Numerical methods for strong solutions of stochastic differential equations: an overview

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena. Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation. Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions. Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process. Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior. This article is an overview of numerical solution methods for SDEs. The solutions are stochastic processes that represent diffusive dynamics, a common modeling assumption in many application areas. We include a description of fundamental numerical methods and the concepts of strong and weak convergence and order for SDE solvers. In addition, we briefly discuss the extension of SDE solvers to coupled systems driven by correlated noise. WIREs Comput Stat 2013. doi: 10.1002/wics.1272This article is categorized under: Applications of Computational Statistics > Computational Finance Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Algorithms and Computational Methods > Numerical Methods

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schroedinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

Dependence on the boundary condition for linear stochastic differential equations in the plane

A computing method of heat transfer in heterogeneous structures is proposed in the paper. The problem is mathematically described as the parabolic boundary value problem with discontinuous coefficients. The generalized solution of this parabolic problem can be approached by the solution of the parabolic boundary problem with smoothed coefficients. To obtain estimates of the solution of the problem with smoothed coefficients the method of numerical solution of stochastic differential equations (SDE) is applied. Some numerical results for honeycomb structures are presented.

In this paper, we shall discuss the uniqueness ”pathwise uniqueness” of the solutions of stochastic partial differential equations (SPDEs) with non-local initial condition,We shall use the Yamada-Watanabe condition for ”pathwise uniqueness” of the solutions of the stochastic differential equation; this condition is weaker than the usual Lipschitz condition. The proof is based on Bihari’sinequality.

On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary

Detailed microscale simulation is typically too computationally expensive for the long time simulations necessary to explore macroscale dynamics. Projective integration uses bursts of the microscale simulator, on microscale time steps, and then computes an approximation to the system over a macroscale time step by extrapolation. Projective integration has the potential to be an effective method to compute the long time dynamic behaviour of multiscale systems. However, many multiscale systems are significantly influenced by noise. By a maximum likelihood estimation, we fit a linear stochastic differential equation to short bursts of data. The analytic solution of the linear stochastic differential equation then estimates the solution over a macroscale, projective integration, time step. We explore how the noise affects the projective integration in two different methods. Monte Carlo simulation suggests design parameters offering stability and accuracy for the algorithms. The algorithms developed here may be applied to compute the long time dynamic behaviour of multiscale systems with noise and to exploit parallel computation. References Yacine Ait-Sahalia. Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica, 70(1):223--262, 2002. Yacine Ait-Sahalia. Likelihood inference for diffusions: a survey. In Jianqing Fan and Hira L. Koul, editors, Frontiers in Statistics: in Honor of Peter J. Bickel's 65th Birthday, chapter 17, pages 369--405. Imperial College Press, 2006. Yacine Ait-Sahalia and Robert Kimmel. Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics, 83(2):413--452, 2007. doi:10.1016/j.jfineco.2005.10.006 C. P. Calderon. Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics. Molecular Simulation, 33(9--10):713--731, August 2007. doi:10.1080/08927020701344740 Radek Erban, Ioannis G. Kevrekidis, and Hans G. Othmer. An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal. Physica D: Nonlinear Phenomena, 215(1):1--24, 2006. doi:10.1016/j.physd.2006.01.008 C. W. Gear, I. G. Kevrekidis, and C. Theodoropoulos. `Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Computers and Chemical Engrg, 26:941--963, 2002. C. W. Gear and Ioannis G. Kevrekidis. Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum. SIAM Journal on Scientific Computing, 24(4):1091--1106, 2003. doi:10.1137/S1064827501388157 Dror Givon and Ioannis G. Kevrekidis. Coarse-grained projective schemes for certain singularly perturbed stochastic problems. Technical report, Princeton University, 2008. Ioannis G. Kevrekidis and Giovanni Samaey. Equation-free multiscale computation: Algorithms and applications. Annu. Rev. Phys. Chem., 60:321--44, 2009. doi:10.1146/annurev.physchem.59.032607.093610 P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics. Springer--Verlag, 1992. C. I. Siettos, M. D. Graham, and I. G. Kevrekidis. Coarse brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation. J. Chemical Physics, 118(22):10149--10156, 2003. doi:10.1063/1.1572456

For theoretical and practical analysis of the self-thinning process we use stochastic differential equation, which take the form: dN (t) = N (t) (α - β ln N (t))dt + μN (t)dW (t), N(t0) = N0, t0 ≤ t ≤ T, where N – tree per hectare (stem/ha), t – stand age, W(t) – scalar standard Brownian motion, N0 – not random, α, β and μ are parameters – real constants. In this paper from a practical viewpoint we apply a simple numerical method for solution of the stochastic differential equations by the Milstein's higher order method. The programs for numerical simulation are written on MAPLE. The convergence of this model is explored too.

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

The Khasminskii theorem have come to play an important role in the nonexplosion solutions of stochastic differential equations (SDEs) without the linear growth condition. In this paper, by using Peng's G-expectation theory, we establish an even more general Khasminskii-type test for stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) that cover a wide class of highly nonlinear G-SDDEs.

This paper deals with some problems of accuracy of algorithms for the numerical solutions of stochastic differential equations (SDEs) versus the size of the ensemble of trajectories simulated. The problems of accuracy arise in estimating functionals of SDE solutions with increasing variance, highly asymmetric distributions, and an indefinite time of arrival of trajectories of solutions at the boundaries of given domains. Some ways of parallelization of statistical algorithms on a multiprocessor cluster are described. Results of numerical experiments performed on a supercomputer available at the Siberian Supercomputer Center are presented.

. In this article, we show the existence of a universal Skorohod measurable functional representation for a large class of semimartingale-driven stochastic differential equations. For this, we prove that paths of the strong solutions of stochastic differential equations can be written as measurable functions of the paths of their driving processes into the space of all càdlàg functions equipped with the Borel sigma-field generated by all open sets with respect to the Skorohod metric. This result can be applied to calculate Malliavin derivatives for SDEs driven by pure-jump Lévy processes with drift.

We consider the stochastic differential equation (SDE) f(s,w, X.(w))dZs((w), where V and Z are vector valued process indexed by . The assumptions we make on Z and on the increasing process controlling Z are satisfied by certain classes of square integrable martingales, by processes of finite variation and by mixtures of these types of processes. Existence, uniqueness and the possibility of explosions of a strong solution X are investigated under Lipschitz conditions on f. A well-known sufficient condition for non-explosion is shown to work also in the multiparameter case and stability of X under perturbation of V, f and Z is proved. Finally more special SDE without Lipschitz conditions are considered, including a class of SDE of the Tsirel'son type.

Roughly speaking, stochastic differential equations are differential equations driven by Gaussian white noise. Here, we are using the term “stochastic differential equations” in a restricted sense and not merely to denote differential equations with some probabilistic aspects. The importance of. stochastic differential equations is largely due to the fact that the solution of such an equation is a sample-continuous Markov process, and conversely, a large and important class of sample-continuous Markov processes can be modeled by the solutions of stochastic differential equations. From the point of view of applications, this is a direct benefit of using white noise as a noise model, and this fact accounts for its popularity. After all, white noise is, at best, a tolerable abstraction and is never a completely faithful representation of a physical noise source. Its raison d’être is the simplicity of analysis that it brings about. We have seen this in connection with Kalman filtering, and we shall see it again in the Markovian nature of solutions to stochastic differential equations.