Adaptive stepsize based on control theory for stochastic differential equations

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.

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.

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.

We introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations (SDEs). The method, which is dependent on a representation of Brownian paths as binary trees, involves estimation of local errors and of their contribution to the global error. We advocate controlling the variance of the one-step errors, conditional on knowledge of the Brownian path, in such a way that after propagation along the trajectory the error over each step will provide an equal contribution to the variance of the global error. Discretization schemes can be chosen that reduce the mean local error so that it is negligible beside the standard deviation. We show that to obtain convergence of variable step size methods for SDEs, in general it is not sufficient to evaluate the Brownian path only at the points in time where one tries to approximate the solution. We prove that convergence of such methods is guaranteed if the Lévy area is approximated well enough by further subdivision of the Brownian path and the discretization scheme employed uses appropriately both the approximate Lévy areas and increments of the Brownian path.

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.

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

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.

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.

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

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.

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.

A representation of solution of stochastic differential equations

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

This paper uses stochastic differential equation and risk theory to study the probability model of insurance company's bankruptcy function and the proportion model of financial insurance investment. It also studies the application of stochastic probability to the investment decision-making of institutional investors. Taking the four major insurance industries in China as an example, it gives the strategy of venture capital.