The three dimensional globally modified Navier-Stokes equations: Recent developments

The globally modified Navier-Stokes equations (GMNSE) were introduced by Caraballo, Kloeden & Real in 2006 and have been investigated in a number of papers since then, both for their own sake and as a means of obtaining results about the 3-dimensionalNavier-Stokes equations. These results were reviewed by Kloeden et al, which was published in 2009, but there have been some important developments since then, which will be reviewed here

Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in the Foundations of Computational Mathematics 201

Viscous motion in an oceanic circulation model

The barotropic motion of a viscous fluid in a laboratory simulation of ocean circulation may be modelled by Beards ley's vorticity equations. It is established here that these equations have unique smooth solutions which depend continuously on initial conditions. To avoid a boundary condition which involves an integral operator, the vorticity equations are replaced by an equivalent system of momentum equations. The system resembles the two-dimensional incompressible Navier-Stokes equations in a rotating reference frame. The existence of unique generalized solutions of the system in a square domain is established by modifying arguments used by Ladyzhenskaya for the Navier-Stokes equations. Smoothness of the solutions is then established by modifying Golovkin's arguments, again originally for the Navier- Stokes equations. A numerical procedure for solving the vorticity equations is discussed, as are the effects of reentrant corners in the domain modelling islands and peninsulae

Cubature on Wiener space in infinite dimension

We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of L\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.Comment: revised version, accepted for publication in Proceedings Roy. Soc.

Digitization of nonautonomous control systems

AbstractMethods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals

Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \xi_n. We establish results on \as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning the rate of decay of X_n to zero.Comment: 22 pages; corrected more typos, fixed LaTeX macro

Approximate solutions of stochastic differential delay equations with Markovian switching

Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching (SDDEwMSs) and to establish the convergence theory for the Euler-Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching

A Delayed Black and Scholes Formula I

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure

On the Influence of Stochastic Moments in the Solution of the Neutron Point Kinetics Equation

On the Influence of Stochastic Moments in the Solution of the Neutron Point Kinetics EquationComment: 12 pages, 2 figure

An introduction to multilevel Monte Carlo for option valuation

Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation. In 2008, Giles proposed a remarkable improvement to the approach of discretizing with a numerical method and applying standard Monte Carlo. His multilevel Monte Carlo method offers an order of speed up given by the inverse of epsilon, where epsilon is the required accuracy. So computations can run 100 times more quickly when two digits of accuracy are required. The multilevel philosophy has since been adopted by a range of researchers and a wealth of practically significant results has arisen, most of which have yet to make their way into the expository literature. In this work, we give a brief, accessible, introduction to multilevel Monte Carlo and summarize recent results applicable to the task of option evaluation.Comment: Submitted to International Journal of Computer Mathematics, special issue on Computational Methods in Financ