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Register a new userOn explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients
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A conjecture appears in cite{milsteinscheme}, in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficients. We answer this conjecture affirmatively for the case of order 1.5 approximations and show that the suggested methodology works. Moreover, we explore the case of having H{o}lder continuous derivatives for the diffusion coefficients.
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A new class of explicit Milstein schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in $mathcal L^p$ to the solution of the corresponding SDEs with optimal rate.
Motivated by the results of cite{sabanis2015}, we propose explicit Euler-type schemes for SDEs with random coefficients driven by Levy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Levy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $mathcal{L}^2$-convergence which is consistent with the corresponding results available in the literature.
Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussches article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussches article are the use of appropriately modifie
The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although $L^q$ convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider $L^2$ convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.
We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in `{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of $L^p-$viscosity solution introduced in cite{CCKS}. We use BSDEs techniques to establish the existence of $L^p-$viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of $L^p$-viscosity solution.
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