We propose a novel approach to numerically approximate McKean–Vlasov stochastic differential equations (MV-SDEs) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPSs) and the associated simulation costs required to achieve the ‘propagation of chaos’ limit. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional sub-space. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD-based method compared with the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.

Generalized orthogonal linear derivative (GOLD) estimates were proposed to correct a problem of correlated estimation errors in generalized local linear approximation (GLLA). This paper shows that GOLD estimates are related to GLLA estimates by the Gram–Schmidt orthogonalization process. Analytical work suggests that GLLA estimates are derivatives of an approximating polynomial and GOLD estimates are linear combinations of these derivatives. A series of simulation studies then further investigates and tests the analytical properties derived. The first study shows that when approximating or smoothing noisy data, GLLA outperforms GOLD, but when interpolating noisy data GOLD outperforms GLLA. The second study shows that when data are not noisy, GLLA always outperforms GOLD in terms of derivative estimation. Thus, when data can be smoothed or are not noisy, GLLA is preferred whereas when they cannot then GOLD is preferred. The last studies show situations where GOLD can produce biased estimates. In spite of these possible shortcomings of GOLD to produce accurate and unbiased estimates, GOLD may still provide adequate or improved model estimation because of its orthogonal error structure. However, GOLD should not be used purely for derivative estimation because the error covariance structure is irrelevant in this case. Future research should attempt to find orthogonal polynomial derivative estimators that produce accurate and unbiased derivatives with an orthogonal error structure.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

In Gauthier, Manolaki, and Nestoridis (2021, Advances in Mathematics 381, 107649), in order to correct a false Mergelyan-type statement given in Gamelin and Garnett (1969, Transactions of the American Mathematical Society 143, 187–200) on uniform approximation on compact sets K in $\mathbb C^d$, the authors introduced a natural function algebra $A_D(K)$ which is smaller than the classical one $A(K)$. In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set K. Finally, we introduce a notion of O-hull of K and strengthen known results.

We develop a novel approach for pricing cyber insurance contracts. The considered cyber threats, such as viruses and worms, diffuse in a structured data network. The spread of the cyber infection is modeled by an interacting Markov chain. Conditional on the underlying infection, the occurrence and size of claims are described by a marked point process. We introduce and analyze a new polynomial approximation of claims together with a mean-field approach that allows to compute aggregate expected losses and prices of cyber insurance. Numerical case studies demonstrate the impact of the network topology and indicate that higher order approximations are indispensable for the analysis of non-linear claims.

Many problems in engineering sciences can be described by linear, inhomogeneous, m-th order ordinary differential equations (ODEs) with variable coefficients. For this wide class of problems, we here present a new, simple, flexible, and robust solution method, based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be expected to perform (i) similar to the Runge-Kutta method, when applied to stiff initial value problems, and (ii) significantly better than the finite difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum, steady-state heat transfer through a cooling web, and the structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

For domains which are star-shapedw.r.t. at least one point, we give new bounds on theconstants in Jackson-inequalities in Sobolev spaces. Forconvex domains, these bounds do not depend on theeccentricity. For non-convex domains with a re-entrantcorner, the bounds are uniform w.r.t. the exteriorangle. The main tool is a new projection operator ontothe space of polynomials.

A database for thermodynamic properties of group-III nitrides and relevant species involved into growth of these materials is developed in this paper. Standard formation enthalpies of materials and coefficients of polynomial approximations of the reduced Gibbs free energies are collected in the tables. They allow one to determine the Gibbs free energy, enthalpy, entropy and specific heat of a species as a function of temperature. The database covers solid and gaseous group-III nitrides, elemental species, gaseous metal-organic compounds, chlorides and hydrides of group-III elements, nitrogen containing precursors and organic byproducts of various chemical reactions proceeding during growth processes. Thermodynamic properties of adducts which can be formed in the vapor phase while mixing ammonia and metal-organic compounds are presented in the database as well. Much of the data given in this paper is presented for the first time. All the data are checked for self-consistency and therefore can be used for thermodynamic calculations.

A pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].