25 research outputs found
Mean-square A -stable diagonally drift-implicit integrators of weak second order for stiff ItĂ´ stochastic differential equations
We introduce two drift-diagonally-implicit and derivative-free integrators for stiff systems of Itô stochastic differential equations with general non-commutative noise which have weak order 2 and deterministic order 2, 3, respectively. The methods are shown to be mean-square A-stable for the usual complex scalar linear test problem with multiplicative noise and improve significantly the stability properties of the drift-diagonally-implicit methods previously introduced (Debrabant and Rößler, Appl. Numer. Math. 59(3-4):595-607, 2009
Explicit stabilized integration of stiff determinisitic or stochastic problems
Explicit stabilized methods for stiff ordinary differential equations have a long history. Proposed in the early 1960s and developed during 40 years for the integration of stiff ordinary differential equations, these methods have recently been extended to implicit-explicit or partitioned type methods for advection-diffusion-reaction problems, and to efficient explicit solvers for stiff mean-square stable stochastic problems. After a short review on the basic stabilized methods we discuss some recent developments
An Invitation to Stochastic Differential Equations in Healthcare
An important problem in finance is the evaluation of the value in the future of assets (e.g., shares in company, currencies, derivatives, patents). The change of the values can be modeled with differential equations. Roughly speaking, a typical differential equation in finance has two components, one deterministic (e.g., rate of interest of bank accounts) and one stochastic (e.g., values of stocks) that is often related to the notion of Brownian motions. The solution of such a differential equation needs the evaluation of Riemann–Stieltjes’s integrals for the deterministic part and Ito’s integrals for the stochastic part. For A few types of such differential equations, it is possible to determine an exact solution, e.g., a geometric Brownian motion. On the other side for almost all stochastic differential equations we can only provide approximations of a solution. We present some numerical methods for solving stochastic differential equations
Stabilized multilevel Monte Carlo method for stiff stochastic differential equations
A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods, the computational cost for stiff systems is significantly reduced, while keeping the computational algorithm fully explicit and easy to implement. Numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized MLMC method and corroborate our theoretical findings. (C) 2013 Elsevier Inc. All rights reserved
Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
We introduce a new algebraic framework based on a modification (called
exotic) of aromatic Butcher-series for the systematic study of the accuracy of
numerical integrators for the invariant measure of a class of ergodic
stochastic differential equations (SDEs) with additive noise. The proposed
analysis covers Runge-Kutta type schemes including the cases of partitioned
methods and postprocessed methods. We also show that the introduced exotic
aromatic B-series satisfy an isometric equivariance property.Comment: 33 page
NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping
Large time-stepping is important for efficient long-time simulations of
deterministic and stochastic Hamiltonian dynamical systems. Conventional
structure-preserving integrators, while being successful for generic systems,
have limited tolerance to time step size due to stability and accuracy
constraints. We propose to use data to innovate classical integrators so that
they can be adaptive to large time-stepping and are tailored to each specific
system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based
schemes adaptive to large time-stepping. The NySALT has optimal parameters for
each time step learnt from data by minimizing the one-step prediction error.
Thus, it is tailored for each time step size and the specific system to achieve
optimal performance and tolerate large time-stepping in an adaptive fashion. We
prove and numerically verify the convergence of the estimators as data size
increases. Furthermore, analysis and numerical tests on the deterministic and
stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal
admissible step size of linear stability, and quadruples the time step size of
the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of
accuracy.Comment: 26 pages, 7 figure