144 research outputs found
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
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
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
Stochastic B-series analysis of iterated Taylor methods
For stochastic implicit Taylor methods that use an iterative scheme to
compute their numerical solution, stochastic B--series and corresponding growth
functions are constructed. From these, convergence results based on the order
of the underlying Taylor method, the choice of the iteration method, the
predictor and the number of iterations, for It\^o and Stratonovich SDEs, and
for weak as well as strong convergence are derived. As special case, also the
application of Taylor methods to ODEs is considered. The theory is supported by
numerical experiments
Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise
A new class of third order Runge-Kutta methods for stochastic differential
equations with additive noise is introduced. In contrast to Platen's method,
which to the knowledge of the author has been up to now the only known third
order Runge-Kutta scheme for weak approximation, the new class of methods
affords less random variable evaluations and is also applicable to SDEs with
multidimensional noise. Order conditions up to order three are calculated and
coefficients of a four stage third order method are given. This method has
deterministic order four and minimized error constants, and needs in addition
less function evaluations than the method of Platen. Applied to some examples,
the new method is compared numerically with Platen's method and some well known
second order methods and yields very promising results.Comment: Two further examples added, small correction
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