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On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

Isotropic Gaussian random fields on the sphere are characterized by Karhunen–Loève expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen–Loève expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.

We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean-field games and mean-field control problems in moderate dimension. We analyze the numerical behavior of our algorithms on several multidimensional examples including non linear quadratic models.

Understanding traffic flow dynamics is crucial for modeling real-world scenarios, where stochastic factors often introduce variability and unpredictability. This study explores the impact of stochastic influences on traffic flow, focusing on the relationship between flow and density in both deterministic and stochastic models. Using the Lighthill-Whitham-Richards (LWR) framework, the study examines a stochastic partial differential equation (SPDE) to simulate traffic behavior under varying conditions. The lognormal random numbers are incorporated for the Brownian motion for stochasticity after analysing its normalitynusing various tests. Numerical solutions were obtained through Godunov’s scheme, incorporating boundary conditions to capture stochastic effects. The findings show that the stochastic approach enhances predictive accuracy by capturing real-world traffic uncertainties.

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O (△t) with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of ∊ from O(∊⁻³) to O(∊⁻²). However, in general, to obtain a rate of strong convergence higher than O (△t½) requires simulation, or approximation, of Lévy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of Lévy areas and still achieve an O (△t²) multilevel correction variance for smooth payoffs, and almost an O (△t3/2) variance for piecewise smooth payoffs, even though there is only O(∊t1/2) strong convergence. This results in an O(∊⁻²) complexity for estimating the value of European and Asian put and call options.

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to z and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\\frac{1}{2}$ - ε for all ε > 0. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.

This paper is concerned with the numerical approximations for stochastic differential equations with non-Lipschitz drift or diffusion coefficients. A modified truncated Euler-Maruyama discretization scheme is developed. Moreover, by establishing the criteria on stochastic C-stability and B-consistency of the truncated Euler-Maruyama method, we obtain the strong convergence and the convergence rate of the numerical method. Finally, numerical examples are given to illustrate our theoretical results.

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with Hölder continuous coefficients.

In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton’s methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques.