19,637 research outputs found
Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position
Impulse methods are generalized to a family of integrators for Langevin
systems with quadratic stiff potentials and arbitrary soft potentials. Uniform
error bounds (independent from stiff parameters) are obtained on integrated
positions allowing for coarse integration steps. The resulting integrators are
explicit and structure preserving (quasi-symplectic for Langevin systems)
Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients
In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models
Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients
We are interested in the strong convergence and almost sure stability of
Euler-Maruyama (EM) type approximations to the solutions of stochastic
differential equations (SDEs) with non-linear and non-Lipschitzian
coefficients. Motivation comes from finance and biology where many widely
applied models do not satisfy the standard assumptions required for the strong
convergence. In addition we examine the globally almost surely asymptotic
stability in this non-linear setting for EM type schemes. In particular, we
present a stochastic counterpart of the discrete LaSalle principle from which
we deduce stability properties for numerical methods
Simulation of stochastic network dynamics via entropic matching
The simulation of complex stochastic network dynamics arising, for instance,
from models of coupled biomolecular processes remains computationally
challenging. Often, the necessity to scan a models' dynamics over a large
parameter space renders full-fledged stochastic simulations impractical,
motivating approximation schemes. Here we propose an approximation scheme which
improves upon the standard linear noise approximation while retaining similar
computational complexity. The underlying idea is to minimize, at each time
step, the Kullback-Leibler divergence between the true time evolved probability
distribution and a Gaussian approximation (entropic matching). This condition
leads to ordinary differential equations for the mean and the covariance matrix
of the Gaussian. For cases of weak nonlinearity, the method is more accurate
than the linear method when both are compared to stochastic simulations.Comment: 23 pages, 6 figures; significantly revised versio
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
In this paper we introduce a randomized version of the backward Euler method,
that is applicable to stiff ordinary differential equations and nonlinear
evolution equations with time-irregular coefficients. In the finite-dimensional
case, we consider Carath\'eodory type functions satisfying a one-sided
Lipschitz condition. After investigating the well-posedness and the stability
properties of the randomized scheme, we prove the convergence to the exact
solution with a rate of in the root-mean-square norm assuming only that
the coefficient function is square integrable with respect to the temporal
parameter.
These results are then extended to the numerical solution of
infinite-dimensional evolution equations under monotonicity and Lipschitz
conditions. Here we consider a combination of the randomized backward Euler
scheme with a Galerkin finite element method. We obtain error estimates that
correspond to the regularity of the exact solution. The practicability of the
randomized scheme is also illustrated through several numerical experiments.Comment: 37 pages, 3 figure
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