stochastic differential equations analyze the L p -distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this is multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate 0.5 with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order. [ABSTRACT FROM AUTHOR]

Comonotonicity has been successfully applied to derive various approximations in the single-factor mortality projection model proposed by Lee and Carter (1992), after Denuit and Dhaene (2007). However, this approach appears to lead to inaccurate approximations in the multi-factor mortality projection models developed by Cairns et al. (2006). Therefore, we propose in this paper an alternative approach which consists in keeping the correlation structure within the time factors unchanged for every calendar year. Proceeding in this way, we provide accurate approximations useful for insurance practitioners avoiding them the simulations within simulations problem whatever the size of their portfolios. [ABSTRACT FROM AUTHOR]

In this paper, we propose two proximal gradient algorithms with variance reduction for stochastic mixed variational inequality problems. One is a proximal extragradient algorithm and another is a proximal forward–backward–forward algorithm. Under the monotonicity assumption on the mapping F and other moderate conditions, we derive some asymptotic convergence properties and O (1 / k) convergence rate in terms of the restricted gap function values for the proposed algorithms. Furthermore, under the bounded metric subregularity condition, we investigate the linear convergence rate and oracle complexity bounds for the proposed algorithms when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of ⌈ k + 1 ⌉ − s with s > 0 , the mean-squared distance of the iterates to the solution set decays at a corresponding polynomial rate, while the iterations and oracle complexities to obtain an ε -solution are O (1 / ε 1 / s) and O (1 / ε 1 + 1 / s) respectively. Finally, some numerical experiments on stochastic network games and traffic assignment problems indicate that the proposed algorithms are efficient. • We study a class of stochastic mixed variational inequality problems. • We present two variance-based proximal algorithms for the considered problem. • We establish asymptotic convergence and oracle complexity of the proposed algorithms. • We carry out numerical experiments on network games and traffic assignment problems. [ABSTRACT FROM AUTHOR]

This work focuses on stability analysis of numerical solutions to jump diffusions and jump diffusions with Markovian switching. Due to the use of Poisson processes, using asymptotic expansions as in the usual approach of treating diffusion processes does not work. Different from the existing treatments of Euler–Maruyama methods for solutions of stochastic differential equations, we use techniques from stochastic approximation. We analyze the almost sure exponential stability and exponential p -stability. The benchmark test model in numerical solutions, namely, one-dimensional linear scalar jump diffusion is examined first and easily verifiable conditions are presented. Then Markovian regime-switching jump diffusions are dealt with. Moreover, analysis of stability of numerical methods for linearizable and multi-dimensional jump diffusions is carried out. [ABSTRACT FROM AUTHOR]

Numerical approximation of stochastic Stokes–Darcy equations usually requires repeated sampling of the random hydraulic conductivity tensor and then simulating flow ensembles. In this setting, we propose an efficient, second order, ensemble algorithm for fast computation of the whole set of realizations of the stochastic Stokes–Darcy model corresponding to different random hydraulic conductivity tensor samples. The ensemble algorithm only requires the solution of two linear systems that have the same constant coefficient matrices for all realizations. We give a complete long time stability and convergence analysis for the method. Numerical experiments are presented to support theoretical results and demonstrate the application of the method. [ABSTRACT FROM AUTHOR]

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4 / 2 model, α -Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract: This work focuses on finding optimal barrier policy for an insurance risk model when the dividends are paid to the share holders according to a barrier strategy. A new approach based on stochastic optimization methods is developed. Compared with the existing results in the literature, more general surplus processes are considered. Precise models of the surplus need not be known; only noise-corrupted observations of the dividends are used. Using barrier-type strategies, a class of stochastic optimization algorithms are developed. Convergence of the algorithm is analyzed; rate of convergence is also provided. Numerical results are reported to demonstrate the performance of the algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm. [Copyright &y& Elsevier]

We study a spatially semidiscrete approximation of nonlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise under weak assumptions on the coefficients avoiding the standard global Lipschitz assumption in the literature. The discretization in space is done by a piecewise linear finite element method. Under some suitable regularity assumption for the exact solution, we present here a rigorous convergence error analysis for the proposed semidiscrete scheme—showing that the numerical scheme admits a convergent rate depending on the regularity of the exact solution. [ABSTRACT FROM AUTHOR]

Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using a spatial discontinuous Galerkin scheme and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and apply it to the compressible Euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties. The performance in multiple stochastic dimensions is compared to the non-intrusive Stochastic Collocation method. The numerical results underline the strength of our method, especially if discontinuities are present in the uncertainty of the solution. [ABSTRACT FROM AUTHOR]

In this paper, the cubic B-spline collocation method is used for solving the stochastic integro-differential equation of fractional order. we show that stochastic integro-differential equation of fractional order is equivalent to a modified stochastic integral equation. Then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation. Using this method, the problem solving turns into a linear system solution of equations. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient. [ABSTRACT FROM AUTHOR]