Abstract:
The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

Abstract:
In this paper we investigate the numerical solution of stochastic partial differential equations (SPDEs) for a wider class of stochastic equations. We focus on non-diagonal colored noise instead of the usual space-time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen $\&$ Kloeden, we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.

Abstract:
Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675x compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively.

Abstract:
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed and a local error analysis is provided. We consider the extension of the method to forward-backward stochastic differential equations (FBSDEs) and reflected FBSDEs. Numerical examples are considered from finance demonstrating the performance of the method.

Abstract:
The paper deals with the numerical solution of the nonlinear Ito stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrodinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.

Abstract:
In this paper we are interested in the numerical solution of stochastic differential equations with non negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super linear problems and the Tamed-Euler method does not preserve positivity. In that direction, we use the Semi-Discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super linear with non negative solution. In this class of stochastic differential equations belongs the Heston $3/2$-model that appears in financial mathematics, for which we prove %theoretically and through numerical experiments the "optimal" order of strong convergence at least $1/2$ of the Semi-Discrete method.

Abstract:
We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma W_{xt}.$ Such equations, which transform to linear and nonlinear wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, are related to what Zimmerman (1972) called a diffusion equation. An explicit numerical scheme is employed in both deterministic and stochastic examples. The scheme is checked for accuracy against known exact analytical solutions for deterministic equations. In the stochastic case with $F=0$, solutions yield sample paths for the Brownian sheet whose statistics match well exact values. Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have been shown to lead to existence and uniqueness of solutions. For the linear case solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms with no signal. Equations considered with nonlinear $F$ included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with $\sigma \le 0.25$ but they tended to be shattered at larger values of $\sigma$. Previous work on stochastic sine-Gordon equations is briefly reviewed.

Abstract:
This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the spatial increment of the approximation can be bounded uniformly in space, which guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension permits to analyze the complexity of the approximation. Using the theory of semigroups of linear operators on Banach spaces, we show that the approximation is (weakly) accurate in representing finite and infinite-time statistics, with an order of accuracy identical to that of its generator. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.

Abstract:
In this paper we study the numerical solutions of the stochastic functional differential equations of the following form $du(x,t) = f(x,t,u_t)dt + g(x,t,u_t)dB(t),~ t>0$ with initial data $u(x,0)= u_0(x)=xi in L^p_{F_0}([- au,0];R^n)$ Here $x in R^n$ ($R^n$ is the $ u$-dimenional Euclidean space), $f: C([- au,0]; R^n ) imes R^{ u + 1} ightarrow R^n$ $g: C([- au,0];R^n) imes R^{ u + 1} ightarrow R^{n imes m } u(x,t)in R^n$ for each $t$, $u_t = {u(x,t+ heta ):- auleq hetaleq 0}in C([- au,0];R^n)$ and $B(t)$ is an m-dimensional Brownian motion.

Abstract:
We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.