Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in [Giles & Reisinger(2011)] for a certain linear stochastic partial differential equation (SPDE), to semi- and fully implicit timestepping as introduced by [Szpruch(2010)] for SDEs. We combine standard finite difference Fourier analysis for PDEs with the linear stability analysis in [Buckwar & Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The results show that Crank-Nicolson timestepping for the principal part of the drift with a partially implicit but negatively weighted double It\^o integral gives unconditional stability over all parameter values, and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries

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

Stochastic differential algebraic equations of index 1 and applications in circuit simulation

AbstractWe discuss differential-algebraic equations driven by Gaussian white noise, which are assumed to have noise-free constraints and to be uniformly of DAE-index 1.We first provide a rigorous mathematical foundation of the existence and uniqueness of strong solutions. Our theory is based upon the theory of stochastic differential equations (SDEs) and the theory of differential-algebraic equations (DAEs), to each of which our problem reduces on making appropriate simplifications.We then consider discretization methods; implicit methods are necessary because of the differential-algebraic structure, and we consider adaptations of such methods used for SDEs. The consequences of an inexact solution of the implicit equations, roundoff and truncation errors, are analysed by means of the mean-square numerical stability of general drift-implicit discretization schemes for SDEs. We prove that the convergence properties of our drift-implicit Euler scheme, split-step backward Euler scheme, trapezoidal scheme and drift-implicit Milstein scheme carry over from the corresponding properties of these methods applied to SDEs.Finally, we show how the theory applies to the transient noise simulation of electronic circuits

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients: applications to financial models

A novel class of implicit Milstein type methods is devised and analyzed in the present work for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters $\theta, \eta \in [0, 1]$ into both the drift and diffusion parts, the new schemes can be viewed as a kind of double implicit methods, which also work for non-commutative noise driven SDEs. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with $\theta \in [\tfrac12, 1]$, solving general SDEs in various circumstances. As applications, some of the proposed schemes are also applied to solve two scalar SDE models arising in mathematical finance and evolving in the positive domain $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($\theta = \eta = 1$) is utilized to approximate the Heston $\tfrac32$-volatility model and the semi-implicit Milstein method ($\theta =1, \eta = 0$) is used to solve the Ait-Sahalia interest rate model. With the aid of the previously obtained error bounds, we reveal a mean-square convergence rate of order one of the positivity preserving schemes for the first time under more relaxed conditions, compared with existing relevant results for first order schemes in the literature. Numerical examples are finally reported to confirm the previous findings.Comment: 36 pages, 3 figure

Strong Convergence of a GBM Based Tamed Integrator for SDEs and an Adaptive Implementation

We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches are illustrated by considering some well-known SDE models

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur