Explicit approximation of the invariant measure for SDDEs with the nonlinear diffusion term

To our knowledge, the existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with the nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler-Maruyama segment process (TEMSP) and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying one of SDDE in the Fortet-Mourier distance. Finally, we give an example and numerical simulations to support our theory.Comment: 31 pages, 2 figure

Hybrid stochastic functional differential equations with infinite delay : approximations and numerics

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay

Explicit approximation of the invariant measure for stochastic delay differential equations with the nonlinear diffusion term

To our knowledge, existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler–Maruyama segment process and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying invariant measure of the SDDE in the Fortet–Mourier distance. Finally, we give an example and numerical simulations to support our theory

An explicit approximation for super-linear stochastic functional differential equations

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler–Maruyama (EM) scheme for SFDEs, and obtain the boundedness and convergence in Lp (p≥2). We also prove the convergence rate with 1/2 order. Different from some previous works (Mao, 2003; Zhang et al., 2018), we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory

Cloning and characterization of a novel diacylglycerol acyltransferase from the diatom Phaeodactylum tricornutum

In this study, a cDNA encoding a novel acyl-CoA:diacylglycerol acyltransferase (DGAT)-like protein is identified and isolated from the diatom microalga Phaeodactylum tricornutum (PtDGAT3). Analysis of the sequence reveals that ptDGAT3 cDNA encodes a protein of 504 amino acids with a molecular mass of 64.5 KDa. The putative ptDGAT3 protein has two catalytic domains: a wax ester synthase-like acyl-CoA acyltransferase domain and a bacteria-specific acyltransferase domain, which shows higher similarity to the DGAT3 of Acinetobacter calcoaceticus than reported DGAT1 or DGAT2 from high plants or algae. Its activity was confirmed by heterologous expression of PtDGAT3 in a neutral lipid-deficient quadruple mutant yeast Saccharomyces cerevisiae H1246. The recombinant yeast restored the formation of a lipid body and displayed a preference to the incorporation of unsaturated C-18 fatty acids into triacyglycerol (TAG). This is the first characterized algal DGAT3 gene, giving further evidence to the occurrence of a DGAT3-mediated TAG biosynthesis pathway