In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The c...

We introduce a variable timestepping algorithm for the Milstein scheme applied to SDEs with a scalar stochastic forcing. Multiple local error estimates are used, corresponding to different terms appearing in the Taylor series of the local truncation error. The timesteps are then chosen so as to bound the standard deviation of the contribution of each of these terms over a unit time interval, an...

In this paper we look at several (trigonometric) stochastic differential equations, find the general form for such nonlinear equation by using I'to formula. Then exact solution different trigonometric equations use of integrals. Ilustrate approach with various examples. (Precise Ito integral formula) and approximate (numerical approximation (the Euler-Maruyama technique Milstein method) were co...

In this paper we propose a Bayesian method for estimating hyperbolic diffusion models. The approach is based on the Markov Chain Monte Carlo (MCMC) method after discretization via the Milstein scheme. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the financial econometrics literature, such as a slowly declining aut...

In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and revers...

We recalculate the amplitude for photon splitting in a strong magnetic eld below the pair production threshold, using the worldline path integral variant of the Bern{Kosower formalism. Numerical comparison (using programs that we have made available for public access on the Internet) shows that the results of the recalculation are identical to the earlier calculations of Adler and later of Ston...

Abstract We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need fulfill commutativity condition for noise term. The newly developed differs significantly from schemes are appropriate case commutative noise. As key result, new needs only two stages specifically tailored based on technique that, comp...

We present a novel scheme for the solution of linear differential equation systems on parallel computers. The Implicit Pipeline (ImP) method uses an implicit timeintegration scheme coupled with an iterative linear solver to solve the resulting differential algebraic system. The ImP method then allows for two independent mechanisms for parallelism: pipelining of the solution of several timesteps...

We use the Wong–Zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations. By approximating the Brownian motion with its truncated spectral expansion and then using different discretizations in time, we present three schemes: a predictor-corrector scheme, a midpoint scheme, and a Milstein-like scheme. We prove that the predictor-correcto...